Question:
Which rule explains why these triangles are congruent?

- AAS
- SAS
- ASA
- SSS
- These triangles cannot be proven congruent.

Answer: - These triangles cannot be proven congruent.

1. Given ∠JGH ≅ ∠KNM, ∠GHJ ≅ ∠NMK, ∠HJG ≅ ∠MKN, according to the angle symbols in the diagram, it can be concluded that these three angles are respectively equal in the two triangles.

2. The sum of the exterior angles in a triangle is 180°. The corresponding angles in the two triangles are respectively equal.

3. According to the congruence theorems of triangles (SSS, SAS, ASA, AAS), it can be seen that at least one pair of sides must be equal to apply these theorems.

4. In the given diagram, there are no symbols or segments indicating any equal sides, so the relationship of equal sides between the triangles cannot be explained.

5. Since no information about equal side lengths is provided, it is not possible to use AAS (Angle-Angle-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), or SSS (Side-Side-Side) to prove the congruence of the two triangles.

6. Based on the above reasoning, the final conclusion is that these triangles cannot be proven congruent.

Naming of Angles

An angle can be named using three points, with the vertex point located in the middle, or it can be named solely by the vertex.

<note>∠JGH</note> ≅ <note>∠KNM</note>, <note>∠GHJ</note> ≅ <note>∠NMK</note>, <note>∠HJG</note> ≅ <note>∠MKN</note> indicates that the corresponding angles in the two triangles are equal.

Definition of Triangle

In the figure of this problem, <note>triangle JGH</note> is a geometric figure composed of <note>three non-collinear points</note> <note>J, G, H</note> and their connecting line segments <note>JG, GH, HJ</note>. <note>Points J, G, H are the three vertices of the triangle</note>, and <note>line segments JG, GH, HJ are the three sides of the triangle</note>. <note>Triangle KNM</note> is a geometric figure composed of <note>three non-collinear points</note> <note>K, N, M</note> and their connecting line segments <note>KN, NM, MK</note>. <note>Points K, N, M are the three vertices of the triangle</note>, and <note>line segments KN, NM, MK are the three sides of the triangle</note>.

Triangular Congruence Theorem (SAS)

<note>Lack of identification and description of equal sides</note>, unable to apply <note>SAS</note> to prove congruence.

If two angles and the included side of one triangle are respectively equal to two angles and the included side of another triangle, then the two triangles are congruent.

In the figure of this problem, in triangles <note>GJH</note> and <note>KNM</note>, angle <note>GJH equals angle KNM</note>, angle <note>JGH equals angle NMK</note>, and angle <note>HJG equals angle MKN</note>. Since the three pairs of corresponding angles of these two triangles are equal respectively, but no sides are marked as equal, the <note>Angle-Side-Angle (ASA) Criterion for Congruence of Triangles</note> cannot be used to prove congruence.

Triangle Congruence Theorem (SSS)

In <note>triangle GHJ</note> and <note>triangle KNM</note>, there are no known equal side relationships, so the <note>Triangle Congruence Theorem (SSS)</note> cannot be applied.


Question:
Look at this figure:What is the shape of its bases?

- heptagon
- hexagon
- rectangle
- octagon

Answer: - hexagon

1. From the figure, it can be seen that this is a hexagonal prism. The definition of a prism is: a geometric solid with two identical and parallel polygonal bases, and other faces are parallelograms.

2. Regular hexagon: A hexagon is a polygon composed of six straight lines.

3. The two bases of the hexagonal prism shown in the figure conform to the shape characteristics of a hexagon, that is, each base has six sides.

4. Based on the shape in the figure and the definition of a hexagonal prism, it is concluded that the base is a hexagon.

5. Through the above reasoning, the final answer is hexagon.

Definition of Prism

A prism is a polyhedron with two parallel and congruent polygonal bases, and all other faces are parallelograms.

<note>The two hexagonal bases of the hexagonal prism are parallel and identical in shape</note>, which conforms to the description of the properties of a prism.

Regular Hexagon

A regular hexagon is a hexagon in which all interior angles are equal, and all sides are of the same length.

<note>In a regular hexagon, each interior angle is equal and each side length xx is equal</note>. Specifically, <note>each interior angle of a regular hexagon is 120 degrees</note>, <note>each side length is equal</note>.


Question:
The graph shows pentagons MNOPQ and M'N'O'P'Q'.

Which sequence of transformations maps MNOPQ onto M'N'O'P'Q'?

- a rotation 90° clockwise around the origin followed by a translation up 6 units
- a translation left 12 units followed by a rotation 90° counterclockwise around the origin
- a reflection across the x-axis followed by a rotation 90° counterclockwise around the origin

Answer: - a reflection across the x-axis followed by a rotation 90° counterclockwise around the origin

1. Observe pentagon MNOPQ and M'N'O'P'Q', and notice that MNOPQ is in the lower right corner, while M'N'O'P'Q' is in the upper left corner.

2. Considering the first step, reflect MNOPQ over the x-axis, this transformation flips it along the x-axis, and the y-coordinates of corresponding points are negated. For example, (x, y) transforms to (x, -y).

3. Label the pentagon obtained after reflecting MNOPQ as M''N''O''P''Q'', which is located below the original pentagon.

4. The reflected pentagon M''N''O''P''Q'' can be made to coincide with the target pentagon M'N'O'P'Q' by rotating it 90° counterclockwise.

5. Using the origin as the center of rotation, rotate pentagon M''N''O''P''Q'' 90° counterclockwise, where the x-coordinate becomes -y and the y-coordinate becomes x.

6. After the above two transformations, the pentagon obtained by reflecting and then rotating 90° counterclockwise coincides completely with M'N'O'P'Q'.

7. Based on the above reasoning steps, the only transformation sequence that fits is first reflecting over the x-axis, then rotating 90° counterclockwise.

Reflection Transformation

A reflection transformation is a type of geometric transformation that flips a figure over a specific line known as the line of reflection. After the reflection transformation, the coordinates of each point on the figure are changed to the coordinates of its corresponding point symmetrically across the line of reflection.

<note>Through reflection transformation, the pentagon MNOPQ is reflected along the x-axis</note>, each point (x, y) of the original figure becomes (x, -y). For example, <note>point M(9, 3) becomes M''(9, -3)</note>, <note>point N(10, 0) becomes N''(10, 0)</note>, and so on to obtain the new pentagon <note>M''N''O''P''Q''</note>.

Rotation Transformation

A rotation involves rotating a geometric figure around a fixed point, known as the center of rotation, through a specified angle. The new coordinates of the points after rotation can be obtained using specific transformation formulas.

In the figure of this problem, perform a <note>90° counterclockwise rotation</note> on the reflected pentagon M''N''O''P''Q''. The coordinates of all points (x, y) change to <note>(-y, x)</note>. For example, <note>point M''(9, -3) becomes M'(3, 9)</note>, <note>point N''(10, 0) becomes N'(0, 10)</note>, and so on to obtain the new pentagon <note>M'N'O'P'Q'</note>.


Question:
The graph shows triangles LMN and L'M'N'.

Which sequence of transformations maps LMN onto L'M'N'?

- a reflection across the x-axis followed by a translation left 12 units
- a translation left 1 unit and up 9 units followed by a rotation 90° counterclockwise around the origin
- a rotation 90° clockwise around the origin followed by a reflection across the x-axis

Answer: - a rotation 90° clockwise around the origin followed by a reflection across the x-axis

1. First, observe the positions of triangle LMN and triangle L'M'N' in the figure. Triangle LMN is located in the fourth quadrant, while triangle L'M'N' is located in the second quadrant.

2. To map triangle LMN to triangle L'M'N', certain transformations are needed. First, consider rotating triangle LMN 90° clockwise around the origin. The rule for rotating 90° clockwise is to convert each point (x, y) to (y, -x).

3. Apply the rotation transformation to point L (10, -7): (10, -7) becomes (-7, -10).

4. Apply the rotation transformation to point M (6, -6): (6, -6) becomes (-6, -6).

5. Apply the rotation transformation to point N (5, -4): (5, -4) becomes (-4, -5).

6. The coordinates of the points after rotation are L''(-7, -10), M''(-6, -6), N''(-4, -5).

7. The next transformation is reflection along the x-axis. The rule for reflection along the x-axis is to convert each point (x, y) to (x, -y).

8. Apply the reflection transformation to point L''(-7, -10): (-7, -10) becomes (-7, 10).

9. Apply the reflection transformation to point M''(-6, -6): (-6, -6) becomes (-6, 6).

10. Apply the reflection transformation to point N''(-4, -5): (-4, -5) becomes (-4, 5).

11. The coordinates of the points after reflection are L'(-7, 10), M'(-6, 6), N'(-4, 5), which match the vertex coordinates of triangle L'M'N'.

12. Therefore, by first rotating 90° clockwise around the origin and then reflecting along the x-axis, triangle LMN is mapped to triangle L'M'N'.

13. Based on the above reasoning, the final answer is: first rotate 90° clockwise around the origin, then reflect along the x-axis.

<note>Rotation transformation rotates the vertices of triangle LMN, L (10, -7), M (6, -6), N (5, -4) 90° clockwise about the origin</note>, resulting in new points <note>L''(-7, -10), M''(-6, -6), N''(-4, -5)</note>. The rule of rotation is to transform each point <note>(x, y) to (y, -x)</note>.

Reflection Transformation

In the figure of this problem, the reflection transformation reflects triangle <note>L'' (-7, -10), M'' (-6, -6), N'' (-4, -5)</note> around the <note>x-axis</note>, resulting in new points <note>L' (-7, 10), M' (-6, 6), N' (-4, 5)</note>. The rule of reflection is to transform each point <note>(x, y)</note> into <note>(x, -y)</note>.


Question:
Which rule explains why these triangles are congruent?

- AAS
- SSS
- SAS
- ASA
- These triangles cannot be proven congruent.

Answer: - ASA

1. Given that line segments PS and PT are equal, i.e., \overline{PS} = \overline{PT}.

2. Given that angles PSQ and PTU are equal, i.e., \angle PSQ = \angle PTU.

3. It is observed that angles QPS and TPU are vertical angles. According to the definition of vertical angles, we have \angle QPS = \angle TPU.

4. Therefore, one angle (\angle PSQ), one side (\overline{PS}), and another adjacent angle (\angle QPS) correspondingly equal in \triangle QPS and \triangle TPU.

5. According to the Angle-Side-Angle (ASA) postulate, if one angle, one included side, and another included angle of two triangles are correspondingly equal, then the two triangles are congruent.

6. In conclusion, \triangle QPS and \triangle TPU are congruent.

Definition of Congruent Triangles

Two triangles are congruent if and only if their corresponding sides are equal and their corresponding angles are equal.

In the figure of this problem, <note>\triangle QPS and \triangle TPU are congruent triangles</note>, <note>the corresponding sides and corresponding angles of \triangle QPS are equal to those of \triangle TPU</note>, namely: <note>side PS = side PT</note>, and the corresponding angles are also equal: <note>angle PSQ = angle PTU</note>, <note>angle QPS = angle TPU</note>.

Definition of Vertical Angles

Vertical angles are the pair of opposite angles formed when two lines intersect. These angles are equal in measure.

<note>In the diagram of this problem, two intersecting lines TS and QU intersect at point P</note>, forming four angles: <note>angle QPS, angle TPU, angle SPU, and angle QPT</note>. According to the definition of vertical angles, <note>angle QPS and angle TPU are vertical angles</note>, <note>angle SPU and angle QPT are vertical angles</note>. Since the angles of vertical angles are equal, <note>angle QPS = angle TPU, angle SPU = angle QPT</note>.

In the figure of this problem, <note>\triangle QPS and \triangle TPU</note> satisfy the conditions of the Angle-Side-Angle (ASA) Criterion for Congruence of Triangles: <note>\angle PSQ = \angle PTU</note>, <note>PS = PT</note>, and <note>\angle QPS = \angle TPU</note>. Therefore, according to the ASA theorem, <note>\triangle QPS and \triangle TPU are congruent</note>.


Question: Find p.Write your answer as a whole number or a decimal. Do not round.p =  $\Box$  meters

Answer: 4 meters

1. Observing the given triangles EFH and GIH, it can be found that triangle EFH and triangle GIH are similar triangles because their corresponding three angles are equal.

