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

B'( $\Box$ , $\Box$ )

C'( $\Box$ , $\Box$ )

D'( $\Box$ , $\Box$ )

Answer:
B'(9,-1)
C'(0,-1)
D'(9,-5)

1. Given the coordinates of point B as (-1, -9). According to the 2D plane rotation formula, for any point (x, y) rotated 90° counterclockwise about the origin, the new coordinates are (-y, x). Therefore, the coordinates of B' are (-(-9), -1), which is (9, -1).

2. For point C, given the coordinates as (-1, 0). Using the same 2D plane rotation formula, the new coordinates of C' are (-(0), -1), which is (0, -1).

3. For point D, given the coordinates as (-5, -9). Using the 2D plane rotation formula, the new coordinates of D' are (-(-9), -5), which is (9, -5).

4. Based on the above reasoning, the coordinates of points B, C, and D after a 90° counterclockwise rotation are B'(9, -1), C'(0, -1), and D'(9, -5).

Coordinates of a Point

In the figure of this problem, <note>the coordinates of point B are (-1, -9)</note>, <note>the coordinates of point C are (-1, 0)</note>, <note>the coordinates of point D are (-5, -9)</note>. These points can be represented by their coordinates, namely <note>B(-1, -9)</note>, <note>C(-1, 0)</note>, <note>D(-5, -9)</note>.

Rotation Transformation

<note>Points B, C, D rotate 90° counterclockwise around the origin</note> as an example. <note>Rotation center: origin</note>; <note>Rotation angle: 90°</note>. The new coordinates of the points after rotation: derived through the formula <note>B'(9, -1)</note>, <note>C'(0, -1)</note>, <note>D'(9, -5)</note>.

2D Plane Rotation Formula

<note>All points are rotated counterclockwise by 90° around the origin</note>, according to the formula to obtain the coordinates of the rotated points. <note>The original coordinates of point B are (-1, -9), and the rotated coordinates B' = (9, -1)</note>. <note>The original coordinates of point C are (-1, 0), and the rotated coordinates C' = (0, -1)</note>. <note>The original coordinates of point D are (-5, -9), and the rotated coordinates D' = (9, -5)</note>.


Question:
Write the coordinates of the vertices after a rotation 270° counterclockwise around the origin.

D'( $\Box$ , $\Box$ )

E'( $\Box$ , $\Box$ )

F'( $\Box$ , $\Box$ )

Answer:
D'(2,-2)
E'(2,-9)
F'(4,-3)

1. Given points D(2, 2), E(9, 2), F(3, 4), we need to rotate them counterclockwise by 270° around the origin O(0, 0).

2. According to the 2D plane rotation formula, x' = x*cos(θ) - y*sin(θ), y' = x*sin(θ) + y*cos(θ), θ=270°. Also, since cos270°=0, sin270°=-1.

3. By calculation, for any point (x, y) rotated 270° counterclockwise, the resulting point is (y, -x).

4. Substituting point D(2, 2) into the 2D plane rotation formula, we get the coordinates of point D' as (2, -2).

5. Substituting point E(9, 2) into the 2D plane rotation formula, we get the coordinates of point E' as (2, -9).

6. Substituting point F(3, 4) into the 2D plane rotation formula, we get the coordinates of point F' as (4, -3).

7. Through the above reasoning, the final answer is D'(2, -2), E'(2, -9), F'(4, -3).

Coordinates of a Point

In a Cartesian coordinate system, the position of a point is represented by an ordered pair (x, y), where x denotes the horizontal (x-coordinate) and y denotes the vertical (y-coordinate).

<note>The coordinates of point D are (2, 2)</note>, <note>The coordinates of point E are (9, 2)</note>, <note>The coordinates of point F are (3, 4)</note>. They respectively represent the positions of these points on the x-axis and y-axis.

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.

<note>Rotate points D, E, and F 270° counterclockwise around the origin O</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).

Original text: <note>θ=270°</note>, <note>cos270°=0</note>, <note>sin270°=-1</note>, so rotating 270° counterclockwise, the formula is <note>x' = y</note>, <note>y' = -x</note>. For point <note>D(2, 2)</note>, substituting into the rotation formula gives <note>D' coordinates: x' = 2, y' = -2</note>, so <note>D' coordinates are (2, -2)</note>. Similarly for <note>D, E, F</note>. Finally, the rotated vertex coordinates of quadrilateral <note>DEF</note> are: <note>D'(2, -2)</note>, <note>E'(2, -9)</note>, <note>F'(4, -3)</note>.


Question:
Are △BCD and △WXY congruent?

- yes
- no

Answer: - no

1. First, we need to find the lengths of the three sides of △BCD. Since point B(–1,–1) and point D(–1,–10) have the same x-coordinate, the length of segment BD is the absolute value of the difference in y-coordinates, which is |–10–(–1)|=9.

2. Then, we find the length of segment BC. Since point B(–1,–1) and point C(–10,–1) have the same y-coordinate, the length of segment BC is the absolute value of the difference in x-coordinates, which is |–10–(–1)|=9.

3. Next, we calculate the length of segment CD. Since point C(–10,–1) and point D(–1,–10) have different x-coordinates and y-coordinates, we need to use the distance formula: distance formula √((x2-x1)^2+(y2-y1)^2).

4. Substituting C(–10,–1) as (x1,y1) and D(–1,–10) as (x2,y2) into the formula, we calculate: CD=√((–1+10)^2+(–10+1)^2)=√(9^2+9^2)=√(81+81)=√162=9√2.

5. The lengths of the three sides of △BCD are BD=9, BC=9, CD=9√2.

6. Next, we calculate the lengths of the three sides of △WXY. First, find the length of segment WY. Since W(–10,10) and Y(–1,10) have the same y-coordinate, the length of segment WY is the absolute value of the difference in x-coordinates, which is |–1–(–10)|=9.

7. Next, we find the length of segment WX. Since W(–10,10) and X(–10,2) have the same x-coordinate, the length of segment WX is the absolute value of the difference in y-coordinates, which is |2–10|=8.

8. Then, we calculate the length of segment XY. Since X(–10,2) and Y(–1,10) have different x-coordinates and y-coordinates, we use the distance formula: XY=√((–1+10)^2+(10–2)^2)=√(9^2+8^2)=√(81+64)=√145.

9. The lengths of the three sides of △WXY are WY=9, WX=8, XY=√145.

10. To determine whether the two triangles are congruent, we use the congruent triangles theorem (SSS). This theorem requires that the corresponding sides of the two triangles are equal in length.

11. From the calculation results, we can see that the side lengths of △BCD are 9, 9, and 9√2, while the side lengths of △WXY are 9, 8, and √145. Clearly, the three sides of these two triangles do not correspond to equal lengths.

12. Therefore, according to the side-side-side theorem, △BCD and △WXY are not congruent.

13. Based on the above reasoning, the final answer is 'no'.

Distance Formula Between Two Points

In a Cartesian coordinate plane, the distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

<note>The coordinates of point C are (–10,–1)</note>, <note>The coordinates of point D are (–1,–10)</note>. Using the distance formula, we get: CD = √((–1+10)^2+(–10+1)^2) = √(9^2+9^2) = √(81+81) = √162 = 9√2. When calculating <note>the length of segment XY, the coordinates of point X are (–10,–2)</note>, <note>The coordinates of point Y are (–1,10)</note>. Using the distance formula, we get: XY = √((–1+10)^2+(–10+2)^2) = √(9^2+8^2) = √(81+64) = √145 = √145. Therefore, the definition is applied in this example to calculate the distance between the two points.

Triangle Congruence Theorem (SSS)

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

In <note>triangle BCD</note> and <note>triangle WXY</note>, <note>side BD is equal to side WY</note>, <note>side BC is not equal to side WX</note>, and <note>side CD is not equal to side XY</note>. According to the Triangle Congruence Theorem (SSS), when the three sides of two triangles are respectively equal, the two triangles are congruent. Therefore, <note>triangle BCD is not congruent to triangle WXY</note>.