2. Specifically, ∠EHF = ∠IHG because these two triangles share angle H; ∠EFH = ∠IGH because they are both right angles; ∠HEF = ∠HIG because the remaining angles are still equal to each other.

3. Since triangle EFH and triangle GIH are similar, according to the proportionality of corresponding sides of similar triangles.

4. For similar triangles, we have EF/IG = FH/GH.

5. The problem gives EF = 8 meters, FH = 6 meters, GH = 3 meters, and IG = p meters.

6. Substitute the values into the proportion EF/IG = FH/GH: 8/p = 6/3.

7. By cross-multiplying, we get the equation: 8 * 3 = 6 * p.

8. Simplify the equation to get 24 = 6p.

9. Divide both sides of the equation by 6 to solve for p, we get p = 24/6.

11. Through the above reasoning, the final answer is 4.

Similarity Theorem for Triangles (AA)

Two triangles are similar if two of their corresponding angles are congruent.

<note>In the diagram of this problem, in triangles EFH and GIH, angle EHF is equal to angle IH(G (both share angle FHI)</note>, and <note>angle EFH is equal to angle IGH (both are right angles)</note>, so <note>triangle EFH is similar to triangle IGH</note>.

Definition of Similar Triangles

Two triangles are similar if and only if their corresponding angles are equal and their corresponding sides are proportional.

<note>Triangles EFH and GIH are similar triangles</note>. According to the definition of similar triangles: <note>∠EFH = ∠IGH</note>, <note>∠EHF = ∠IHG</note>, <note>∠HEF = ∠HIG</note>; <note>EF/IG = FH/GH</note>.


Question:
Which rule explains why these triangles are congruent?

- SAS
- AAS
- ASA
- SSS
- These triangles cannot be proven congruent.

Answer: - These triangles cannot be proven congruent.

1. Given line segment \overline{NQ} = \overline{NT}.

2. For angles \angle PNQ and \angle SNT, since the angles formed by intersecting lines are vertical angles, and vertical angles are always equal, according to the 'definition of vertical angles', we have \angle PNQ = \angle SNT.

3. In triangles \triangle PNQ and \triangle SNT, it is known that \overline{NQ} = \overline{NT} (one angle opposite one side), and \angle PNQ = \angle SNT has been proven.

4. If no other equivalent angles or sides can be found, then a specific congruence theorem (SAS, ASA, SSS) cannot be applied to prove these two triangles are congruent.

5. Therefore, based on the above analysis, these two triangles cannot be proven congruent because there is a lack of known conditions matching a complete congruence theorem.

6. Through the above reasoning, the final answer is These triangles cannot be proven congruent.

Definition of Vertical Angles

In the diagram of this problem, segments <note>QT</note> and <note>SP</note> intersect at point <note>N</note>, forming four angles: <note>angle PNQ</note>, <note>angle QNS</note>, <note>angle SNT</note>, and <note>angle TNP</note>. According to the definition of vertical angles, <note>angle PNQ and angle SNT are vertical angles</note>, <note>angle QNS and angle TNP are vertical angles</note>. Since vertical angles are equal in measure, <note>angle PNQ = angle SNT</note>, <note>angle QNS = angle TNP</note>.

Congruence Theorems for Triangles

Two triangles are congruent if and only if one of the following conditions is satisfied:
1) Side-Side-Side (SSS): All three corresponding sides are equal.
2) Side-Angle-Side (SAS): Two corresponding sides and the included angle are equal.
3) Angle-Side-Angle (ASA): Two corresponding angles and the included side are equal.

<note>In the diagram of this problem, if the two triangles PNQ and SNT satisfy any of the following conditions, then they are congruent:</note> (1. Side-Side-Side (SSS): <note>The sides PN, PQ, and NQ of triangle PNQ are respectively equal to the sides SN, ST, and NT of triangle SNT, i.e., PN=SN, PQ=ST, NQ=NT</note>.)(2. Side-Angle-Side (SAS): <note>The sides PN, QN, and the included angle PNQ of triangle PNQ are respectively equal to the sides SN, NT, and the included angle SNT of triangle SNT, i.e., PN=SN, QN=NT, angle PNQ=angle SNT</note>.)(3. Angle-Side-Angle (ASA): <note>The angles SNT, STN, and the included side NT of triangle SNT are respectively equal to the angles PNQ, PQN, and the included side NQ of triangle PNQ, i.e., angle SNT=angle PNQ, angle STN=angle PQN, NT=NQ</note>.)


Question:
Which rule explains why these triangles are congruent?

- SSS
- ASA
- SAS
- AAS
- These triangles cannot be proven congruent.

Answer: - These triangles cannot be proven congruent.

1. Given that segment AB = FG and segment AC = GH, it shows that two pairs of corresponding sides of these two triangles are equal.

2. Given that angle ∠C = ∠H, it shows that two corresponding angles of these two triangles are equal.

3. Try to find an applicable theorem for triangle congruence. Based on the given information, we can correspond two sides and one angle, and this angle is not the included angle between the two sides, so it cannot be applied to Side-Angle-Side (SAS) or Angle-Side-Angle (ASA).

4. Further analysis shows that this is a combination of two sides and a non-included angle, which meets the conditions of SSA. SSA is not a criterion for triangle congruence because the combination does not meet the triangle congruence conditions (SSS, ASA, SAS, AAS).

5. Due to the lack of information, we cannot apply the criteria for triangle congruence to determine that these triangles are congruent. Therefore, it cannot be determined that these two triangles are congruent.

6. After the above reasoning, the final conclusion is that these triangles cannot be proven to be congruent.

Definition of Congruent Triangles

In the figure of this problem, the corresponding sides and angles of triangles ABC and FGH are as follows: side AB = side FG, side AC = side FH, corresponding angle ∠C = ∠H. These two triangles have two pairs of corresponding sides <note>(AB = FG, AC = FH)</note> and one corresponding angle <note>(∠C = ∠H)</note> equal, but lack sufficient conditions to prove congruence, so congruence cannot be proven.

Triangle Congruence Theorem (SSS)

Two triangles are congruent if their three pairs of corresponding sides are equal in length.

In the diagram of this problem, in triangles ABC and FGH, <note>side AB is equal to side FG</note>, <note>side AC is equal to side FH</note>, but there is no information about the third pair of corresponding sides being equal, so the Triangle Congruence Theorem (SSS) cannot be applied.

Triangular Congruence Theorem (SAS)

If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

In the figure of this problem, it is known that <note>line segment AB = FG</note> and <note>line segment AC = FH</note> as well as <note>angle ∠C = ∠H</note>, but <note>this angle is not the included angle between the given two sides</note>, so the <note>Triangular Congruence Theorem (SAS)</note> cannot be applied.

In the figure of this problem, in triangles ABC and FGH, <note>angle ∠ACB is equal to angle ∠FHG</note>, <note>side AB is equal to side FG</note>, <note>side AC is equal to side FH</note>. It does not meet ASA, so the Angle-Side-Angle (ASA) Criterion for Congruence of Triangles cannot be applied.

Congruence Theorem for Triangles (AAS)

Two triangles are congruent if two angles and the included side of one triangle are respectively equal to two angles and the included side of the other triangle.

<note>AB = FG</note> and <note>AC = FH</note> and <note>∠C = ∠H</note>, but there is no information about the second pair of angles to prove the AAS case, so the AAS theorem cannot be applied.


Question: Find m.Write your answer as a whole number or a decimal. Do not round.m =  $\Box$  meters

Answer: 6 meters

1. In △DCB and △ACE given in the figure, we observe that ∠DCB = ∠ACE = 90°, so these two angles are congruent.

2. Given: ∠B=∠E, ∠D=∠A, and according to the similarity theorem (AA), we can establish △DCB ~ △ACE.

3. Since △DCB ~ △ACE, according to the corresponding sides proportionality theorem, we have:

   DC/AC = CB/CE.

4. Combining the given side lengths, DC = 4 m, AC = 8 m, CB = 3 m, we get the proportion equation:

   4/8 = 3/m.

5. By simplifying the proportion 4/8, we get 1/2, thus the proportion equation becomes:

   1/2 = 3/m.

6. Cross-multiplying to eliminate the proportion, we get:

   m = 3 * 2.

7. Performing the multiplication, we get m = 6.

8. Through the above reasoning, we finally get the answer as 6 meters.

Similarity Theorem for Triangles (AA)

The two angles of <note>△DCB and △ACE</note>, <note>∠DCB and ∠ACE are both 90°</note>, and it is known from the diagram that <note>∠B=∠E</note> and <note>∠D=∠A</note>. Therefore, according to the <note>Similarity Theorem for Triangles (AA)</note>, <note>△DCB ~ △ACE</note>.

Definition of Similar Triangles

<note>△DCB and △ACE are similar triangles</note>. According to the definition of similar triangles: <note>∠DCB = ∠ACE</note>, <note>∠B = ∠E</note>; <note>DC/AC = CB/CE</note>. In this problem, it is known that <note>DC = 4 meters</note>, <note>AC = 8 meters</note>, <note>CB = 3 meters</note>, therefore this theorem is used to establish a proportional relationship and ultimately solve for m.


Question:
The graph shows triangles UVW and U'V'W'.

Which sequence of transformations maps UVW onto U'V'W'?

- a reflection across the y-axis followed by a translation left 9 units and up 10 units
- a rotation 180° around the origin followed by a translation left 9 units and down 2 units
- a translation right 10 units and up 12 units followed by a rotation 90° counterclockwise around the origin

Answer: - a reflection across the y-axis followed by a translation left 9 units and up 10 units

1. First, check the coordinates of points U, V, W and U', V', W' in the figure to confirm the positions of the initial and target points.

2. The coordinates of the original triangle UVW are U(-2,-6), V(-5,-8), W(-5,-6); the coordinates of the target triangle U'V'W' are U'(-7,4), V'(-4,2), W'(-4,4).

3. First, consider the reflection operation. Reflect triangle UVW with the y-axis as the axis of symmetry. At this time, the new coordinates of points U, V, W are U''(2,-6), V''(5,-8), W''(5,-6).

4. Comparing the coordinate differences between U''V''W'' and U'V'W', we find that U'(-7,4) is obtained by moving U''(2,-6) 9 units to the left and 10 units up, and V'(-4,2) and W'(-4,4) are also obtained through this transformation.

5. Through the above analysis, we confirm that the sequence of reflection plus translation can correctly and completely map triangle UVW to triangle U'V'W'.

6. Therefore, choose the sequence of reflection plus translation: first reflect over the y-axis, then translate 9 units to the left and 10 units up.

7. This method is consistent with the options given in the problem, so it is the correct answer.

Reflection Transformation

<note>In the figure of this problem, during the reflection transformation, points U(-2, -6), V(-5, -8), W(-5, -6) are symmetrically transformed along the y-axis to U''(2, -6), V''(5, -8), W''(5, -6)</note>.<note>The coordinates of the vertices of the triangle reflect this transformation relationship on the axis of symmetry</note>, where the distance of each point U, V, W from the y-axis remains unchanged.

Definition of Translation

A translation is a geometric transformation where a figure is moved in the plane along a certain direction, without altering its shape and orientation.

Original: <note>U''(2, -6)</note>, <note>V''(5, -8)</note>, <note>W''(5, -6)</note> moved 9 units to the left and 10 units up, transforming to <note>U'(-7, 4)</note>, <note>V'(-4, 2)</note>, <note>W'(-4, 4)</note>. The coordinates of each vertex of the triangle reflect this transformation relationship on the coordinate axis, where each point's <note>x coordinate decreased by 9</note> and <note>y coordinate increased by 10</note>.

Reflection Transformation

Original: <note>The vertices of triangle UVW are U(-2, -6), V(-5, -8), W(-5, -6)</note> reflected along the <note>y-axis</note> to obtain new vertices <note>U''(2, -6), V''(5, -8), W''(5, -6)</note>. The reflection theorem explains the changes in the coordinates of vertices <note>U, V, W</note>, i.e., <note>x-coordinates are transformed to the opposite direction, y-coordinates remain unchanged</note>.

Translation Invariance Theorem

After a translation transformation, the shape and size of the figure remain unchanged, but its position is altered.

In the figure of this problem, after the triangle is reflected, the vertices <note>U''(2, -6), V''(5, -8), W''(5, -6)</note> undergo a translation transformation <note>(a=-9, b=10)</note> to the new vertices <note>U'(-7, 4), V'(-4, 2), W'(-4, 4)</note>. The Translation Invariance Theorem explains the changes in the coordinates of the vertices <note>U'', V'', W''</note>, specifically the vertex <note>U''</note>'s <note>x-coordinate minus 9, y-coordinate plus 10</note>.


Question:
Which rule explains why these triangles are congruent?

- SAS
- ASA
- AAS
- SSS
- These triangles cannot be proven congruent.

Answer: - These triangles cannot be proven congruent.

1. According to the problem statement, it is known that side UV is equal to side UX, denoted as: \( \overline{UV}=\overline{UX} \).

2. The problem statement also gives that angle UVW is equal to angle UXW, denoted as: \( \angle UVW = \angle UXW \).

3. Additionally, note that segment UW is the common side of the two triangles △UVW and △UXW, so according to the reflexive property of congruence (i.e., any geometric figure is congruent to itself), we have \( \overline{UW} = \overline{UW} \).

4. Organize the obtained congruence relationships:

   - \( \overline{UV} = \overline{UX} \) (corresponding sides)

   - \( \overline{UW} = \overline{UW} \) (common side)

   - \( \angle UVW = \angle UXW \) (corresponding angles)

5. Observing the arrangement, the above congruence relationships satisfy the sequence SSA (side-side-angle), that is, the two triangles have one pair of sides, one pair of angles adjacent to these sides, and another pair of sides equal.

6. However, SSA is not a standard condition for proving triangle congruence. According to the conditions for triangle congruence, commonly used methods are SAS (side-angle-side), ASA (angle-side-angle), AAS (angle-angle-side), and SSS (side-side-side); therefore, these triangles cannot be concluded to be congruent based on the given information.

Definition of Congruent Triangles

In the diagram of this problem, to determine if △UVW and △UXW are congruent, their corresponding sides and angles need to be checked. This includes: <note>\(\overline{UV}\cong\overline{UX}\)</note>, <note>\(\angle UVW\cong\angle UXW\)</note>, and <note>\(\overline{UW}\cong\overline{UW}\)</note>. Although the diagram meets these conditions, the order of conditions is SSA (side-side-angle), which is not a standard criterion for congruence, thus congruence cannot be determined.

Triangular Congruence Theorem (SAS)

<note>\(\overline{UV}\cong\overline{UX}\)</note>, <note>\(\overline{UW}\cong\overline{UW}\)</note> and <note>\(\angle UVW\cong\angle UXW\)</note>, but they do not satisfy the SAS theorem because <note>\(\angle UVW\) is not the angle between \(\overline{UW}\) and \(\overline{UV}\)</note>.

In the figure of this problem, there is no known equality relationship between two angles and their included side. The known conditions are the non-included angle <note>\(\angle UVW\cong\angle UXW\)</note> and the corresponding side <note>\(\overline{UV}\cong\overline{UX}\)</note>, thus the ASA criterion is not satisfied.

Congruence Theorem for Triangles (AAS)

<note>In the diagram of this problem, two angles \(\angle\) are not provided as equal, only \(\angle UVW \cong \angle UXW\) is given</note>.

Triangle Congruence Theorem (SSS)

In the figure of this problem, only two pairs of equal sides are provided <note>\(\overline{UV}\cong\overline{UX}\)</note>, <note>\(\overline{UW}\cong\overline{UW}\)</note> rather than three pairs. Therefore, it does not satisfy the SSS theorem.


Question: Find k.Write your answer as a whole number or a decimal. Do not round.k =  $\Box$  feet

Answer: 10 feet

1. According to the figure given in the problem, it is known that △IJK and △ILM, ∠KIL and ∠KIJ are vertical angles, and ∠IJK and ∠ILM are right angles. Since the three corresponding angles of the two triangles are equal respectively, it can be determined that △IJK ∼ △ILM (Similarity Theorem of Triangles (AA)).

2. In similar triangles, the corresponding sides are proportional, so the corresponding proportional relationship can be written: JK/LM = KI/MI.

3. Through the given conditions in the problem, the side lengths JK = 3 ft, LM = 6 ft, and KI = 5 ft. Set MI = k ft.

4. Substitute into the proportional relationship to get: 3/6 = 5/k.

5. Simplify the proportional equation: 1/2 = 5/k.

6. Through cross-multiplying the equation, we get: 1 × k = 2 × 5, thus obtaining k = 10.

7. Through the above reasoning, the final answer is 10 ft.

Definition of Similar Triangles

In the problem diagram, <note>triangle IJK and triangle ILM are similar triangles</note>. According to the <note>definition of similar triangles</note>, we have: <note>∠IJK = ∠ILM</note>, <note>∠IKJ = ∠IML</note>, <note>∠KIJ = ∠MIL</note>; <note>JK/LM = KI/MI</note>. In the given conditions of the problem, <note>side lengths JK = 3 feet, LM = 6 feet, and KI = 5 feet</note>. Set <note>MI = k feet</note>, thus obtaining the proportional relationship: <note>3/6 = 5/k</note>. Simplifying this proportional relationship, we get: <note>1/2 = 5/k</note>. Finally, through the cross-multiplication equation <note>1 × k = 2 × 5</note>, we obtain <note>k = 10 feet</note>.

Similarity Theorem for Triangles (AA)

<note>Angle KIJ is equal to angle LIM</note>, and <note>angle KJI is equal to angle ILM</note>, so <note>triangle IKJ is similar to triangle ILM</note>.

Definition of Vertical Angles

<note>In the figure of this problem, two intersecting lines LJ and MK intersect at point I</note>, forming four angles: <note>angle LIK</note>, <note>angle MIJ</note>, <note>angle LIM</note>, and <note>angle KIJ</note>. According to the definition of vertical angles, <note>angle LIK and angle MIJ are vertical angles</note>, <note>angle LIM and angle KIJ are vertical angles</note>. Since vertical angles are equal in measure, <note>angle LIK = angle MIJ</note>, <note>angle LIM = angle KIJ</note>.


Question:
The graph shows pentagons GHIJK and G'H'I'J'K'.

Which sequence of transformations maps GHIJK onto G'H'I'J'K'?

- a reflection across the x-axis followed by a rotation 90° clockwise around the origin
- a reflection across the y-axis followed by a rotation 90° counterclockwise around the origin
- a translation right 8 units and down 1 unit followed by a rotation 90° counterclockwise around the origin

Answer: - a translation right 8 units and down 1 unit followed by a rotation 90° counterclockwise around the origin

1. The coordinates of the vertices of pentagon GHIJK are G(-6,-9), H(-8,-7), I(-5,-3), J(-3,-4), K(-2,-8).

2. The coordinates of the vertices of pentagon G'H'I'J'K' are G'(10,2), H'(8,0), I'(4,3), J'(5,5), K'(9,6).

3. Check the first transformation: Translate GHIJK 8 units to the right, i.e., increase the x-coordinate of each vertex by 8, resulting in new vertices G''(2,-9), H''(0,-7), I''(3,-3), J''(5,-4), K''(6,-8).

4. Check the second transformation: Translate the transformed pentagon G''H''I''J''K'' 1 unit down, i.e., decrease the y-coordinate of each vertex by 1, resulting in G'''(2,-10), H'''(0,-8), I'''(3,-4), J'''(5,-5), K'''(6,-9).

5. Determine the third transformation: Rotate counterclockwise 90° around the origin. The rotation formula is (x, y) → (-y, x), thus the new vertices are calculated as:

- G'''(2,-10) after rotation becomes G'(10,2).

- H'''(0,-8) after rotation becomes H'(8,0).

- I'''(3,-4) after rotation becomes I'(4,3).

- J'''(5,-5) after rotation becomes J'(5,5).

- K'''(6,-9) after rotation becomes K'(9,6).

6. Applying these three transformations step by step to pentagon GHIJK indeed transforms it into G'H'I'J'K'. Therefore, the correct transformation steps are translating 8 units to the right, then 1 unit down, and finally rotating counterclockwise 90° around the origin.

7. Based on the above reasoning, the final answer is: Translate 8 units to the right, then 1 unit down, and then rotate counterclockwise 90° around the origin.

Definition of Translation

In the figure of this problem, the vertices of pentagon GHIJK <note>G(-6,-9)</note>, <note>H(-8,-7)</note>, <note>I(-5,-3)</note>, <note>J(-3,-4)</note>, <note>K(-2,-8)</note> are translated 8 units to the right to become <note>G''(2,-9)</note>, <note>H''(0,-7)</note>, <note>I''(3,-3)</note>, <note>J''(5,-4)</note>, <note>K''(6,-8)</note>. Then, these vertices are translated 1 unit downward to become <note>G'''(2,-10)</note>, <note>H'''(0,-8)</note>, <note>I'''(3,-4)</note>, <note>J'''(5,-5)</note>, <note>K'''(6,-9)</note>. After the second translation, the pentagon formed maintains the original shape and orientation.

Rotation Transformation

<note>After two translations, the pentagon G'''H'''I'''J'''K''' (coordinates are G'''(2,-10), H'''(0,-8), I'''(3,-4), J'''(5,-5), K'''(6,-9))</note> rotates <note>counterclockwise around the origin by 90°</note>. Using the <note>rotation formula (x, y) → (-y, x)</note>, the new vertices become <note>G'(10,2), H'(8,0), I'(4,3), J'(5,5), K'(9,6)</note>. These <note>new coordinates correspond to vertices G', H', I', J', K' distributed on the pentagon G'H'I'J'K'</note>.

2D Plane Rotation Formula

Consider a rotation transformation in a two-dimensional Euclidean plane about the origin. If a point (x, y) is rotated counterclockwise by an angle θ to a new position (x', y'), the coordinates of the new point are given by the formulas x' = x*cos(θ) - y*sin(θ) and y' = x*sin(θ) + y*cos(θ). Specifically, when θ = 90°, the formulas simplify to x' = -y and y' = x. Therefore, the formula for a 90° counterclockwise rotation is (x, y) -> (-y, x).

In the diagram of this problem, the vertices of pentagon G'''H'''I'''J'''K''' G'''(2,-10), H'''(0,-8), I'''(3,-4), J'''(5,-5), K'''(6,-9) are calculated according to the theorem formula to obtain new coordinates G'(10,2), H'(8,0), I'(4,3), J'(5,5), K'(9,6). Therefore, using this theorem proves the correctness of the rotation transformation.


Question:
What is OP?

OP= $\Box$

Answer: OP=9

1. Since the problem mentions that the circle is an inscribed circle, and both LQ and PQ are tangent segments from point Q, according to the tangent length theorem, we get LQ=PQ=3.

2. From the given conditions, the length of MQ is 8. We obtain the equation MQ=LM+LQ. Therefore, substituting the known lengths, we can calculate LM=8-3=5.

3. Since it is an inscribed circle, MN and LM are tangent segments from point M to the circle, according to the tangent length theorem, we get MN=LM=5.

4. From the given conditions, the length of MO is 14. We obtain the equation MO=NO+MN. Therefore, substituting the known lengths, we can calculate NO=14-5=9.

5. Since it is an inscribed circle, OP and NO are tangent segments from point O to the circle, according to the tangent length theorem, we get OP=NO=9.

6. Through the above reasoning, we finally obtain the answer OP=9.

Tangent-Segment Theorem

If two tangent segments are drawn to a circle from an external point, then the lengths of these tangent segments are equal. Moreover, the line segment connecting the center of the circle to the external point bisects the angle formed by the two tangent segments.

In the diagram of this problem, from an external point <note>Q</note>, two tangents to the circle <note>LQ</note> and <note>PQ</note> are drawn, and their tangent lengths are equal, i.e., <note>LQ = PQ = 3</note>. From an external point <note>M</note>, two tangents to the circle <note>MN</note> and <note>LM</note> are drawn, and their tangent lengths are equal, i.e., <note>MN = LM = 5</note>. From an external point <note>O</note>, two tangents to the circle <note>OP</note> and <note>NO</note> are drawn, and their tangent lengths are equal, i.e., <note>OP = NO = 9</note>.


Question:
Can you show that this quadrilateral is a parallelogram?

- yes
- no

Answer: - yes

1. First observe the figure, the diagonals of the quadrilateral are marked as bisected, i.e., the diagonals bisect each other.

2. According to the theorem for determining a parallelogram: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

3. In this problem, assume the vertices of the quadrilateral are A, B, C, and D, and the diagonals AC and BD intersect at point O.