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

S'( $\Box$ , $\Box$ )

T'( $\Box$ , $\Box$ )

U'( $\Box$ , $\Box$ )

V'( $\Box$ , $\Box$ )

Answer:
S'(-1,-7)
T'(-10,-7)
U'(-10,-2)
V'(-1,-2)

1. Given the coordinates of the vertices of the rectangle are S(-7, 1), T(-7, 10), U(-2, 10), V(-2, 1).

2. We need to rotate these points counterclockwise by 90° around the origin (0, 0).

3. According to the 2D plane rotation formula: (x', y') = (-y, x), where (x, y) are the original coordinates, and (x', y') are the transformed coordinates.

4. Applying the rotation formula to point S(-7, 1), we get the coordinates of S': (x', y') = (-1, -7).

5. Applying the rotation formula to point T(-7, 10), we get the coordinates of T': (x', y') = (-10, -7).

6. Applying the rotation formula to point U(-2, 10), we get the coordinates of U': (x', y') = (-10, -2).

7. Applying the rotation formula to point V(-2, 1), we get the coordinates of V': (x', y') = (-1, -2).

8. Through the above reasoning, we finally obtain the coordinates of the four vertices after rotation: S'(-1, -7), T'(-10, -7), U'(-10, -2), V'(-1, -2).

Rotation Transformation

<note>In the figure of this problem, the fixed point of rotation is the origin (0, 0)</note>, we rotate the vertices of the rectangle <note>(S, T, U, V)</note> counterclockwise around this fixed point <note>by 90°</note>, thus obtaining the new vertices <note>(S', T', U', V')</note>.

2D Plane Rotation Formula

In this problem, <note>the coordinates of point S are (-7, 1)</note>, we need to <note>rotate point S counterclockwise by 90° around the origin O</note> to get <note>the coordinates of point S' (-1, -7)</note>. Similarly, <note>the coordinates of point T are (-7, 10)</note>, we need to <note>rotate point T counterclockwise by 90° around the origin O</note> to get <note>the coordinates of point T' (-10, -7)</note>. Similarly, <note>the coordinates of point U are (-2, 10)</note>, we need to <note>rotate point U counterclockwise by 90° around the origin O</note> to get <note>the coordinates of point U' (-10, -2)</note>. Similarly, <note>the coordinates of point V are (-2, 1)</note>, we need to <note>rotate point V counterclockwise by 90° around the origin O</note> to get <note>the coordinates of point V' (-1, -2)</note>.


Question:
Does this picture have rotational symmetry?

- yes
- no

Answer: - yes

1. Observing the image, we can see that it contains a regular polygon, which is composed of 10 identical sectors.

2. Label each sector in the image as A1, A2, ..., A10. According to the formula for the central angle and interior angle of a regular polygon, the central angle of each sector is 360°/10 = 36°.

3. The smallest rotation that maps a geometric figure onto itself is called rotational symmetry. If it can coincide at an angle less than 360°, it has rotational symmetry.

4. For this figure, when we rotate it 36° clockwise or counterclockwise, due to the identical sectors, the figure looks exactly the same as it did initially.

5. Therefore, this figure has rotational symmetry at 36° and other multiples of 36° (such as 72°, 108°, ..., 324°).

6. Based on the rotational invariance theorem and the analysis of this figure, it is determined that this figure has rotational symmetry.

Definition of Regular Polygon

A polygon is a regular polygon if and only if all its sides are of equal length and all its interior angles are equal.

<note>In the figure of this problem</note>, the entire circular shape contains a regular polygon composed of <note>10 identical sectors</note>, that is, a <note>regular decagon</note>. Each sector has <note>equal side lengths</note>, and each sector's <note>central angle is 36°</note>.

Definition of Central Angle

An angle formed by two radii connecting two points on the circumference of a circle to the center of the circle is called a central angle.

<note>In the figure of this problem, the central angle of each sector is 36°</note>, calculated as 360°/10 = 36°. <note>In the figure, the central angle of each sector is formed by the center O and the sides of two adjacent sectors.</note>

Rotation Invariance Theorem

Certain geometric figures will coincide with their original position after being rotated by a specific angle. This property is known as rotational invariance.

<note>In this problem diagram, when the figure is rotated 36° clockwise or counterclockwise, the figure returns to its original appearance</note>. This satisfies the definition of rotational symmetry, indicating that the figure has rotational symmetry. It can also be seen that <note>every multiple of 36° such as 72°, 108°, ..., 324°, the figure can coincide with itself</note>.

Formulas for the Central Angle and Interior Angle of a Regular Polygon

For a regular polygon with \( n \) sides, the measure of each central angle is given by \( \frac{360^\circ}{n} \). The measure of each interior angle is given by \( \frac{(n - 2) \cdot 180^\circ}{n} \).

<note>A regular polygon has 10 sides</note>, in which <note>the degree of the central angle is 360 degrees divided by the number of sides</note>, and each <note>interior angle degree is (number of sides - 2) multiplied by 180 degrees then divided by the number of sides</note>.


Question:
If $\overline{PQ}$ is not parallel to $\overline{RS}$ , what is m $\angle $ R?

m $\angle $ R= $\Box$ °

Answer: m \$\angle \$ R=106°

1. According to the problem description, quadrilateral PQRS has one pair of parallel sides and another pair of equal sides. We first confirm that the parallel sides are \\overline{PS} and \\overline{QR}, and the sides between the two diagonals with the same angle are \\overline{PQ} and \\overline{RS}. This information suggests that quadrilateral PQRS is an isosceles trapezoid.

2. An isosceles trapezoid is a quadrilateral with one pair of parallel sides (bases) and another pair of equal sides (legs). Here, it is known that \\overline{PQ} and \\overline{RS} are equal, and \\overline{PS} and \\overline{QR} are parallel, thus PQRS meets the definition of an isosceles trapezoid.

3. An important property of an isosceles trapezoid is that the base angles are equal. In the isosceles trapezoid PQRS, we know that \\angle QPS and \\angle PSR are base angles, and \\angle SRQ and \\angle RQP are also base angles.

4. According to the properties of an isosceles trapezoid, the base angles \\angle QRS and \\angle PQR are equal. Therefore, \\angle R = \\angle Q.

5. From the figure, we know that \\angle Q = 106°.

6. Since the base angles are equal, we can infer that \\angle R = 106°.

7. Through the above reasoning, we finally conclude that the answer is 106°.

Properties of an Isosceles Trapezoid

In an isosceles trapezoid, the base angles are equal.

<note>In the isosceles trapezoid PQRS, side PS and side QR are the two bases of the isosceles trapezoid, and side PQ and side RS are the two legs of the isosceles trapezoid</note>. According to the properties of an isosceles trapezoid, <note>the base angle ∠QPS and the base angle ∠PSR are equal</note>, <note>the base angle ∠PQR and the base angle ∠QRS are equal</note>.

Definition of Isosceles Trapezoid

A trapezoid is isosceles if and only if its non-parallel sides (legs) are congruent (∅).

<note>Side SP and side RQ are parallel</note>, <note>side SR and side PQ are the legs of the trapezoid</note>. According to the definition of an isosceles trapezoid, <note>side SR and side PQ are equal</note>. Therefore, <note>trapezoid SPQR is an isosceles trapezoid</note>.


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

- yes
- no

Answer: - yes

1. Given that one pair of opposite sides of the quadrilateral are parallel and equal in length to 33.

2. According to the theorem 3 for determining parallelograms, if a quadrilateral has one pair of opposite sides that are parallel and equal, then the quadrilateral is a parallelogram.

3. Therefore, according to this theorem, the given quadrilateral is a parallelogram.

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 diagram of this problem, <note>one pair of opposite sides of the quadrilateral are parallel and both have a length of 33</note>. This confirms that <note>this pair of sides are parallel and equal</note>, which exactly meets the definition of a parallelogram.

Parallelogram Criterion Theorem 3

If one pair of opposite sides of a quadrilateral are both parallel (∥) and congruent (≅), then the quadrilateral is a parallelogram.

<note>Opposite sides are parallel and equal</note>, so the quadrilateral is a <note>parallelogram</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,-2)
L'(3,-2)
M'(3,-9)

1. Given that the coordinates of point J are (−9, −8), we need to rotate it 90° counterclockwise about the origin. According to the 2D plane rotation formula, the new coordinates of (x, y) after a 90° counterclockwise rotation are (−y, x).

2. Using the above rotation formula, transforming J(−9, −8) gives J' (8, −9).

3. Next, we consider point K with coordinates (−2, −8). Applying the same rotation formula, we get K' (8, −2).

4. For point L with coordinates (−2, −3), the new coordinates after rotation are L' (3, −2).

5. Finally, for point M with coordinates (−9, −3), applying the same rotation formula gives M' (3, −9).

6. Through the above reasoning, the final coordinates of the vertices after rotation are: J'(8, −9), K'(8, −2), L'(3, −2), M'(3, −9).

Coordinates of a Point

<note>The coordinates of point J are J(x, y) = J(-9, -8)</note>, <note>the coordinates of point K are K(x, y) = K(-2, -8)</note>, <note>the coordinates of point L are L(x, y) = L(-2, -3)</note>, <note>the coordinates of point M are M(x, y) = M(-9, -3)</note>.

2D Plane Rotation Formula

<note>The coordinates of point J (−9, −8) after applying the rotation transformation rule, rotating 90° counterclockwise around the origin, become J' (8, −9)</note>。<note>The coordinates of point K (−2, −8) after applying the rotation transformation rule become K' (8, −2)</note>。<note>The coordinates of point L (−2, −3) after applying the rotation transformation rule become L' (3, −2)</note>。<note>The coordinates of point M (−9, −3) after applying the rotation transformation rule become M' (3, −9)</note>。


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

- yes
- no

Answer: - yes

1. Given a quadrilateral, one pair of opposite angles are equal, each measuring 56°, and another angle measures 124°.

2. Using the polygon interior angle sum theorem: The sum of the interior angles of a polygon is equal to (n-2) × 180°, where n is the number of sides of the polygon. For a quadrilateral, the interior angle sum is (4-2) × 180° = 360°.

3. Let the measure of the unknown angle be x, then we have 56° + 124° + 56° + x = 360°.

5. To find x, subtract 236° from both sides, yielding x = 124°.

6. It is calculated that the other pair of opposite angles in the quadrilateral are equal, each measuring 124°. Since in a quadrilateral, if both pairs of opposite angles are equal (56° and 124°), according to the parallelogram theorem 5, if both pairs of opposite angles in a quadrilateral are equal, the quadrilateral is a parallelogram. Thus, the quadrilateral is a parallelogram.

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

Definition of Parallelogram

<note>The four angles of the quadrilateral are 56°, 124°, 56°, and 124°</note>. Because <note>the two pairs of opposite angles are equal (56° and 124°)</note>, this quadrilateral is a parallelogram.

Polygon Interior Angle Sum Theorem

The sum of the interior angles of a polygon is equal to (n - 2) * 180°, where n represents the number of sides of the polygon.

In the figure of this problem, <note>the sum of the interior angles of a quadrilateral is (4 - 2) * 180° = 2 * 180° = 360°</note>. The formula used is <note>the sum of the interior angles of a quadrilateral formula</note>. In the figure, it is known that <note>the degrees of three angles are 56°, 124°, and 56° respectively</note>, let the fourth angle be x degrees. According to <note>the sum of the interior angles formula</note>, we have <note>56° + 124° + 56° + x = 360°</note>.

Parallelogram Criterion Theorem 5

A quadrilateral is a parallelogram if and only if each pair of opposite angles are congruent.

<note>One pair of opposite angles in the quadrilateral are equal and measure 56°</note>, <note>the other pair of opposite angles are equal and measure 124°</note>, so the quadrilateral is a parallelogram.


Question:
Are △EFG and △UVW congruent?

- yes
- no

Answer: - yes

1. Given △EFG with vertices E(-1,10), F(-1,0), G(-10,10) and △UVW with vertices U(10,-9), V(0,-9), W(10,0), prove whether these two triangles are congruent.

2. Use the coordinate calculation formula to calculate the side lengths of △EFG. When calculating side EF, E and F have the same x-coordinate, so the length of EF is the absolute value of the difference in y-coordinates, i.e., EF=|0-10|=10.

3. When calculating side EG, E and G have the same y-coordinate, so the length of EG is the absolute value of the difference in x-coordinates, i.e., EG=|-10-(-1)|=9.

4. When calculating side FG, F and G have different x and y coordinates, so use the distance formula to calculate the length of FG. The distance formula is: √((x2-x1)^2 + (y2-y1)^2), substituting F(-1,0), G(-10,10), FG=√((-10-(-1))^2 + (10-0)^2)=√(81+100)=√181.

5. The lengths of the three sides of △EFG are EF=10, EG=9, FG=√181.

6. Use the same method to calculate the side lengths of △UVW. When calculating side UV, U and V have the same y-coordinate, so the length of UV is the absolute value of the difference in x-coordinates, i.e., UV=|0-10|=10.

7. When calculating side UW, U and W have the same x-coordinate, so the length of UW is the absolute value of the difference in y-coordinates, i.e., UW=|0-(-9)|=9.

8. When calculating side VW, V and W have different x and y coordinates, so use the distance formula to calculate the length of VW. Substituting V(0,-9), W(10,0), VW=√((10-0)^2 + (0-(-9))^2)=√(100+81)=√181.

9. The lengths of the three sides of △UVW are UV=10, UW=9, VW=√181.

10. According to the congruent triangles theorem (SSS), if the three sides of two triangles are respectively equal, then the two triangles are congruent. Here EF=UV, EG=UW, FG=VW.

11. After the above reasoning, the final answer is: △EFG and △UVW are congruent.

Distance Formula Between Two Points

<note>The coordinates of point F are (-1,0)</note>, <note>The coordinates of point G are (-10,10)</note>, so <note>The distance FG is √((x2 - x1)^2 + (y2 - y1)^2)</note> = √((-10 - (-1))^2 + (10 - 0)^2) = √(81 + 100) = √181. <note>The coordinates of point V are (0,-9)</note>, <note>The coordinates of point W are (10,0)</note>, so <note>The distance VW is √((x2 - x1)^2 + (y2 - y1)^2)</note> = √((10 - 0)^2 + (0 - (-9))^2) = √(100 + 81) = √181.

Triangle Congruence Theorem (SSS)

In the diagram of this problem, triangles <note>△EFG</note> and <note>△UVW</note>, <note>side EF is equal to side UV</note>, <note>side EG is equal to side UW</note>, <note>side FG is equal to side VW</note>. According to the triangle congruence condition <note>SSS (side-side-side)</note>, when the three sides of two triangles are respectively equal, these two triangles are congruent. Therefore, <note>triangle △EFG is congruent to triangle △UVW</note>.


Question:
Is parallelogram DEFG a rhombus?

- yes
- no

Answer: - yes

1. Given that the four angles of parallelogram DEFG are equal, i.e., m∠DEG = m∠DGE = m∠EGF = m∠FEG = 30°, thus segment EG bisects ∠DEF and ∠DGF.

2. Since m∠DEG = m∠DGE = m∠EGF = m∠FEG = 30°, according to the principle of equal sides and equal angles, DE = DG and GF = GE. At this point, DEFG is an isosceles trapezoid and satisfies the condition of equal diagonals.

3. In parallelogram DEFG, the property that segment EG bisects two adjacent angles (i.e., ∠DEF and ∠DGF) indicates that it is a diagonal and bisects these two angles. Therefore, according to the properties of parallelograms, all sides of the quadrilateral are equal, DE = EF = FG = GD.

4. According to the definition of a rhombus (a parallelogram with four equal sides), quadrilateral DEFG meets the conditions of this definition, i.e., DE = EF = FG = GD. Therefore, parallelogram DEFG is a rhombus.

5. Based on the above reasoning, the final answer is yes.

Definition of Rhombus

A quadrilateral is a rhombus if and only if all four sides are of equal length, and its diagonals are perpendicular bisectors of each other.