4. From the problem, we know that AO = OC and BO = OD, i.e., point O is the midpoint of AC and BD.

5. Through the properties of a parallelogram, it can be concluded that the quadrilateral ABCD satisfies the conditions of a parallelogram.

6. Through the above reasoning, the final answer is yes.

Definition of Parallelogram

A quadrilateral is a parallelogram if and only if it has two pairs of opposite sides that are both parallel (∥) and congruent (≅).

In the figure of this problem, <note>quadrilateral ABCD is a parallelogram</note>, <note>side AB is parallel to and equal to side CD</note>, <note>side AD is parallel to and equal to side BC</note>.

Parallelogram Determination Theorem 4

A quadrilateral is a parallelogram if and only if its diagonals bisect each other.

The diagonals <note>AC</note> and <note>BD</note> of the quadrilateral <note>ABCD</note> bisect each other, so the quadrilateral <note>ABCD</note> is a parallelogram.


Question: Find d.Write your answer as a whole number or a decimal. Do not round.d =  $\Box$  meters

Answer: 4 meters

1. First, according to the definition of vertical angles, we have ∠OKL and ∠MKN as vertical angles and ∠OKL = ∠MKN.

2. From the figure, we know ∠KLO = ∠KNM.

3. According to the AA criterion for similar triangles, we know that if two triangles have two angles respectively equal, then these two triangles are similar. Therefore, triangle OKL and triangle MKN are similar.

4. According to the definition of similar triangles, the ratios of corresponding sides of similar triangles are equal. Therefore, we have LO/NM = KL/KN.

5. Based on the given conditions, we get LO = 3 m, NM = 6 m, KL = 2 m, and KN = d.

6. Substitute the given conditions into the ratio: 3/6 = 2/d.

7. Simplify the ratio: 1/2 = 2/d.

8. Cross-multiply to get 1 * d = 2 * 2.

9. Calculate to get d = 4.

10. Through the above reasoning, we finally get the answer d = 4.

Similarity Theorem for Triangles (AA)

<note>Angle ∠OKL is equal to angle ∠MKN</note>, and <note>angle ∠KLO is equal to angle ∠KNM</note>, so <note>triangle OKL is similar to triangle MKN</note>.

Definition of Vertical Angles

<note>Two intersecting lines OM and LN intersect at point K</note>, forming four angles: <note>∠OKL, ∠MKN, ∠OKN, and ∠MKL</note>. According to the definition of vertical angles, <note>∠OKL and ∠MKN</note> are vertical angles, <note>∠OKN and ∠MKL</note> are vertical angles. Since vertical angles are equal, <note>∠OKL = ∠MKN</note>.

Definition of Similar Triangles

<note>Triangles OKL and MKN are similar triangles</note>. According to the <note>definition of similar triangles</note>, we have: <note>LO/NM = KL/KN = OK/MK</note>.


Question: Find g.Write your answer as a whole number or a decimal. Do not round.g =  $\Box$  meters

Answer: 4 meters

1. According to the given figure, we see that △FGH and △FIJ both contain right angles, and the included angle ∠HFG=∠JFI.

2. Because ∠FGH and ∠FIJ are both right angles, through the triangle sum theorem we know ∠GFH=∠JFI.

3. According to the similarity triangle determination theorem (AA), we can conclude that △FGH ∼ △FIJ because they have two corresponding angles that are equal.

4. According to the definition of similar triangles, the corresponding sides of similar triangles are proportional, we can establish the proportion: FG/FI = GH/IJ.

5. According to the given conditions, FI = 6 m, FG = 3 m and IJ = 8 m. Substituting into the proportion relationship, we get 3/6 = g/8.

6. Simplifying the proportion: 3/6 = g/8 becomes 1/2 = g/8.

7. By cross-multiplying, we get 2g = 1 * 8.

8. Calculating, we get g = 4.

9. Through the above reasoning, the final answer is 4 meters.

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

<note>In triangle △FGH, angle ∠G is a right angle (90 degrees)</note>, thus triangle △FGH is a right triangle. <note>Side FG and side GH are the legs, and side FH is the hypotenuse</note>. <note>In triangle △FIJ, angle ∠I is a right angle (90 degrees)</note>, thus triangle △FIJ is a right triangle. <note>Side FI and side IJ are the legs, and side FJ is the hypotenuse</note>.

Similarity Theorem for Triangles (AA)

In <note>triangle △FGH and triangle △FIJ</note>, if <note>angle ∠HFG is equal to angle ∠JFI</note>, and <note>angle ∠GHF is equal to angle ∠FJI</note>, then <note>triangle △FGH is similar to triangle △FIJ</note>.

Definition of Similar Triangles

<note>Triangles FGH and FIJ are similar triangles</note>. According to the definition of similar triangles: <note>FG/FI = GH/IJ</note>.


Question:
Which rule explains why these triangles are congruent?

- ASA
- SSS
- SAS
- AAS
- These triangles cannot be proven congruent.

Answer: - These triangles cannot be proven congruent.

1. According to the markings in the figure, we can observe that: in △FBC and △HBG, there are ∠BFC=∠BGH, ∠BCF=∠BHG. Based on the definition of vertical angles, we get ∠FBC=∠GBH.

2. To study whether the two triangles are congruent, we need to judge based on the conditions for congruence. The conditions for triangle congruence are: the SSS congruence theorem, the SAS congruence theorem, the ASA congruence theorem, and the AAS congruence theorem. These axioms require at least one pair of corresponding sides to be equal.

3. By observing the given markings and arrangement of the three angles, we find that the information about one pair of corresponding sides being equal between the two triangles is missing.

4. Since all congruence conditions require at least one pair of corresponding sides to be equal, and such information is not provided in this problem, we cannot determine the congruence of the triangles using the above conditions.

5. Therefore, based on the provided information, it can be inferred that these two triangles cannot be proven congruent.

Definition of Congruent Triangles

In the figure of this problem, to determine whether △FBC and △HBG are congruent, we need to compare their corresponding sides and corresponding angles. According to the definition of congruent triangles, <note>two triangles are congruent if their corresponding sides are equal and their corresponding angles are equal.</note> The problem provides information that three pairs of corresponding angles are equal: <note>∠BFC=∠BGH</note>, <note>∠BCF=∠BHG</note>, <note>∠FBC=∠GBH</note>, but does not provide information that the corresponding sides are equal, so it cannot be proven that these two triangles are congruent.

Triangle Congruence Theorem (SSS)

<note>△FBC</note> and <note>△HBG</note> have three pairs of corresponding sides that are equal.

Triangular Congruence Theorem (SAS)

<note>In the given diagram</note>, the <note>Side-Angle-Side (SAS) Congruence Theorem</note> cannot be applied because <note>the problem does not provide the equality of two corresponding sides and the included angle of △FBC and △HBG</note>.

If two angles and the included side of one triangle are respectively equal to two angles and the included side of another triangle, then the two triangles are congruent.

In the original figure, the <note>Angle-Side-Angle Criterion</note> cannot be applied because the problem does not provide the <note>equality of two corresponding angles and the included side of △FBC and △HBG</note>.

Congruence Theorem for Triangles (AAS)

<note>In the problem diagram</note>, the <note>Angle-Angle-Side Theorem</note> cannot be applied because <note>the problem does not provide the equality of two corresponding angles and the non-included side of △FBC and △HBG</note>.

Definition of Vertical Angles

<note>Two intersecting lines FG and HC intersect at point B</note>, forming four angles: <note>angle FBC</note>, <note>angle GBH</note>, <note>angle FBH</note>, and <note>angle GBC</note>. According to the definition of vertical angles, <note>angle FBC and angle GBH are vertical angles</note>, <note>angle FBH and angle GBC are vertical angles</note>. Since the angles of vertical angles are equal, <note>angle FBC = angle GBH</note>, <note>angle FBH = angle GBC</note>.


Question: Find v.Write your answer as a whole number or a decimal. Do not round.v =  $\Box$  feet

Answer: 8 feet

1. From the figure in the problem, it can be observed that ∠QPO and ∠NMO are both right angles, ∠POQ = ∠MON.

2. △QPO and △NMO have two angles that are respectively equal, so according to the theorem of similar triangles (AA), △QPO ∽ △NMO.

3. Based on the definition of similar triangles, we get:

   The corresponding sides of △QPO and △NMO are QP and NM, PO and MO, and QO and NO, thus:

   QP / NM = PO / MO = QO / NO.

4. The problem states that QP = 3 ft, NM = 6 ft, PO = 4 ft, MO = v.

5. Therefore, we can set up the proportion: 3 / 6 = 4 / v.

6. Simplifying this proportion, we get: 1 / 2 = 4 / v.

7. By cross-multiplication, we get: v * 1 = 2 * 4.

8. Solving for v, we get v = 8.

9. Through the above reasoning, the final answer is 8 ft.

Definition of Similar Triangles

<note>Triangles QPO and NMO</note> are similar triangles. According to the definition of similar triangles: <note>angle QPO = angle NMO</note>, <note>angle QOP = angle MON</note>, <note>angle OQP = angle ONM</note>; <note>QP/NM = PO/MO = QO/NO</note>. Specifically, according to the problem conditions, it is known that <note>QP = 3 feet</note>, <note>NM = 6 feet</note>, <note>PO = 4 feet</note>, <note>MO = v</note>, we get <note>3/6 = 4/v</note>.

Similarity Theorem for Triangles (AA)

<note>∠QPO = ∠NMO (right angle, i.e., 90°)</note> and <note>∠POQ = ∠NOM</note>, thus, <note>△QPO ∽ △NMO</note>.


Question:
Find a.

Write your answer in simplest radical form.
 $\Box$  meters

Answer: 9 meters

1. Given that the triangle is an isosceles right triangle because it has two 45° angles. This means the two legs of the right triangle are equal.

2. Let the legs of this isosceles right triangle be 'a', and the hypotenuse be 9√2 meters.

3. According to the properties of an isosceles right triangle, the length of the hypotenuse is equal to the length of a leg multiplied by √2.

4. Based on this property, we can establish the equation: hypotenuse = √2 · leg, i.e., 9√2 = √2 · a.

5. Dividing both sides by √2, we get: a = 9.

6. Through the above reasoning, the final answer is 9 meters.

Properties of Isosceles Right Triangle

The length of the hypotenuse is equal to the length of one of the legs multiplied by √2.

<note>Triangle ABC is an isosceles right triangle</note>, <note>hypotenuse AC = leg BC * √2 = 9√2</note>.


Question:
If JL=24 and LM=10, what is KL?

Write your answer as a whole number or as a decimal rounded to the nearest hundredth.
KL =  $\Box$

Answer: KL = 15.49

1. Observe the figures △JLK and △MLK, noting that these triangles both have a right angle and share the common angle ∠L.

2. Since both triangles △JLK and △MLK have a right angle and share the common angle ∠L, according to the similarity criterion for triangles (AA), which states that if two triangles have two corresponding angles equal, then the triangles are similar, △JLK and △MLK are similar.

3. According to the definition of similar triangles, the corresponding sides of the triangles are proportional. Therefore, the proportion is JL/KL = KL/LM.

4. Substitute the known values into the proportion: 24/KL = KL/10.

5. Cross-multiply to solve the equation: KL² = 24 * 10.

6. Perform the multiplication and take the square root: KL² = 240, thus KL = √240.

7. Calculate the square root, yielding KL ≈ 15.4919.

8. Finally, round to the nearest hundredth to get the result KL ≈ 15.49.

9. Through the above reasoning, the final answer is KL = 15.49.

Definition of Right Triangle

<note>Angle ∠JKL is a right angle (90 degrees)</note>, therefore triangle JLK is a right triangle. Side JK and side LK are the legs, side JL is the hypotenuse. <note>Angle ∠LMK is a right angle (90 degrees)</note>, therefore triangle MLK is a right triangle. Side ML and side KM are the legs, side KL is the hypotenuse.

Similarity Theorem for Triangles (AA)

In <note>triangle JLK</note> and <note>triangle MLK</note>, <note>angle JLK is equal to angle MLK</note>, and <note>angle LKJ is equal to angle LMK</note>, so <note>triangle JLK is similar to triangle MLK</note>.

Definition of Similar Triangles

<note>Triangle JLK</note> and <note>Triangle MLK</note> are similar triangles. According to the definition of similar triangles: <note>JL/KL = KL/LM</note>.


Question:
Which rule explains why these triangles are congruent?

- AAS
- ASA
- SSS
- SAS
- These triangles cannot be proven congruent.

Answer: - AAS

1. Given that \( \angle MPN \cong \angle MSQ \), this is the condition provided by the problem.

2. Observing the figure, according to the definition of vertical angles, we have \( \angle NMP \cong \angle QMS \).

3. The condition provided by the problem is \( \overline{MN} \cong \overline{MQ} \).

4. Given two pairs of angles \( \angle MPN \cong \angle MSQ \) and \( \angle NMP \cong \angle QMS \), and one non-included side \( \overline{MN} \cong \overline{MQ} \), which satisfies the angle-angle-side congruence theorem. According to the congruent triangles determination theorem (AAS), the two triangles \( \triangle MNP \) and \( \triangle MQS \) are congruent.

5. Based on the above reasoning process, it can be concluded that \( \triangle MNP \cong \triangle MQS \) is proven by the angle-angle-side (AAS) congruence theorem.

Definition of an Angle

An angle is a geometric figure formed by two rays sharing a common endpoint. This common endpoint is called the vertex of the angle, and the two rays are referred to as the sides of the angle.

In the diagram of this problem, <note>\( \angle MPN \)</note> is a geometric figure formed by rays <note>\( \overline{MP} \)</note> and <note>\( \overline{PN} \)</note>, which share a common endpoint <note>\( P \)</note>. This common endpoint <note>\( P \)</note> is called the vertex of angle <note>\( \angle MPN \)</note>, and rays <note>\( \overline{MP} \)</note> and <note>\( \overline{PN} \)</note> are called the sides of angle <note>\( \angle MPN \)</note>. Similarly, <note>\( \angle MSQ \)</note> is a geometric figure formed by rays <note>\( \overline{MS} \)</note> and <note>\( \overline{SQ} \)</note>, which share a common endpoint <note>\( S \)</note>. This common endpoint <note>\( S \)</note> is called the vertex of angle <note>\( \angle MSQ \)</note>, and rays <note>\( \overline{MS} \)</note> and <note>\( \overline{SQ} \)

Definition of Line Segment

A line segment is a portion of a straight line that includes two endpoints and all the points between them.

In the diagram of this problem, <note>line segment \( \overline{MN} \)</note> is a part of a straight line, containing <note>endpoint \( M \)</note> and <note>endpoint \( N \)</note> and all points between them. <note>Line segment \( \overline{MQ} \)</note> is a part of a straight line, containing <note>endpoint \( M \)</note> and <note>endpoint \( Q \)</note> and all points between them. <note>Line segment \( \overline{MN} \)</note> and <note>line segment \( \overline{MQ} \)</note> have two endpoints, which are <note>\( M \)</note> and <note>\( N \)</note>, <note>\( M \)</note> and <note>\( Q \)</note> respectively, and every point on <note>line segment \( \overline{MN} \)</note> and <note>line segment \( \overline{MQ} \)</note> is located between <note>endpoint \( M \)</note> and <note>endpoint \( N \)</note>, <note>endpoint \( M \)</note> and <note>endpoint \( Q \)</note>.

Definition of Vertical Angles

Vertical angles are the pair of opposite angles formed when two lines intersect. These angles are equal in measure.

In the diagram of this problem, two intersecting lines <note>\( \overline{MN} \)</note> and <note>\( \overline{MQ} \)</note> intersect at point <note>M</note>, forming four angles: <note>\( \angle NMP \)</note>, <note>\( \angle QMS \)</note>, <note>\( \angle NMS \)</note>, and <note>\( \angle PMQ \)</note>. According to the definition of vertical angles, <note>\( \angle NMP \)</note> and <note>\( \angle QMS \)</note> are vertical angles, and <note>\( \angle NMS \)</note> and <note>\( \angle PMQ \)</note> are vertical angles. Since vertical angles are equal, <note>\( \angle NMP \cong \angle QMS \)</note>, <note>\( \angle NMS \cong \angle PMQ \)</note>.

Congruence Theorem for Triangles (AAS)

In the figure of this problem, <note>\( \angle MPN \cong \angle MSQ \)</note> and <note>\( \angle NMP \cong \angle QMS \)</note>, and <note>the included side \( \overline{MN} \cong \overline{MQ} \)</note>. According to this theorem, it can be determined that <note>\( \triangle MNP \) and \( \triangle MQS \) are congruent</note>.


Question: Find d.Write your answer as a whole number or a decimal. Do not round.d =  $\Box$  feet

Answer: 3 feet

1. First, observe the figure in the problem. It can be found that △EDC and △FDB are two right triangles, and ∠BFD and ∠CED are right angles.

2. Since both triangles contain ∠BDF and each has a right angle, it can be determined that the three corresponding angles of △EDC and △FDB are equal. According to the similarity theorem (AA), they are similar triangles.

3. According to the definition of similar triangles, the corresponding sides of two similar triangles are proportional, i.e.,:

   (i) \( \frac{ED}{FD} = \frac{DC}{DB} = \frac{CE}{BF} \)

4. According to the given conditions, BC = 5 ft, DC = 5 ft, CE = d ft, BF = 6 ft, DB = BC + DC = 5 + 5 = 10.

5. Substitute the known conditions into the proportional relationship:

   (i) \( \frac{5}{10} = \frac{d}{6} \)

6. To solve for d, cross-multiply to get \( 5 \times 6 = 10 \times d \), i.e., \( 30 = 10d \).

7. Divide both sides by 10 to get \( d = \frac{30}{10} \).

8. Through the above reasoning, the final answer is 3.

Similarity Theorem for Triangles (AA)

Two triangles are similar if two of their corresponding angles are congruent.

<note>△EDC and △FDB are two right triangles</note>. According to the problem-solving process, <note>∠BFD = ∠CED = 90°</note>, and the corresponding angles of △EDC and △FDB are <note>∠EDC and ∠FDB, ∠DEC and ∠DFB, ∠ECD and ∠FBD</note>. Additionally, according to the properties of similar triangles, their corresponding sides are proportional: <note>\( \frac{ED}{FD} = \frac{DC}{DB} = \frac{CE}{BF} \)</note>.

Definition of Similar Triangles

Two triangles are similar if and only if their corresponding angles are equal and their corresponding sides are proportional.

<note>In the figure of this problem, triangles △EDC and △FDB are similar triangles</note>. According to the definition of similar triangles: <note>∠EDC = ∠FDB</note>, <note>∠ECD = ∠FBD</note>, <note>∠CED = ∠BFD</note>; <note>\( \frac{ED}{FD} = \frac{DC}{DB} = \frac{CE}{BF} \)</note>. Specifically, <note>BC = 5 ft</note>, <note>DC = 5 ft</note>, <note>CE = d ft</note>, <note>BF = 6 ft</note>, <note>DB=BC+DC=5+5=10</note>. According to the proportional relationship, <note>\( \frac{5}{10} = \frac{d}{6} \)</note>, after cross-multiplying, we get <note>\( 30 = 10d \)</note>, dividing both sides by 10, the result is <note>\( d = 3 \)</note>.

Definition of Right Triangle

In the figure of this problem, in triangle FDB, <note>angle ∠BFD is a right angle (90 degrees)</note>, thus triangle FDB is a right triangle. <note>Side FB and side FD are the legs, and side BD is the hypotenuse</note>. In triangle EDC, <note>angle ∠DEC is a right angle (90 degrees)</note>, thus triangle EDC is a right triangle. <note>Side EC and side ED are the legs, and side CD is the hypotenuse</note>.


Question: Find n.Write your answer as a whole number or a decimal. Do not round.n =  $\Box$  feet

Answer: 5 feet

1. Observing the two triangles △KIJ and △KLH in the figure, based on the AA criterion for similar triangles, we can confirm that △KIJ and △KLH are similar triangles.

2. According to the AA criterion for similar triangles, ∠KIJ = ∠KHL, ∠IKJ = ∠LKH, ∠KJI = ∠KLH. Through the AA criterion for similar triangles; since it is known that ∠HKL of triangle △KLH and ∠IKL of triangle △KIJ are vertical angles, we can conclude △KIJ ∽ △KLH.

3. According to the definition of similar triangles, the ratios of corresponding sides of the triangles are equal, that is, we can obtain the following proportional relationships: a. Comparing side KJ with side KH, and side IJ with side LH, that is: IJ / LH = KJ / KH. b. Given IJ = 8 ft, LH = 4 ft, JK = 10 ft, and the value we need to find is n (HK). Thus, we can set up the proportion: 8 / 4 = 10 / n.

4. From the above equation 8 / 4 = 10 / n, we can cross-multiply to get: 8 * n = 4 * 10.

5. Simplifying the above equation: 8n = 40.

6. To solve for n, divide both sides by 8 to get: n = 40 / 8.

7. Simplifying further, we get: n = 5.

8. Through the above reasoning, we finally obtain the answer as 5.

Similarity Theorem for Triangles (AA)

In triangles <note>△KIJ</note> and <note>△KLH</note>, if <note>∠IKJ is equal to ∠LKH</note> and <note>∠KJI is equal to ∠KHL</note>, then <note>△KIJ is similar to △KLH</note>.

Definition of Vertical Angles

<note>Two intersecting lines IL and JH intersect at point K</note>, forming four angles: <note>angle IKH, angle IKJ, angle JKL, and angle HKL</note>. According to the definition of vertical angles, <note>angle IKH and angle JKL are vertical angles</note>, <note>angle IKJ and angle HKL are vertical angles</note>. Since vertical angles are equal, <note>angle IKH = angle JKL</note>, <note>angle IKJ = angle HKL</note>.

Definition of Similar Triangles

<note>Triangles △KIJ and △KLH are similar triangles</note>. According to the definition of similar triangles: <note>angle ∠KIJ = ∠KLH</note>, <note>angle ∠IKJ = ∠LKH</note>, <note>angle ∠KJI = ∠KHL</note>; <note>IJ/LH = KI/KL = JK/HK</note>.


Question:
If KM=14 and LM=10, what is MN?

Write your answer as a whole number or as a decimal rounded to the nearest hundredth.
MN =  $\Box$

Answer: MN = 7.14

1. Given △LNM and △KLM, angles ∠LNM and ∠KLM are both right angles, and angle ∠LMN is a common angle, so △LNM and △KLM have two equal angles.

2. According to the similarity theorem (AA) (two angles are equal, two triangles are similar), we get △LNM ∼ △KLM.

3. According to the definition of similar triangles, the corresponding sides of similar triangles are proportional, so MN/LM = LN/KL = LM/KM.

4. Given KM = 14 and LM = 10.

5. In the side ratio formula of similar triangles MN/LM = LM/KM, substitute the known values to get MN/10 = 10/14.

6. Solve the equation MN/10 = 10/14 to find MN, and get MN = (10/14) × 10.

7. Calculate to get MN = 100/14 = 7.142857....

8. Round MN to two decimal places to get MN ≈ 7.14.

9. Through the above reasoning, the final answer is MN ≈ 7.14.

Definition of Right Triangle

In the figure of this problem, in triangle LNM, <note>angle LNM is a right angle (90 degrees)</note>, so triangle LNM is a right triangle. <note>Side LN and side NM are the legs</note>, <note>side LM is the hypotenuse</note>. In triangle KLM, <note>angle KLM is a right angle (90 degrees)</note>, so triangle KLM is a right triangle. <note>Side KL and side LM are the legs</note>, <note>side KM is the hypotenuse</note>.

Similarity Theorem for Triangles (AA)

<note>In the diagram of this problem, in triangles LNM and KLM, angle LMN is equal to angle LMK (common angle)</note>, <note>and angle LNM is equal to angle KLM (both are right angles)</note>, so <note>triangle LNM is similar to triangle KLM</note>.

Definition of Similar Triangles

<note>In the figure of this problem, triangle LNM and triangle KLM are similar triangles</note>. According to the definition of similar triangles: <note>angle LNM = angle KLM</note>, <note>angle MLN = angle MLK</note>; <note>MN/LM = LN/KL = LM/KM</note>. Substituting the known values <note>KM=14</note> and <note>LM=10</note>, we can obtain <note>MN/LM=LM/KM</note>.


Question:
Find s.

Write your answer in simplest radical form.
 $\Box$  centimeters

Answer: 3 centimeters

1. Given that this is a 45°-45°-90° isosceles right triangle, because in this isosceles right triangle, the 45° angles are equal. Assume the two legs of the isosceles right triangle are s, and let the hypotenuse be h, it is known that the length of the hypotenuse h is 3√2 cm.