<note>In quadrilateral DEFG, all sides DE, EF, FG, and GD are equal</note>, thus quadrilateral DEFG is a rhombus. Additionally, <note>the diagonals EG and DF of quadrilateral DEFG are perpendicular bisectors of each other</note>, that is, <note>the diagonals EG and DF intersect at point O</note>, and <note>angle EOF is a right angle (90 degrees)</note>, and <note>DE=EF=FG=GD</note>.

Definition of Isosceles Triangle

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

<note>In triangle DEG, side DE and side DG are equal</note>, therefore triangle DEG is an isosceles triangle; similarly, <note>In triangle FEG, side GF and side GE are equal</note>, therefore triangle FEG is an isosceles triangle.

Properties of Parallelogram Theorem

In a parallelogram, opposite angles are equal, opposite sides are equal, and the diagonals bisect each other.

In the diagram of this problem, in parallelogram DEFG, <note>the opposite angles ∠DEF and ∠DGF are equal, and the opposite angles ∠DGE and ∠EGF are equal</note>; <note>sides DE and EF are equal, and sides FG and GD are equal</note>; <note>the diagonals EG and DF bisect each other, meaning the intersection point divides diagonal EG into two equal segments EG and GE, and divides diagonal DF into two equal segments DF and FD</note>.

Definition of Rhombus

<note>All sides DE, EF, FG, GD are equal</note>, therefore quadrilateral DEFG is a rhombus. In addition, the diagonals EG and DF of quadrilateral DEFG are perpendicular bisectors of each other, that is, <note>diagonals EG and DF intersect at point O, and angle EOF is a right angle (90 degrees)</note>, and <note>DE=EF=FG=GD</note>.


Question:
Are △GHI and △DEF congruent?

- yes
- no

Answer: - no

1. Based on the coordinates provided in the problem, determine the lengths of the sides of △GHI: G(2, -9), H(-8, -9), I(-8, 0).

2. Side GH is a horizontal segment because the y-coordinates of G and H are the same. Calculate the length of GH as |x_H - x_G| = |-8 - 2| = 10.

3. Side HI is a vertical segment because the x-coordinates of H and I are the same. Calculate the length of HI as |y_I - y_H| = |0 - (-9)| = 9.

4. Side GI is a diagonal segment. Use the distance formula between two points to calculate: GI = sqrt((x_I - x_G)^2 + (y_I - y_G)^2) = sqrt((-8 - 2)^2 + (0 - (-9))^2) = sqrt(10^2 + 9^2) = sqrt(100 + 81) = sqrt(181).

5. The lengths of the three sides of △GHI are GH = 10, HI = 9, GI = sqrt(181).

6. Next, determine the lengths of the sides of △DEF: D(1, -3), E(1, 8), F(10, 8).

7. Side DE is a vertical segment because the x-coordinates of D and E are the same. Calculate the length of DE as |y_E - y_D| = |8 - (-3)| = 11.

8. Side EF is a horizontal segment because the y-coordinates of E and F are the same. Calculate the length of EF as |x_F - x_E| = |10 - 1| = 9.

9. Use the distance formula to calculate the length of side DF: DF = sqrt((x_F - x_D)^2 + (y_F - y_D)^2) = sqrt((10 - 1)^2 + (8 - (-3))^2) = sqrt(9^2 + 11^2) = sqrt(81 + 121) = sqrt(202).

10. The lengths of the three sides of △DEF are DE = 11, EF = 9, DF = sqrt(202).

11. Compare the lengths of the sides of △GHI and △DEF: GH = 10 ≠ DE = 11, HI = 9 = EF = 9, GI = sqrt(181) ≠ DF = sqrt(202).

12. According to the congruence theorem (SSS), two triangles are congruent if their corresponding sides are equal. From the comparison above, △GHI and △DEF do not satisfy the SSS theorem.

13. Based on the above analysis, the conclusion is: △GHI and △DEF are not congruent.

Distance Formula Between Two Points

In the figure of this problem, the length of <note>side GI</note> is calculated using the <note>Distance Formula Between Two Points</note>, specifically <note>GI = sqrt((x_I - x_G)^2 + (y_I - y_G)^2) = sqrt((-8 - 2)^2 + (0 - (-9))^2) = sqrt(10^2 + 9^2) = sqrt(100 + 81) = sqrt(181)</note>. Similarly, the length of <note>side DF</note> is also calculated using this formula, specifically <note>DF = sqrt((x_F - x_D)^2 + (y_F - y_D)^2) = sqrt((10 - 1)^2 + (8 - (-3))^2) = sqrt(9^2 + 11^2) = sqrt(81 + 121) = sqrt(202)</note>.

Triangle Congruence Theorem (SSS)

In the figure of this problem, in triangles GHI and DEF, <note>side GH is not equal to side ED</note>, <note>side HI is equal to side EF</note>, <note>side GI is not equal to side DF</note>. According to the Triangle Congruence Theorem (SSS), when the three sides of two triangles are respectively equal, the two triangles are congruent. Therefore, <note>triangle GHI is not congruent to triangle DEF</note>.


Question:
Are △QRS and △DEF congruent?

- yes
- no

Answer: - yes

1. Based on the coordinates given in the problem, calculate the side lengths of △QRS. First, calculate side QS. Points Q(-8, 1) and S(-8, 10) have the same x-coordinate, so the length of side QS is the absolute value of the difference between their y-coordinates, which is |10-1|=9.

2. Next, calculate side RS. Points R(1, 10) and S(-8, 10) have the same y-coordinate, so the length of side RS is the absolute value of the difference between their x-coordinates, which is |-8-1|=9.

3. Finally, calculate side QR. Q(-8, 1) and R(1, 10) have different x-coordinates and y-coordinates, so use the distance formula to calculate QR. The distance formula is: √((x2-x1)^2 + (y2-y1)^2). Substituting x1=-8, y1=1, x2=1, y2=10, we get QR = √((1 - (-8))^2 + (10 - 1)^2) = √(9^2 + 9^2) = √162 = 9√2.

4. Based on the coordinates given in the problem, calculate the side lengths of △DEF. First, calculate side DF. Points D(10, -8) and F(1, -8) have the same y-coordinate, so the length of side DF is the absolute value of the difference between their x-coordinates, which is |1-10|=9.

5. Next, calculate side EF. Points E(1, 1) and F(1, -8) have the same x-coordinate, so the length of side EF is the absolute value of the difference between their y-coordinates, which is |-8-1|=9.

6. Finally, calculate side DE. D(10, -8) and E(1, 1) have different x-coordinates and y-coordinates, so use the distance formula to calculate DE. The distance formula is: √((x2-x1)^2 + (y2-y1)^2). Substituting x1=10, y1=-8, x2=1, y2=1, we get DE = √((1 - 10)^2 + (1 - (-8))^2) = √((-9)^2 + 9^2) = √162 = 9√2.

7. Through calculation, the three side lengths of △QRS are 9, 9, 9√2, and the three side lengths of △DEF are also 9, 9, 9√2.

8. Using the congruent triangles theorem (SSS), this theorem states that if the three corresponding sides of two triangles are equal, then the two triangles are congruent. Since the corresponding sides of triangles QRS and DEF are equal, we can confirm △QRS ≅ △DEF.

9. Based on the above reasoning, the final answer is yes.

Distance Formula Between Two Points

In the diagram of this problem, the calculation of QR uses the <note>Distance Formula Between Two Points</note>, and the calculation process of QR is as follows: <note>QR = √((1 - (-8))^2 + (10 - 1)^2)</note> = √(9^2 + 9^2) = √162 = 9√2. In addition, <note>DE is also calculated using this formula</note>: <note>DE = √((1 - 10)^2 + (1 - (-8))^2)</note> = √((-9)^2 + 9^2) = √162 = 9√2.

Triangle Congruence Theorem (SSS)

<note>Side QS is equal to side DF</note>, <note>side RS is equal to side EF</note>, <note>side QR is equal to side DE</note>. According to the Triangle Congruence Theorem (SSS), when the three sides of two triangles are respectively equal, the two triangles are congruent. Therefore, <note>triangle QRS is congruent to triangle DEF</note>.


Question:
Are △TUV and △ABC congruent?

- yes
- no

Answer: - no

1. First, calculate the side length of △TUV. Points T(1,8) and V(1,-1) have the same x-coordinate, so TV is the absolute value of the difference of their y-coordinates. Therefore, TV=|-1-8|=9.

2. Next, calculate TU. Points T(1,8) and U(10,8) have the same y-coordinate, so TU is the absolute value of the difference of their x-coordinates. Therefore, TU=|10-1|=9.

3. Finally, calculate UV. Points U(10,8) and V(1,-1) have neither the same x-coordinate nor the same y-coordinate, so use the distance formula between two points to calculate UV. The distance formula is: d=√((x2-x1)^2+(y2-y1)^2). Substituting U(10,8) as (x1,y1) and V(1,-1) as (x2,y2), we get: UV=√((1-10)^2+(-1-8)^2)=√((-9)^2+(-9)^2)=√(81+81)=√162=9√2.

4. Thus, the side lengths of △TUV are TV=9, TU=9, UV=9√2.

5. Now, calculate the side length of △ABC. Points A(-2,10) and B(-2,1) have the same x-coordinate, so AB is the absolute value of the difference of their y-coordinates. Therefore, AB=|1-10|=9.

6. Next, calculate AC. Points A(-2,10) and C(-10,10) have the same y-coordinate, so AC is the absolute value of the difference of their x-coordinates. Therefore, AC=|-10-(-2)|=|-10+2|=8.

7. Since none of the side lengths of triangle TUV are 8, while the side length AC of triangle ABC is 8, according to the congruent triangle theorem (SSS), △TUV and △ABC are not congruent.

8. Based on the above reasoning, the final answer is No.

Distance Formula Between Two Points

The distance between <note>point U (10,8)</note> and <note>point V (1,-1)</note> can be calculated using the <note>distance formula</note>. After substituting the coordinates, the distance <note>UV = √((1 - 10)^2 + (-1 - 8)^2) = √((-9)^2 + (-9)^2) = √(81 + 81) = √162 = 9√2</note>.

Triangle Congruence Theorem (SSS)

In the problem diagram, it is calculated that <note>the side lengths of △TUV are TV = 9, TU = 9, UV = 9√2</note>, while <note>the side lengths of △ABC are AB = 9, AC = 8</note>, <note>BC needs to be calculated</note>. Since <note>none of the side lengths of triangle TUV are 8</note>, and <note>the length of side AC of triangle ABC is 8</note>, therefore, by the Side-Side-Side Congruence Theorem, <note>△TUV and △ABC are not congruent</note>.


Question:
Is parallelogram QRST a rectangle?

- yes
- no

Answer: - yes

1. Given PQ≅PR≅PS≅PT, it can be deduced that the distances from point P to each vertex of the parallelogram are equal.

2. QS and RT are the diagonals of the parallelogram QRST. Combining with step 1, it can be deduced that QS=RT, meaning the diagonals of the parallelogram QRST bisect each other and are equal.

3. According to the rectangle determination theorem 2: A quadrilateral with diagonals that bisect each other and are equal is a rectangle. Therefore, the parallelogram QRST is a rectangle.

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

Definition of Parallelogram

In the figure of this problem, the quadrilateral QRST is a <note>parallelogram</note>, <note>side QR is parallel and equal to side ST</note>, <note>side QT is parallel and equal to side RS</note>.

Definition of Rectangle

A quadrilateral is called a rectangle if and only if each of its interior angles is a right angle (90°) and its opposite sides are both parallel (∥) and congruent (≅).

In the figure of this problem, quadrilateral QRST is a rectangle, whose <note>interior angles ∠QRS, ∠RST, ∠STQ, ∠TQR are all right angles (90 degrees)</note>, and <note>side QR is parallel and equal in length to side ST</note>, <note>side QT is parallel and equal in length to side RS</note>.

Properties of Parallelogram Theorem

In the parallelogram <note>QRST</note>, <note>the angles ∠QRS and ∠QTS are equal</note>, <note>the angles ∠RQT and ∠RST are equal</note>; <note>sides QR and ST are equal</note>, <note>sides QT and RS are equal</note>; <note>the diagonals QS and RT bisect each other</note>, that is, the intersection point P <note>divides the diagonal QS into two equal segments QP and PS</note>, <note>divides the diagonal RT into two equal segments RP and PT</note>.

Rectangle Identification Theorem 2

A quadrilateral is a rectangle if and only if its diagonals bisect each other and are equal in length.

In this problem diagram, <note>the diagonals RT and QS of quadrilateral QRST bisect each other and are equal</note>, so <note>quadrilateral QRST is a rectangle</note>.


Question:
Which two triangles are congruent by the SAS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △EFG≅△RTS

1. Observing △EFG and △RTS, we can see the marked sides and angles.

2. It is known that side FG ≅ side TS, this is the first pair of corresponding sides in the two triangles.

3. It is known that side EG ≅ side RS, this is the second pair of corresponding sides in the two triangles.

4. It is known that ∠EGF ≅ ∠RST, this is the corresponding included angle in the two triangles.

5. From the above known conditions, it can be seen that △EFG and △RTS have two pairs of corresponding sides and their included angle equal, which meets the criteria of the congruent triangles theorem (SAS).

6. According to the congruent triangles theorem (SAS), if two triangles have two sides and their included angle equal respectively, then these two triangles are congruent. In this problem, △EFG and △RTS are congruent because they meet the criteria of the congruent triangles theorem (SAS).

7. The complete notation for congruent triangles is: △EFG ≅ △RTS.

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.

<note>Side EG ≅ Side RS</note>, <note>Side FG ≅ Side TS</note>, and <note>Angle ∠EGF ≅ ∠RST</note>, therefore according to the Triangular Congruence Theorem (SAS), <note>these two triangles are congruent</note>, i.e., <note>△EFG ≅ △RTS</note>.


Question:
Is parallelogram UVWX a rectangle?

- yes
- no

Answer: - yes

1. Given that quadrilateral UVWX is a parallelogram, according to the properties of parallelograms, UV is parallel to WX, and UW is parallel to VX.

2. Given UT = VT = WT = XT = 34, according to the properties of triangle side lengths, the four segments UT, VT, WT, XT are equal.

3. Since UT = VT = WT = XT, point T is the intersection point of the diagonals of quadrilateral UVWX, according to the property of equality, T is the midpoint of the diagonals.

4. Connect UW and VX, and note that UW = VX = 68, according to the conditions given in the problem and the hypotenuse lookup, UWT and VXT are isosceles right triangles.

5. According to the properties of isosceles right triangles, ∠UWT = ∠VXT = 90°, because in an isosceles right triangle, the angle between two equal sides is 90°.

6. Since ∠UWT = ∠VXT = 90°, this means that each angle inside quadrilateral UVWX is a right angle.

7. Therefore, according to the definition of a rectangle, UVWX is a rectangle.

8. After the above reasoning, the final answer is yes.

Definition of Parallelogram

In the figure of this problem, the quadrilateral <note>UVWX</note> is a parallelogram, with side <note>UV</note> parallel and equal to side <note>WX</note>, and side <note>UW</note> parallel and equal to side <note>VX</note>.

Definition of Rectangle

<note>In the figure of this problem, quadrilateral UVWX is a rectangle</note>, whose <note>interior angles ∠VUW, ∠UVX, ∠VXW, ∠XWV are all right angles (90 degrees)</note>, and <note>sides UV and WX are parallel and equal in length</note>, <note>sides UW and VX are parallel and equal in length</note>.

Properties of Parallelogram Theorem

Original: <note>In parallelogram UVWX, the opposite angles ∠UVX and ∠UWX are equal</note>, <note>the opposite angles ∠VUW and ∠VWX are equal</note>; <note>sides UV and WX are equal</note>, <note>sides UW and VX are equal</note>; <note>the diagonals UT and VX bisect each other</note>, that is, <note>the intersection point T divides the diagonal UT into two equal segments UT and XT</note>, <note>and divides the diagonal VX into two equal segments VT and WT</note>.