2. According to the property of an isosceles right triangle: in a 45°-45°-90° isosceles right triangle, the ratio of the legs to the hypotenuse is 1:√2. Therefore, h = s√2.

3. Substitute the known hypotenuse length into the equation: 3√2 = s√2.

4. Solve the equation: divide both sides by √2, obtaining s = 3.

5. Through the above reasoning, the final answer is 3 cm.

Definition of Right Triangle

<note>One angle in the triangle is a right angle (90 degrees)</note>, therefore the triangle is a right triangle. The two legs of the right angle are s and s, and the hypotenuse is 3√2 centimeters.

Definition of Isosceles Triangle

A triangle is an isosceles triangle if and only if it has at least two congruent (≅) sides.

In the figure of this problem, in the triangle, <note>the two right-angle sides are equal</note>, therefore the triangle is an <note>isosceles triangle</note>.

Properties of Isosceles Right Triangle

<note>The triangle is an isosceles right triangle</note>, <note>Hypotenuse = leg * √2</note>.


Question:
If PR=35 and RS=12, what is QR?

Write your answer as a whole number or as a decimal rounded to the nearest hundredth.
QR =  $\Box$

Answer: QR = 20.49

1. In the problem, we know that △QSR and △PQR are both right triangles.

2. Because ∠QSR and ∠PQR are both right angles, ∠QSR = 90° and ∠PQR = 90°.

3. Observing the figure, we can see that ∠SRQ and ∠PRQ are common, so these two triangles have two equal angles.

4. According to the similarity theorem (AA), if two triangles have two corresponding angles equal, then the two triangles are similar. Therefore, △QSR ∼ △PQR.

5. Because the corresponding sides of similar triangles are proportional, we get the proportional relationship of the corresponding sides of the similar triangles: QR/PR = RS/QR.

6. Substituting the known side lengths, PR = 35 and RS = 12, we get the proportional relationship equation: QR/35 = 12/QR.

7. Simplifying the proportional relationship: QR * QR = 35 * 12.

8. Further simplifying the equation to QR^2 = 420.

9. Solving for QR, we get QR = √420.

10. Calculating this square root, QR = 20.4939...

11. Rounding the result to two decimal places, QR = 20.49.

12. Through the above reasoning, the final answer is QR = 20.49.

Definition of Right Triangle

<note>In the figure of this problem, in triangle QSR, angle QSR is a right angle (90 degrees)</note>, therefore triangle QSR is a right triangle. <note>Sides QS and RS are the legs, and side QR is the hypotenuse</note>. In triangle PQR, <note>angle PQR is a right angle (90 degrees)</note>, therefore triangle PQR is a right triangle. <note>Sides PQ and QR are the legs, and side PR is the hypotenuse</note>.

Similarity Theorem for Triangles (AA)

<note>In triangles QSR and PQR, ∠QSR is equal to ∠PQR (both are 90°)</note>, and <note>∠SRQ is equal to ∠PRQ</note>, so <note>triangle QSR is similar to triangle PQR</note>.

Definition of Similar Triangles

<note>Triangle QSR and triangle PQR are similar triangles</note>. According to the definition of similar triangles: <note>angle QSR = angle PQR</note>, <note>angle SRQ = angle PRQ</note>; <note>QR/PR = RS/QR</note>.


Question: What is the volume? $\Box$  cubic meters

Answer: 729 cubic meters

1. The geometric figure given in the problem is a cube, with each side length being 9 meters.

2. Since the problem requires solving for the volume, according to the volume formula of a cube V = a^3, where a is the side length.

3. Substitute the cube side length a=9 meters into the formula, and calculate the volume V = 9 meters × 9 meters × 9 meters.

4. After calculation, we get V = 729 cubic meters.

5. Through the above reasoning, the final answer is 729 cubic meters.

Definition of Cube

A cube is a three-dimensional geometric figure with six faces, all of which are squares.

<note>A cube is represented as a solid with six square faces</note>, each edge length is 9 meters, which meets the definition of a cube. Since the problem provides the length of each edge, <note>each face is a square with a side length of 9 meters</note>.

Volume Formula of a Cube

The formula for the volume of a cube is: Volume = side length × side length × side length, i.e., \( V = a^3 \), where \( a \) is the side length of the cube.

Original text: <note>Side length a = 9 meters</note>, the calculation process of volume V is: V = <note>9 meters × 9 meters × 9 meters</note> = 729 cubic meters. By substituting the side length for calculation, it is finally confirmed that the volume is <note>729 cubic meters</note>.


Question:
If KL=22 and KN=17, what is KM?

Write your answer as a whole number or as a decimal rounded to the nearest hundredth.
KM =  $\Box$

Answer: KM = 28.47

1. Given △KNL and △KLM are both right triangles, where ∠LNK and ∠MLK are right angles, and ∠K is the common angle.

2. According to the theorem of similar triangles (AA), if two angles of two triangles correspondingly equal, then the two triangles are similar, therefore △KNL and △KLM are similar.

3. According to the definition of similar triangles, for △KNL and △KLM, the following proportional relationship holds: KM/KL=KL/KN.

4. Given KL=22 and KN=17, substitute the known values into the proportional relationship, we get KM/22=22/17.

5. Solve the proportional equation by multiplying both sides by 22: KM=(22/17) * 22.

6. Calculate to get KM=484/17, KM≈28.470588235294117647058823529411.

7. According to the problem requirement, round KM to two decimal places, we get KM≈28.47.

8. Through the above reasoning, the final answer is KM=28.47.

Definition of Right Triangle

<note>In the figure of this problem, in triangle KNL, angle LNK is a right angle (90 degrees)</note>, therefore triangle KNL is a right triangle. <note>Side KN and side NL are the legs, side KL is the hypotenuse</note>. <note>In triangle KLM, angle MLK is a right angle (90 degrees)</note>, therefore triangle KLM is a right triangle. <note>Side KL and side LM are the legs, side KM is the hypotenuse</note>.

Similarity Theorem for Triangles (AA)

In the figure of this problem, in triangle KNL and triangle KLM, <note>∠LNK is equal to ∠MLK</note>, and <note>∠LKN is equal to ∠LKM</note>, so <note>triangle KNL is similar to triangle KLM</note>.

Definition of Similar Triangles

<note>Triangles KNL and KLM are similar triangles</note>. According to the definition of similar triangles: <note>angle LNK = angle MLK</note>, <note>angle NLK = angle LMK</note>, <note>angle LKN = angle MKL</note>; <note>KM/KL = KL/KN</note>.


Question:
If SV=28 and TV=15, what is UV?

Write your answer as a whole number or as a decimal rounded to the nearest hundredth.
UV =  $\Box$

Answer: UV = 8.04

1. Given △SVT and △TVU are both right triangles, where SV=28 and TV=15.

2. In Rt△SVT and Rt△STU, according to the property of complementary acute angles in right triangles, we have ∠VST + ∠STV = 90° and ∠VUT + ∠VST = 90°.

3. Through the equation ∠VST + ∠STV= ∠VUT + ∠VST, we can obtain ∠STV = ∠VUT.

4. Since △SVT and △TVU have two corresponding angles equal (∠SVT = ∠TVU and ∠STV = ∠VUT), according to the AA criterion for similar triangles, we can determine △SVT ~ △TVU.

5. According to the definition of similar triangles, the corresponding sides of similar triangles are proportional, so we have SV/TV = TV/UV.

6. Substitute the known side lengths into the proportion, that is, 28/15 = 15/UV.

7. Solve the above proportion by cross-multiplying, we get 28 × UV = 15 × 15.

8. Simplify the equation to obtain UV = 225/28.

9. Calculate the value of 225/28 which is approximately 8.0357, rounding to two decimal places we get UV = 8.04.

10. Through the above reasoning, the final answer is UV=8.04.

Definition of Right Triangle

In <note>triangle SVT</note>, <note>angle ∠SVT is a right angle (90 degrees)</note>, so <note>triangle SVT is a right triangle</note>. Side <note>SV</note> and side <note>VT are the legs</note>, and side <note>ST is the hypotenuse</note>. In <note>triangle TVU</note>, <note>angle ∠TVU is a right angle (90 degrees)</note>, so <note>triangle TVU is a right triangle</note>. Side <note>TV</note> and side <note>VU are the legs</note>, and side <note>TU is the hypotenuse</note>. In <note>triangle STU</note>, <note>angle ∠STU is a right angle (90 degrees)</note>, so <note>triangle STU is a right triangle</note>. Side <note>ST</note> and side <note>TU are the legs</note>, and side <note>SU is the hypotenuse</note>.

Definition of Similar Triangles

<note>Triangles SVT and TVU are similar triangles</note>. According to the definition of similar triangles: <note>∠SVT = ∠TVU</note>, <note>∠VST = ∠VTU</note>; <note>SV/TV = TV/UV</note>.

Similarity Theorem for Triangles (AA)

<note>∠STV</note> is equal to <note>∠TUV</note>, and <note>∠SVT</note> is equal to <note>∠TVU</note>, so <note>triangle SVT is similar to triangle TVU</note>.

Complementary Acute Angles in a Right Triangle

In a right triangle, the sum of the two non-right angles is 90°.

<note>In the right triangle SVT, angle SVT is a right angle (90 degrees)</note>, ∠VST and ∠STV are the two acute angles other than the right angle. According to the property of complementary acute angles in a right triangle, <note>the sum of ∠VST and ∠STV is 90 degrees, i.e., ∠VST + ∠STV = 90°</note>. Similarly, <note>in the right triangle STU, ∠VUT + ∠VST = 90°</note>.


Question:
If LM=24 and MN=13, what is KM?

Write your answer as a whole number or as a decimal rounded to the nearest hundredth.
KM =  $\Box$

Answer: KM = 44.31

1. In △LNM, ∠LNM is a right angle, and in △KLM, ∠KLM is also a right angle. Both triangles have the common angle ∠M.

2. According to the theorem of similarity (AA), if two triangles have two corresponding angles equal respectively, then the two triangles are similar. Since △LNM and △KLM both have equal right angles and the common angle ∠M, △LNM ∼ △KLM.

3. Since △LNM ∼ △KLM, the corresponding sides of similar triangles are proportional. Therefore,
 rac{KM}{LM} = rac{LM}{MN}

4. Given LM=24 and MN=13, substitute into the proportion, i.e.,
 rac{KM}{24} = rac{24}{13}

5. Solve the proportion:
 KM = rac{24 	imes 24}{13}

6. Calculation yields:
 KM = rac{576}{13} ≈ 44.307692...

7. Rounding to two decimal places, we get KM≈44.31.

8. Through the above reasoning, the final answer is 44.31.

Definition of an Angle

<note>∠LNM is a right angle formed by ray LN and ray NM through the common endpoint N</note>. Similarly, <note>∠KLM is a right angle formed by ray KL and ray LM through the common endpoint L</note>. And <note>∠LMK is an angle formed by ray ML and ray MK through the common endpoint M</note>.

Definition of Right Triangle

In the figure of this problem, in triangle LNM, <note>angle LNM is a right angle (90 degrees)</note>, so triangle LNM is a right triangle. <note>Side LN and side NM are the legs</note>, <note>side LM is the hypotenuse</note>. In triangle KLM, <note>angle KLM is a right angle (90 degrees)</note>, so triangle KLM is a right triangle. <note>Side KL and side LM are the legs</note>, <note>side KM is the hypotenuse</note>.

Similarity Theorem for Triangles (AA)

In the given figure, triangle LNM and triangle KLM, <note>∠LNM is equal to ∠KLM</note>, and <note>∠M is a common angle</note>, so <note>triangle LNM is similar to triangle KLM</note>.

Definition of Similar Triangles

<note>Triangles LNM and KLM are similar triangles</note>. According to the definition of similar triangles: <note>angle NLM = angle LKM</note>, <note>angle LNM = angle KLM</note>, <note>angle LMN = angle LMK</note>; <note>MN/LM = LN/LK = MN/LM</note>.


Question:
If QR=25 and RS=14, what is PR?

Write your answer as a whole number or as a decimal rounded to the nearest hundredth.
PR =  $\Box$

Answer: PR = 44.64

1. Observe △QSR and △PQR, first notice that in triangle △QSR, ∠QSR is a right angle, and through the given condition in △PQR, ∠PQR is also a right angle.