Definition of Rectangle

<note>Quadrilateral UVWX is a rectangle</note>, whose <note>interior angles ∠VUW, ∠UVX, ∠VXW, ∠XWV are all right angles (90 degrees)</note>, and <note>side UV is parallel and equal in length to side WX</note>, <note>side UW is parallel and equal in length to side VX</note>.


Question:
Are △FGH and △STU congruent?

- yes
- no

Answer: - no

1. Given the vertices of △FGH as F(–10,8), G(–1,8), H(–10,–1) and the vertices of △STU as S(10,–8), T(0,–8), U(10,1). We need to determine if these two triangles are congruent.

2. First, calculate the side lengths of △FGH. Since point F(–10,8) and point H(–10,–1) have the same x-coordinate, FH is the absolute value of the difference in y-coordinates, thus FH = |–1–8| = 9.

3. Since point F(–10,8) and point G(–1,8) have the same y-coordinate, FG is the absolute value of the difference in x-coordinates, thus FG = |–1 – (–10)| = 9.

4. Calculate side GH. Since point G(–1,8) and point H(–10,–1) have different x-coordinates and y-coordinates, use the distance formula to calculate GH:

5. The distance formula is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substituting G(–1,8) as (x1,y1) and H(–10,–1) as (x2,y2), we get:

6. GH = \(\sqrt{(–10 - (–1))^2 + (–1 - 8)^2}\) = \(\sqrt{(–9)^2 + (–9)^2}\) = \(\sqrt{81 + 81}\) = \(\sqrt{162}\) = 9\(\sqrt{2}\).

7. Therefore, the side lengths of △FGH are FH = 9, FG = 9, GH = 9\(\sqrt{2}\).

8. Next, calculate the side lengths of △STU. Since point S(10,–8) and point U(10,1) have the same x-coordinate, SU is the absolute value of the difference in y-coordinates, thus SU = |1 – (–8)| = 9.

9. Since point S(10,–8) and point T(0,–8) have the same y-coordinate, ST is the absolute value of the difference in x-coordinates, thus ST = |0 – 10| = 10.

10. Calculate side UT, use the distance formula to calculate UT:

11. Substituting point T(0,–8) as (x1,y1) and point U(10,1) as (x2,y2), we get:

12. UT = \(\sqrt{(10 - 0)^2 + (1 - (–8))^2}\) = \(\sqrt{10^2 + 9^2}\) = \(\sqrt{100 + 81}\) = \(\sqrt{181}\).

13. The side lengths of △STU are SU = 9, ST = 10, UT = \(\sqrt{181}\).

14. According to the congruent triangles theorem (SSS), if the corresponding sides of two triangles are equal, then the triangles are congruent. In this problem, FH ≠ ST, so the corresponding sides of the two triangles are not all equal.

15. Based on the above reasoning, △FGH and △STU are not congruent.

Coordinates of a Point

In a Cartesian coordinate system, the position of a point is represented by an ordered pair (x, y), where x denotes the horizontal (x-coordinate) and y denotes the vertical (y-coordinate).

<note>The coordinates of point F are (–10, 8)</note>, <note>the coordinates of point G are (–1, 8)</note>, <note>the coordinates of point H are (–10, –1)</note>, <note>the coordinates of point S are (10, –8)</note>, <note>the coordinates of point T are (0, –8)</note>, <note>the coordinates of point U are (10, 1)</note>. The coordinates of each vertex clearly indicate its position in the Cartesian coordinate system.

Distance Formula Between Two Points

In a Cartesian coordinate plane, the distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

<note>In the diagram of this problem</note>, when calculating <note>the length of side GH</note>, we use the distance formula. <note>The coordinates of point G are (–1, 8)</note>, <note>the coordinates of point H are (–10, –1)</note>, according to the distance formula, <note>GH = \(\sqrt{(–10 - (–1))^2 + (–1 - 8)^2}\)</note> = \(\sqrt{81 + 81}\) = \(\sqrt{162}\) = 9\(\sqrt{2}\). When calculating <note>the length of side UT</note>, it is the same. <note>The coordinates of point T are (0, –8)</note>, <note>the coordinates of point U are (10, 1)</note>, <note>UT = \(\sqrt{(10 - 0)^2 + (1 - (–8))^2}\)</note> = \(\sqrt{181}\).

Triangle Congruence Theorem (SSS)

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

<note>In the figure of this problem</note>, in triangles FGH and STU, <note>side FH is equal to side SU</note>, <note>side FG is not equal to side ST</note>, <note>side GH is not equal to side UT</note>. According to the <note>Triangle Congruence Theorem (SSS)</note>, when the three sides of two triangles are respectively equal, the two triangles are congruent. Therefore, triangle FGH is not congruent to triangle STU.


Question:
Which two triangles are congruent by the SAS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △WXY≅△GIH

1. Observe the figure. Based on the lengths and angles marked in the figure, we need to confirm whether there exist two pairs of corresponding equal sides and corresponding equal included angles in the triangles.

2. Observe △WXY and △GHI: It is marked in the figure that side WY is equal in length to side GH, i.e., \( \overline{WY} \cong \overline{GH} \).

3. It is marked in the figure that ∠W is equal to ∠G, i.e., \( \angle W \cong \angle G \).

4. It is marked in the figure that side WX is equal in length to side GI, i.e., \( \overline{WX} \cong \overline{GI} \).

5. According to the triangle congruence theorem (SAS), if two sides and the included angle of one triangle are respectively equal to two sides and the included angle of another triangle, then the two triangles are congruent. In this problem, it can be expressed as: the two sides WY, WX and the included angle ∠W of △WXY are respectively equal to the two sides GH, GI and the included angle ∠G of △GHI.

6. Therefore, according to the triangle congruence theorem (SAS), we can conclude that △WXY ≅ △GIH.

7. Through the above reasoning, the final answer is that the two triangles △WXY and △GIH are congruent.

Triangular Congruence Theorem (SAS)

<note>Side WY is equal to side GH</note>, <note>side WX is equal to side GI</note>, and <note>the included angle ∠WXY is equal to the included angle ∠IGH</note>, therefore, according to the Triangular Congruence Theorem (SAS), these two triangles are congruent.


Question:
Are △FGH and △BCD congruent?

- yes
- no

Answer: - no

1. According to the problem, we need to determine whether △FGH and △BCD are congruent.

2. First, verify if the three sides of the two triangles satisfy the side-side-side congruence theorem: corresponding sides are equal.

3. Calculate the side lengths of △FGH. The coordinates of point F are (9,0), and the coordinates of point H are (9,-9). The length of FH is equal to the absolute value of the difference in y-coordinates: FH=|-9-0|=9.

4. The coordinates of point G are (-2,-9), and the coordinates of point H are (9,-9). The length of GH is equal to the absolute value of the difference in x-coordinates: GH=|9-(-2)|=11.

5. For side FG, the coordinates of point F are (9,0), and the coordinates of point G are (-2,-9). Using the distance formula, which is the distance between two points d=√((x2-x1)²+(y2-y1)²). Substitute the values to calculate: FG=√(((-2)-9)²+((-9)-0)²)=√(((-11)²)+((-9)²))=√(121+81)=√202.

6. The side lengths of △FGH are: FH=9, GH=11, FG=√202.

7. Now calculate the side lengths of △BCD.

8. The coordinates of point B are (-10,9), and the coordinates of point D are (-1,9). The length of BD is equal to the absolute value of the difference in x-coordinates: BD=|-1-(-10)|=9.

9. The coordinates of point C are (-1,-3), and the coordinates of point D are (-1,9). The length of CD is equal to the absolute value of the difference in y-coordinates: CD=|9-(-3)|=12.

10. For side BC, the coordinates of point B are (-10,9), and the coordinates of point C are (-1,-3). Using the distance formula: BC=√(((-1)-(-10))²+((-3)-9)²)=√(9²+(-12)²)=√(81+144)=√225=15.