2. In the two triangles △QSR and △PQR, ∠R is a common angle. Therefore, △QSR and △PQR are similar according to the AA similarity theorem, as each has two corresponding angles equal. Hence, △QSR and △PQR are similar.

3. According to the definition of similar triangles, the corresponding sides of similar triangles are proportional, so PR/QR = QR/RS.

4. Substitute the given conditions QR = 25, RS = 14, to get PR/25 = 25/14.

5. Solve the equation PR/25 = 25/14 by cross-multiplying to get PR x 14 = 25 x 25.

6. Calculate to get PR = 625/14.

7. Perform the division PR = 625/14 to get PR ≈ 44.642857.

8. Round PR ≈ 44.642857 to two decimal places to get PR ≈ 44.64.

9. Through the above reasoning, the final answer is 44.64.

Definition of Right Triangle

In the figure of this problem, in triangle QSR, <note>angle QRS is a right angle (90 degrees)</note>, thus triangle QSR is a right triangle. <note>Side QS and side RS are the legs</note>, <note>side QR is the hypotenuse</note>. In triangle PQR, <note>angle PQR is a right angle (90 degrees)</note>, thus triangle PQR is a right triangle. <note>Side PQ and side QR are the legs</note>, <note>side PR is the hypotenuse</note>.

Similarity Theorem for Triangles (AA)

In the figure of this problem, in triangles QSR and PQR, <note>angle QSR is equal to angle PQR (both are right angles)</note>, and <note>angle R is a common angle</note>, so <note>triangle QSR is similar to triangle PQR</note>.

Definition of Similar Triangles

In the problem diagram, triangles <note>△QSR and △PQR are similar triangles</note>. According to the definition of similar triangles: <note>∠QSR = ∠PQR</note>, <note>∠QRS = ∠QPR</note>; <note>PR/QR = QR/RS</note>. Substituting the given conditions <note>QR = 25, RS = 14</note>, we get <note>PR / 25 = 25 / 14</note>.


Question: What is the volume? $\Box$  cubic feet

Answer: 1,000 cubic feet

1. Given that each side length of the cube is 10 ft.

2. According to the volume formula of a cube, for a cube, the volume is equal to the cube of the side length, i.e., Volume = side length × side length × side length.

3. Substitute the given side length of 10 ft to calculate the volume: Volume = 10 ft × 10 ft × 10 ft.

4. Calculate to get Volume = 1000 cubic ft.

5. Through the above reasoning, the final answer is 1000 cubic ft.

Definition of Cube

<note>In the diagram of this problem, each edge length of the cube is 10 feet, and each face is a square.</note>

Volume Formula of a Cube

Original text: <note>For a cube with a side length of 10 feet</note>, the volume calculation formula is: Volume = <note>10 feet × 10 feet × 10 feet</note>, resulting in a final volume of <note>1000 cubic feet</note>.


Question:
If WX=24 and WY=38, what is WZ?

Write your answer as a whole number or as a decimal rounded to the nearest hundredth.
WZ =  $\Box$

Answer: WZ = 15.16

1. Observe the figure and give the known conditions: WX=24 and WY=38.

2. Analyze the figures △WZX and △WXY, noting that they both contain right angles, i.e., ∠WZX=∠WXY=90°.

3. At the same time, ∠XWZ is a common angle in △WZX and △WXY, thus they are similar triangles. According to the 'AA similarity theorem': If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. This means △WZX ∼ △WXY.

4. Since △WZX and △WXY are similar triangles, according to the definition of similar triangles, we get the proportional relationship of corresponding sides: WZ/WX = WX/WY.

5. Substitute the known side lengths into the proportional relationship: WZ/24 = 24/38.

6. Solve the proportion by cross-multiplication: WZ * 38 = 24 * 24.

7. Calculate: WZ * 38 = 576.

8. Further simplify to get: WZ = 576 / 38.

9. Calculate the quotient: WZ ≈ 15.1578, rounding to two decimal places gives: WZ ≈ 15.16.

10. Through the above reasoning, the final answer is 15.16.

Definition of an Angle

<note>Angle ∠WZX</note> is a geometric figure formed by <note>ray WZ</note> and <note>ray XZ</note>, which share a common endpoint <note>Z</note>. This common endpoint <note>Z</note> is called the <note>vertex of angle ∠WZX</note>, and <note>ray WZ</note> and <note>XZ</note> are called the <note>sides of angle ∠WZX</note>. <note>Angle ∠WXY</note> is a geometric figure formed by <note>ray WX</note> and <note>ray XY</note>, which share a common endpoint <note>X</note>. This common endpoint <note>X</note> is called the <note>vertex of angle ∠WXY</note>, and <note>ray WX</note> and <note>XY</note> are called the <note>sides of angle ∠WXY</note>. <note>Angle ∠XWZ</note> is a geometric figure formed by <note>ray WX</note> and <note>ray WZ</note>, which share a common endpoint <note>W</note>. This common endpoint

Similarity Theorem for Triangles (AA)

In <note>triangle WZX</note> and <note>triangle WXY</note>, if <note>angle WZX is equal to angle WXY</note>, and <note>angle XWZ is equal to angle XWY</note>, then <note>triangle WZX is similar to triangle WXY</note>.

Definition of Similar Triangles

<note>Triangle WZX</note> and <note>Triangle WXY</note> are similar triangles. According to the definition of similar triangles: <note>Angle WZX = Angle WXY</note>, <note>Angle XWZ = Angle XWY</note>; <note>WZ/WX = WX/WY</note>.


Question:
Write the coordinates of the vertices after a rotation 90° counterclockwise around the origin.

J'( $\Box$ , $\Box$ )

K'( $\Box$ , $\Box$ )

L'( $\Box$ , $\Box$ )

M'( $\Box$ , $\Box$ )

Answer:
J'(-8,-9)
K'(-8,-6)
L'(-9,-6)
M'(-9,-9)

1. Given that a point is rotated counterclockwise 90° around the origin, according to the 2D plane rotation formula: if point P(x, y) is rotated counterclockwise 90° around the origin to obtain P'(x', y'), then x' = -y, y' = x.

2. For point J(-9, 8), substituting into the rotation formula gives the coordinates of J': x' = -8, y' = -9, so the coordinates of J' are (-8, -9).

3. For point K(-6, 8), substituting into the rotation formula gives the coordinates of K': x' = -8, y' = -6, so the coordinates of K' are (-8, -6).

4. For point L(-6, 9), substituting into the rotation formula gives the coordinates of L': x' = -9, y' = -6, so the coordinates of L' are (-9, -6).

5. For point M(-9, 9), substituting into the rotation formula gives the coordinates of M': x' = -9, y' = -9, so the coordinates of M' are (-9, -9).

6. Through the above steps, the vertices of quadrilateral JKLM, J, K, L, and M, are converted to new vertices according to the rotation formula, namely J'(-8, -9), K'(-8, -6), L'(-9, -6), M'(-9, -9).

7. Through reasoning and calculation, the coordinates of the vertices of quadrilateral JKLM after a 90° counterclockwise rotation are: J'(-8, -9), K'(-8, -6), L'(-9, -6), M'(-9, -9).

2D Plane Rotation Formula

<note>Point J(-9, 8)</note>, substituting into the rotation formula we get <note>the coordinates of J': x' = -8, y' = -9</note>, so <note>the coordinates of J' are (-8, -9)</note>. Similarly for <note>K, L, M</note>. Finally, we obtain <note>the rotated vertex coordinates of quadrilateral JKLM: J'(-8, -9), K'(-8, -6), L'(-9, -6), M'(-9, -9)</note>.


Question: What is the volume? $\Box$  cubic millimeters

Answer: 486 cubic millimeters

1. Observe the cuboid shown in the problem, measure its length, width, and height as 9 mm, 6 mm, and 9 mm respectively.

2. According to the formula for the volume of a cuboid, Volume V = length × width × height. This formula is derived based on the definition of a cuboid in solid geometry.

3. Substitute the measured length, width, and height of the cuboid into the formula, obtaining V = 9 mm × 6 mm × 9 mm.

4. Calculate to get: V = 486 cubic millimeters.

5. Therefore, through the above reasoning, the final answer is 486 cubic millimeters.

Definition of Rectangular Prism

A rectangular prism is defined as a three-dimensional geometric figure formed by six rectangular faces, where three mutually adjacent faces respectively define the length, width, and height.

<note>In the figure of this problem</note>, the geometric shape shown is a <note>rectangular prism</note>, where the <note>length is 9 millimeters</note>, the <note>width is 6 millimeters</note>, and the <note>height is 9 millimeters</note>. The length, width, and height of the rectangular prism correspond to the <note>lengths marked in the figure</note>.

Volume Formula of Rectangular Prism

The volume of a rectangular prism is equal to the product of its length, width, and height.

<note>In the figure of this problem, this geometric principle is used to calculate the volume of the rectangular prism</note>. <note>The length, width, and height of the rectangular prism are 9 mm, 6 mm, and 9 mm respectively</note>, by calculating <note>Volume V = 9 mm × 6 mm × 9 mm</note>, the result is <note>Volume is 486 cubic millimeters</note>.


Question: Find m.Write your answer as a whole number or a decimal. Do not round.m =  $\Box$  meters

Answer: 3 meters

1. Redraw △JKL and △MNK, and label the known side lengths and angles.

2. Observe that △LKJ and △MKN have congruent interior angles, where ∠LJK and ∠MNK are right angles, and the common angle is ∠K, thus △LKJ ∼ △MKN.

3. According to the definition of similar triangles, the ratios of corresponding sides of similar triangles are equal. Therefore, the ratio of corresponding sides KJ and KN are equal, and the ratio of corresponding sides LJ and MN are equal, KJ = 4 meters, KN = 8 meters.

4. Set up the proportion:

   KJ/KN = LJ/MN = LK/MK

5. Substitute the known side lengths:

   4/8 = m/6

6. Calculate:

   4/8 = m/6 -> 4 * 6 = 8 * m -> 24 = 8m

7. Solve the equation:

   m = 24/8 = 3

8. Through the above reasoning, the final answer is 3 meters.

Similarity Theorem for Triangles (AA)

<note>△JKL and △MNK</note> satisfy <note>∠LJK = ∠MNK = 90°</note>, <note>∠LJK = ∠MNK</note>, and share the angle <note>∠K</note>, therefore <note>△JKL ∼ △MKN</note>.

Definition of Similar Triangles

In the figure of this problem, <note>triangle JKL and triangle MNK are similar triangles</note>. According to the definition of similar triangles: <note>angle LJK = angle MNK</note>, <note>angle JKL = angle NKM</note>, <note>angle KLJ = angle KMN</note>; <note>KJ/KN = LJ/MN = LK/MK</note>.


Question:
Find p.

Write your answer in simplest radical form.
 $\Box$  miles

Answer: 5 miles

1. Given that the triangle is a right triangle, with one acute angle of 30 degrees and the other acute angle of 60 degrees, and the longer leg of the right triangle is 5√3.

2. According to the properties of a 30-60-90 triangle: In a right triangle, if one angle is 30 degrees, then the shorter leg is equal to half of the hypotenuse, and the longer leg is equal to the shorter leg multiplied by √3.

3. In this problem, let the length of the shorter leg be p. According to the properties of a 30-60-90 triangle, the longer leg is p√3.

4. It is given that the longer leg is 5√3, so we can set up the equation: p√3 = 5√3.

5. Divide both sides of the equation by √3 to get p = 5.

6. Through the above reasoning, the final answer is 5 miles.

Definition of Right Triangle

<note>One of the angles is a right angle (90 degrees)</note>, therefore the triangle is a right triangle. Side <note>p</note> and side <note>5√3</note> are the legs, and the remaining longest side is the <note>hypotenuse</note>.

Properties of 30°-60°-90° Triangle

In a 30°-60°-90° triangle, the side opposite the 30-degree angle (the shorter leg) is half the length of the hypotenuse. The side opposite the 60-degree angle (the longer leg) is √3 times the length of the shorter leg.

In the figure of this problem, <note>in a 30°-60°-90° triangle, the side opposite the 30° angle is p</note>, <note>the side opposite the 60° angle is 5√3</note>, <note>the side opposite the right angle is the hypotenuse</note>. According to the properties of a 30°-60°-90° triangle, <note>side p is equal to half of the hypotenuse</note>, <note>side 5√3 is equal to √3 times side p</note>. That is: <note>p = 1/2 * hypotenuse</note>, <note>5√3 = p * √3</note>.