11. The side lengths of △BCD are: BD=9, CD=12, BC=15.

12. Compare the side lengths of the two triangles:

13. The sides of △FGH (FH, GH, FG) have lengths 9, 11, √202,

14. The sides of △BCD (BD, CD, BC) have lengths 9, 12, 15.

15. Clearly, there is no set of three side lengths in △FGH and △BCD that correspond exactly.

16. According to the side-side-side congruence theorem (SSS), since the three sides of △FGH and △BCD do not correspond exactly, they are not congruent.

17. Based on the above reasoning, the final answer is no, △FGH and △BCD are not congruent.

Triangle Congruence Theorem (SSS)

Original: <note>Compare the side lengths of △FGH and △BCD</note>, and find that <note>the sides of △FGH (FH, GH, FG) have lengths of 9, 11, √202</note>, <note>the sides of △BCD (BD, CD, BC) have lengths of 9, 12, 15</note>. None of these sets of side lengths completely correspond to each other, therefore, according to the triangle congruence condition <note>SSS (side-side-side)</note>, when <note>the three sides of two triangles are respectively equal</note>, the two triangles are congruent. <note>△FGH and △BCD are not congruent.</note>


Question:
Are △DEF and △HIJ congruent?

- yes
- no

Answer: - no

1. Given the vertex coordinates of triangle △DEF as D(–4,–10), E(6,–1), F(6,–10), and the vertex coordinates of triangle △HIJ as H(–1,–3), I(–10,6), J(–1,6).

2. By calculating the side lengths of triangle △DEF:

   a. Since the y-coordinates of D(–4,–10) and F(6,–10) are the same, the length of DF is the absolute value of the difference in x-coordinates, i.e., DF=|6+4|=10.

   b. Since the x-coordinates of E(6,–1) and F(6,–10) are the same, the length of EF is the absolute value of the difference in y-coordinates, i.e., EF=|-10+1|=9.

   c. Using the distance formula between two points to calculate the length of DE. The distance formula is: d=√[(x2-x1)^2+(y2-y1)^2]. Treating D(–4,–10) as (x1,y1) and E(6,–1) as (x2,y2), substituting into the formula gives DE=√[(6+4)^2+(-1+10)^2]=√[100+81]=√181.

3. By calculating the side lengths of triangle △HIJ:

   a. Since the x-coordinates of H(–1,–3) and J(–1,6) are the same, the length of HJ is the absolute value of the difference in y-coordinates, i.e., HJ=|6+3|=9.

   b. Since the y-coordinates of I(–10,6) and J(–1,6) are the same, the length of IJ is the absolute value of the difference in x-coordinates, i.e., IJ=|-1+10|=9.

   c. Using the distance formula between two points to calculate the length of HI. Treating H(–1,–3) as (x1,y1) and I(–10,6) as (x2,y2), substituting into the formula gives HI=√[(-10+1)^2+(6+3)^2]=√[81+81]=√162.

4. Obtaining the side lengths of triangle △DEF as DF=10, EF=9, DE=√181; and the side lengths of triangle △HIJ as HJ=9, IJ=9, HI=√162.

5. According to the congruence theorem (SSS), two triangles are congruent if and only if their corresponding side lengths are equal. In this problem, △DEF and △HIJ have different distributions of side lengths, thus not satisfying the congruence theorem (SSS).

6. Based on the above reasoning, the final conclusion is that the triangles are not congruent.

Distance Formula Between Two Points

The distance between <note>D(–4,–10)</note> and <note>E(6,–1)</note> is <note>DE=√[(6+4)^2+(-1+10)^2]=√181</note>; similarly, the distance between <note>H(–1,–3)</note> and <note>I(–10,6)</note> is <note>HI=√[(-10+1)^2+(6+3)^2]=√162</note>.

Triangle Congruence Theorem (SSS)

In the diagram of this problem, compare the lengths of the three sides of triangle <note>△DEF</note> and triangle <note>△HIJ</note>: the side lengths of triangle <note>△DEF</note> are <note>DF=10</note>, <note>EF=9</note>, <note>DE=√181</note>; the side lengths of triangle <note>△HIJ</note> are <note>HJ=9</note>, <note>IJ=9</note>, <note>HI=√162</note>. Because <note>the corresponding side lengths are not all equal (DE is not equal to HI)</note>, according to the triangle congruence condition <note>SSS (Side-Side-Side)</note>, when the three sides of two triangles are respectively equal, the two triangles are congruent. Therefore, triangle <note>△DEF</note> and triangle <note>△HIJ</note> are not congruent.


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. The given triangles are △VTU and △XTW. It is known that ∠VTU and ∠WTX are vertical angles. According to the theorem of vertical angles being equal, we know ∠VTU = ∠XWT.

2. At the same time, it is known that ∠UVT and ∠XWT are corresponding angles because both angles are marked with the same symbol in the geometric diagram. Therefore, we know ∠UVT = ∠XWT.

3. Next, we need to verify if there is at least one pair of corresponding sides that are equal. Observing △VTU and △XTW, we find that no segments are marked as equal. Therefore, we cannot confirm that any pair of sides of △VTU and △XTW are equal.

4. Common triangle congruence theorems are SSS (side-side-side), ASA (angle-side-angle), SAS (side-angle-side), and AAS (angle-angle-side). These theorems all require at least one pair of sides to be equal, but our geometric problem does not provide data supporting any sides being equal.

5. Therefore, based on the geometric information provided in the problem, we cannot conclude that any sides of the two triangles are equal, and thus cannot apply any triangle congruence theorems to prove these two triangles congruent.

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

Definition of Vertical Angles

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

<note>Two intersecting lines VW and UX intersect at point T</note>, forming two angles: <note>∠VTU and ∠WTX</note>. According to the definition of vertical angles, <note>∠VTU and ∠WTX are vertical angles</note>. Since the angles of vertical angles are equal, <note>∠VTU = ∠WTX</note>.


Question:
Which rule explains why these triangles are congruent?

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

Answer: - These triangles cannot be proven congruent.

1. Given segment TU ≅ segment VW, this conclusion is drawn based on the symbol markings in the problem.

2. Given angle TUV ≅ angle TWV, this conclusion is drawn based on the symbol markings in the problem.

3. Observing that triangles TUV and VWT share side TV, we can conclude segment TV ≅ segment TV.

4. Analyzing the known congruent segments and angles in the above two triangles respectively, for triangle TUV: segment TU, angle TUV, segment TV. In triangle TWV, the corresponding ones are segment VW, angle TWV, segment TV.

5. Based on the above analysis, the corresponding relationship formed by the two triangles in order is SSA (Side-Side-Angle).

6. In the congruent triangle determination theorem (SAS), SSA is not a valid method, because it is not possible to prove the congruence of triangles solely by two sides and an angle that is not included between these two sides.

7. After the above analysis, it is not possible to use ASA, SAS, AAS, or SSS to prove the congruence of these two triangles.

8. Therefore, these two triangles cannot be proven to be congruent.

Triangular Congruence Theorem (SAS)

The original: <note>side TU is equal to side VW</note>, <note>side TV is equal to side TV</note>, but <note>angle UTV is equal to angle WVT</note>, therefore according to the Triangular Congruence Theorem (SAS), these two triangles are not congruent.


Question:
Which two triangles are congruent by the SAS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △ABC≅△RTS

1. Observing the given triangle figure, it can be seen that △ABC and △RST are candidates that may satisfy the SAS theorem.

2. In △ABC, mark the visible sides and angles: side AC is equal to side RS, side BC is equal to side ST, and angle ∠ACB is equal to angle ∠RST.

3. According to the congruent triangles determination theorem (SAS), if two triangles have two pairs of corresponding sides equal and the included angle equal, then the two triangles are congruent. Specifically here: side BC = side ST, angle ∠ACB = angle ∠RST, and side AC = side RS.

4. Using the conditions, it is concluded that △ABC and △RST satisfy the conditions of the congruent triangles determination theorem (SAS).

5. According to the definition of congruent triangles, write out the corresponding vertices of the congruent triangles: △ABC ≅△RTS.