Question: Find b.Write your answer as a whole number or a decimal. Do not round.b =  $\Box$  yards

Answer: 3 yards

1. In the given figure, △ILJ and △MLK are two right triangles, where ∠LIJ and ∠LMK are both right angles.

2. It is observed that ∠ILJ and ∠MLK are equal because they are common angles.

3. Therefore, △ILJ and △MLK are similar, according to the similarity theorem of triangles (AA).

4. The definition of similar triangles indicates that corresponding sides of similar triangles are proportional. Therefore, according to the side length ratio formula of similar triangles, we get: IL / ML = JI / KM.

5. Given IL = IM + ML = 4 + 4 = 8 yd, JI = 6 yd, and ML = 4 yd, we can set up the proportion equation: 8 / 4 = 6 / b.

6. Simplifying further, we get: 2 = 6 / b.

7. By cross-multiplying, we get b = 6 / 2, which means b = 3 yd.

8. Through the above reasoning, the final answer is b = 3 yd.

Definition of Right Triangle

<note>In the figure of this problem, in triangle ILJ, angle LIJ is a right angle (90 degrees)</note>, therefore triangle ILJ is a right triangle. Side IJ and side IL are the legs, and side JL is the hypotenuse. <note>In triangle MLK, angle LMK is a right angle (90 degrees)</note>, therefore triangle MLK is a right triangle. Side MK and side ML are the legs, and side KL is the hypotenuse.

Similarity Theorem for Triangles (AA)

In the figure of this problem, <note>△ILJ and △MLK are similar triangles</note>. Because <note>∠ILJ is equal to ∠MLK</note> and <note>∠LIJ and ∠LMK are both right angles</note>, the corresponding angles of the two triangles are equal. According to the definition of similar triangles, we further derive that <note>the corresponding sides are proportional, that is, IL / ML = JI / KM</note>.

Definition of Similar Triangles

<note>△ILJ and △MLK are similar triangles</note>. According to the definition of similar triangles: <note>∠ILJ = ∠MLK</note>, <note>∠IJL = ∠MKL</note>, <note>∠JIL = ∠KML</note>; <note>IL/ML = JI/KM = JL/LK</note>.


Question: What is the volume? $\Box$  cubic yards

Answer: 168 cubic yards

1. According to the problem, this geometric figure is a cuboid, i.e., a rectangular prism.

2. Based on the properties of a cuboid, the volume is equal to the product of its length, width, and height.

3. Observing the figure, the length of the cuboid is 8 yards, the width is 3 yards, and the height is 7 yards.

4. According to the cuboid volume formula: Volume = Length × Width × Height, substitute the known values to calculate, obtaining Volume = 8 yards × 3 yards × 7 yards.

5. Perform the multiplication: 8 × 3 = 24, then 24 × 7 = 168.

6. Therefore, the volume of the cuboid is 168 cubic yards.

Definition of Rectangular Prism

<note>This geometric body is a rectangular prism</note>, <note>all six of its faces are rectangles</note>, <note>each angle is a right angle</note>.

Volume Formula of Rectangular Prism

<note>The length of the rectangular prism is 8 yards</note>, <note>the width is 3 yards</note>, <note>the height is 7 yards</note>. We use the formula: <note>Volume = Length × Width × Height</note>, substituting the known values to get <note>Volume = 8 yards × 3 yards × 7 yards</note>.


Question: What is the volume? $\Box$  cubic meters

Answer: 343 cubic meters

1. Given that each edge length of the cube is 7 meters.

2. According to the cube volume calculation formula: Volume = edge length × edge length × edge length, the volume calculation process can be derived.

3. Substitute the known edge length of 7 meters into the formula: Volume = 7 meters × 7 meters × 7 meters.

4. Calculate the product: 7 × 7 = 49, and 49 multiplied by 7 equals 343.

5. Therefore, according to the formula calculation, the volume of the cube is 343 cubic meters.

6. Through the above reasoning, the final answer is 343 cubic meters.

Definition of Cube

<note>This three-dimensional geometric figure is a cube</note>, with each edge measuring <note>7 meters</note>, thus meeting all the characteristics of the definition of a cube, including equal-length edges and square faces.

Volume Formula of a Cube

In the figure of this problem, it is known that <note>the side length of the cube is 7 meters</note>. According to this formula, the volume calculation process can be obtained: <note>Volume = 7 meters × 7 meters × 7 meters</note>. Through calculation, the final result is <note>the volume of the cube is 343 cubic meters</note>.


Question: The volume of this triangular prism is 80 cubic millimeters. What is the value of z?z =  $\Box$  millimeters

Answer: 4 millimeters

1. Given the volume of the triangular prism is 80 cubic millimeters, the length of the base edge is 8 millimeters, and the length of the prism is 5 millimeters.

2. The formula for the volume of the triangular prism is: Volume = 1/2 × base edge × height × length.

3. According to the problem, substituting the known values, we get: 80 = 1/2 × 8 × z × 5.

4. Simplifying the equation, 1/2 × 8 equals 4, so the equation becomes: 80 = 4 × z × 5.

5. Calculating 4 × 5 gives 20, further simplifying the equation to: 80 = 20 × z.

6. Dividing both sides of the equation by 20, we get z = 80 / 20.

7. Calculating 80 / 20 gives 4, therefore z = 4.

8. Through the above reasoning, the final answer is 4 millimeters.

Definition of Triangular Prism

A triangular prism is a type of hexahedron that is formed by two parallel and congruent triangular bases and three rectangular lateral faces.

The <note>triangular prism</note> is composed of <note>two triangular bases</note> and <note>three rectangular sides</note>, where the <note>base side lengths are 8 millimeters each</note>, the <note>height is z millimeters</note>, and the <note>prism length is 5 millimeters</note>.

Volume Formula of Prism

The volume of a prism is equal to the base area multiplied by the height.

In the figure of this problem, in the prism, <note>let z be the height of the triangular base</note>, then the area of the base is <note>½ * 8 * z</note>, and the height of the prism is <note>5mm</note>. Therefore, according to the volume formula of the prism, the volume of the prism is equal to <note>½ * 8 * z * 5</note>.


Question:
Find s.

Write your answer in simplest radical form.
 $\Box$  centimeters

Answer: 8 centimeters

1. Given a right triangle, its two right-angle sides are equal, and it is an isosceles right triangle.

2. In the figure, the two non-right angles of the right triangle are 45°.

3. According to the properties of an isosceles right triangle, if one right-angle side length is s, then the hypotenuse length is s√2.

4. The length of the hypotenuse in the figure is represented as 8√2 cm, thus we establish the equation hypotenuse = s√2.

5. Substituting the given condition into the equation, we get 8√2 = s√2.

6. Dividing both sides by √2, we get s = 8.

7. Through the above reasoning, the final answer is 8.

Definition of Isosceles Triangle

In the figure of this problem, <note>triangle ABC is an isosceles triangle</note>, where <note>∠BAC = ∠ABC = 45°</note>, and <note>AB = BC = s</note>.

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

In the figure of this problem, in triangle ABC, <note>angle ACB is a right angle (90 degrees)</note>, therefore <note>triangle ABC is a right triangle</note>. <note>Side AB and side BC are the legs</note>, <note>side AC is the hypotenuse</note>.

Properties of Isosceles Right Triangle

In the figure of this problem, the triangle is an isosceles right triangle, <note>hypotenuse 8√2</note> = <note>leg s*√2</note>.


Question: What is the surface area? $\Box$  square feet

Answer: 96 square feet

1. As shown in the figure, the solid figure is a cube, and the edge length of the cube is known to be 4 ft. According to the definition of a cube, a cube is a geometric body with six equal square faces.

2. Calculate the area of each face of the cube. According to the formula for the area of a square, the area is equal to the square of the side length. Each face has a side length of 4 ft, so the area of each square face is 4 ft × 4 ft = 16 square feet.

3. The cube has a total of 6 equal square faces. According to the formula for the surface area of a cube, the surface area of a cube is the sum of the areas of its six faces.

4. Calculate the total surface area. The surface area is the sum of the areas of the 6 square faces, so the surface area = 6 × 16 square feet = 96 square feet.

5. Based on the above reasoning, the final answer is 96 square feet.

Definition of Cube

In the figure of this problem, <note>all faces are squares with a side length of 4 feet</note>. <note>The three-dimensional figure of the cube</note> meets the conditions of the definition of a cube, that is, it has six faces, <note>each face is a square of exactly the same shape with a side length of 4 feet</note>.

Definition of Square

A quadrilateral is a square if and only if all four sides are congruent (≅) and all four interior angles are right angles (90°).

In the figure of this problem, each face of the cube is a <note>square with a side length of 4 feet</note>. Each face <note>meets the conditions of the definition of a square, i.e., all four sides are of equal length (4 feet)</note>, <note>and each interior angle is 90 degrees</note>.

Area Formula for Square

The area of a square is equal to the square of its side length, that is, Area = side length × side length.

<note>The side length of each face of the cube is 4 feet</note>. According to the area formula for a square, <note>the area of the square = 4 feet × 4 feet = 16 square feet</note>.

Surface Area Formula for a Cube

The total surface area of a cube is equal to 6 times the square of the edge length of the cube.

<note>A cube has six faces</note>, each face has an area of <note>16 square feet</note>, therefore the surface area of the cube = <note>6 × 16 square feet = 96 square feet</note>.


Question:
Find q.

Write your answer in simplest radical form.
 $\Box$  millimeters

Answer: 2 millimeters

1. Given that this is a right triangle, one acute angle is 30°, the other acute angle is 60°, and the length of the shorter leg is 1 mm.

2. According to the properties of a 30°-60°-90° triangle: in a 30°-60°-90° right triangle, the hypotenuse is twice the length of the shorter leg. Specifically, if the shorter leg length is a, then the hypotenuse length is 2a.

3. In this problem, the shorter leg length is known to be 1 mm, so according to this theorem, the hypotenuse, which is side q, is 2×1=2 mm.

4. Therefore, through the above reasoning, the final answer is 2 mm.

Definition of Right Triangle

<note>In the diagram of this problem, the angle in the triangle is a right angle (90 degrees)</note>, so the triangle is a right triangle. <note>Side 1 mm and another right-angle side are the legs</note>, <note>side q is the hypotenuse</note>.

Properties of 30°-60°-90° Triangle

<note>The angle opposite to 1mm is 30 degrees</note>, <note>The angle opposite to the other leg is 60 degrees</note>, <note>The angle opposite to side q is 90 degrees</note>. <note>Side q is the hypotenuse</note>, <note>The side of 1mm is opposite the 30-degree angle</note>, <note>The other leg is opposite the 60-degree angle</note>. According to the properties of a 30°-60°-90° triangle, <note>The side of 1mm is equal to half of side q</note>, <note>The other leg is equal to √3 times the side of 1mm</note>. That is: <note>q = 2 * 1mm = 2mm</note>, <note>The other leg = 1mm * √3 = √3mm</note>.


Question: What is the volume of this rectangular pyramid? $\Box$  cubic yards

Answer: 672 cubic yards

1. The problem states that the base of the rectangular pyramid is a square with a side length of 12 yards, and the height of the pyramid is 14 yards.

2. According to the formula for the area of a square, the area of the square base is A = side length × side length = 12 yards × 12 yards = 144 square yards.

3. According to the formula for the volume of a pyramid, the volume of the pyramid can be obtained by the formula V = (1/3) × base area × height.

4. Substituting the values into the formula, the volume V = (1/3) × 144 square yards × 14 yards.

5. Calculating the above formula, V = (1/3) × 2016 cubic yards = 672 cubic yards.

6. Through the above reasoning, the final answer is 672 cubic yards.

Area Formula for Square

In the figure of this problem, the base is a square, <note>with a side length of 12 yards</note>. Therefore, <note>the area of the base A = 12 yards × 12 yards = 144 square yards</note>.

Volume Formula of Pyramid

The volume \( V \) of a pyramid is equal to one third of the product of its base area and its height. Mathematically, this is expressed as: \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \).

<note>The area of the base is the area of a square</note>, which is 144 square yards, <note>the height is 14 yards</note>. Therefore, the volume V = (1/3) × 144 square yards × 14 yards. Substituting into the formula, we get V = 672 cubic yards.