6. Through the above reasoning, the final answer is: △ABC ≅△RTS.

Definition of Congruent Triangles

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

<note>In the figure of this problem, triangle ABC and triangle RST are congruent triangles</note>, the corresponding sides and angles of triangle ABC are equal to those of triangle RST, namely: <note>side AC = side RS</note> <note>side BC = side ST</note> <note>side AB = side RT</note>, and the corresponding angles are also equal: <note>angle ∠ACB = angle ∠RST</note> <note>angle ∠CAB = angle ∠SRT</note> <note>angle ∠ABC = angle ∠RTS</note>.

Triangular Congruence Theorem (SAS)

In the figure of this problem, in triangles ABC and SRT, <note>side AC is equal to side SR</note>, <note>side BC is equal to side RT</note>, and <note>the included angle ∠ACB is equal to the included angle ∠RST</note>. Therefore, according to the Triangular Congruence Theorem (SAS), these two triangles are congruent. Thus, <note>△ABC ≅△SRT</note>.


Question:
Which two triangles are congruent by the SSS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △VWX≅△PQR

1. Given △VWX, △BCD and △PQR, the corresponding sides of these triangles are marked as equal in the figure.

3. According to the figure, side VW and side PQ have the same number of marks, indicating side VW ≅ side PQ.

4. Continuing to observe the figure, side VX and side PR have the same number of marks, indicating side VX ≅ side PR.

5. Finally, side WX and side QR have the same number of marks, indicating side WX ≅ side QR.

6. All three pairs of sides are equal, so according to the Side-Side-Side (SSS) congruence theorem, △VWX ≅ △PQR.

8. In the figure, side BD and side PQ have the same number of marks, indicating side BD ≅ side PQ.

9. Observing the marks on side BC and side PR, we find BC ≅ side PR.

10. Finally, side CD and side QR have the same number of marks, indicating side CD ≅ side QR.

11. Therefore, using the SSS congruence theorem, △BCD ≅ △PQR.

12. Summarizing the above comparisons, we find that △VWX and △PQR are a pair of congruent triangles. According to the problem requirements, match the vertices to obtain the congruent form:

13. Through the above reasoning, the final answer is △VWX ≅ △PQR.

Definition of Congruent Triangles

<note>Triangle VWX and triangle PQR are congruent triangles</note>, the corresponding sides and corresponding angles of triangle VWX are equal to those of triangle PQR, namely: <note>side VW = side PQ</note> <note>side VX = side PR</note> <note>side WX = side QR</note>, and the corresponding angles are also equal: <note>angle VWX = angle PQR</note> <note>angle V = angle P</note> <note>angle X = angle R</note>.

Triangle Congruence Theorem (SSS)

<note>In the figure of this problem, in triangles VWX and PQR, side VW is equal to side PQ, side VX is equal to side PR, and side WX is equal to side QR.</note> According to the Triangle Congruence Theorem (SSS), when the three sides of two triangles are respectively equal, the two triangles are congruent. Therefore, <note>triangle VWX is congruent to triangle PQR.</note>


Question:
Which two triangles are congruent by the SSS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △DEF≅△YXZ

1. Given the theorem of triangle congruence (SSS): If three sides of one triangle are respectively equal to three sides of another triangle, then these two triangles are congruent.

2. As seen from the figure: sides EF, DF, and DE correspondingly have equal lengths to sides XZ, YZ, and XY, respectively, EF≅XZ, DF≅YZ, DE≅XY.

3. According to the theorem of triangle congruence (SSS), the three sides of triangles DEF and XYZ are respectively equal, thus, △DEF≅△YXZ.

4. By observation, vertex D corresponds to vertex Y, vertex E corresponds to vertex X, vertex F corresponds to vertex Z.

5. Therefore, through the above reasoning, it is concluded that triangles DEF and XYZ are congruent, △DEF≅△YXZ.

Definition of Congruent Triangles

<note>Triangle DEF and triangle XYZ are congruent triangles</note>, <note>the corresponding sides and corresponding angles of triangle DEF are equal to those of triangle XYZ</note>, that is: <note>side DE≅side XY</note>, <note>side DF≅side YZ</note>, <note>side EF≅side XZ</note>, meanwhile, <note>the corresponding angles are also equal</note>: <note>angle DEF≅angle YXZ</note>, <note>angle EDF≅angle XYZ</note>, <note>angle DFE≅angle YZX</note>.

Triangle Congruence Theorem (SSS)

<note>The three sides of triangles DEF and XYZ are equal respectively:</note> side DE≅side XY, side DF≅side YZ, side EF≅side XZ. Therefore, according to <note>Triangle Congruence Theorem (SSS)</note>, when the three sides of two triangles are equal respectively, the two triangles are congruent. Therefore, <note>△DEF≅△YXZ</note>.


Question:
Are △WXY and △IJK congruent?

- yes
- no

Answer: - no

1. According to the problem statement, the vertices of triangle WXY are W(1,1), X(1,–8), Y(10,–8). The vertices of triangle IJK are I(1,10), J(–8,10), K(–8,2).

2. First, calculate the side lengths of triangle WXY.

3. Calculate the length of WX. Since the x-coordinates of points W(1,1) and X(1,–8) are the same, the length of WX is the absolute value of the difference in y-coordinates, i.e., WX=|–8–1|=9.

4. Calculate the length of XY. Since the y-coordinates of points X(1,–8) and Y(10,–8) are the same, the length of XY is the absolute value of the difference in x-coordinates, i.e., XY=|10–1|=9.

5. Calculate the length of WY. Since W(1,1) and Y(10,–8) do not have the same x-coordinates or the same y-coordinates, use the distance formula between two points to calculate WY. According to the distance formula, for points W(x1=1,y1=1) and Y(x2=10,y2=–8), WY=√((x2–x1)²+(y2–y1)²)=√((10–1)²+(–8–1)²)=√(81+81)=√162=9√2.

6. Therefore, the side lengths of triangle WXY are WX=9, XY=9, WY=9√2.

7. Next, calculate the side lengths of triangle IJK.

8. Calculate the length of IJ. Since the y-coordinates of points I(1,10) and J(–8,10) are the same, the length of IJ is the absolute value of the difference in x-coordinates, i.e., IJ=|–8–1|=9.

9. Calculate the length of JK. Since the x-coordinates of points J(–8,10) and K(–8,2) are the same, the length of JK is the absolute value of the difference in y-coordinates, i.e., JK=|2–10|=8.

10. Calculate the length of KI. For points K(–8,2) and I(1,10), according to the distance formula between two points, KI=√((1–(–8))²+(10–2)²)=√(9²+8²)=√(81+64)=√145.

11. Therefore, the side lengths of triangle IJK are IJ=9, JK=8, KI=√145.

12. According to the congruence theorem (SSS), if the corresponding three sides of two triangles are equal, then the two triangles are congruent.

13. Comparing the side lengths of the two triangles, it can be found that the three sides of triangle WXY are 9, 9, 9√2, while the three sides of triangle IJK are 9, 8, √145.

14. Since the condition that the corresponding three sides of the two triangles are equal is not satisfied, the SSS congruence condition cannot be used.

15. Therefore, triangles WXY and IJK are not congruent triangles.

16. After the above reasoning, the final answer is no.

Distance Formula Between Two Points

<note>In the figure of this problem</note>, use the distance formula to calculate <note>the distance WY between point W(1,1) and Y(10,–8)</note>=√((10–1)²+(–8–1)²)=√162=9√2, <note>and the distance KI between point K(–8,2) and I(1,10)</note>=√((1–(–8))²+(10–2)²)=√(9²+8²)=√(81+64)=√145.

Triangle Congruence Theorem (SSS)

In the figure of this problem, in triangles WXY and IJK, <note>side WX is equal to side IJ</note>, side XY is not equal to side JK, and side WY is not equal to side KI. According to the Triangle Congruence Theorem (SSS), when the three sides of two triangles are respectively equal, these two triangles are congruent. Therefore, triangle WXY is not congruent to triangle IJK.


Question:
Which two triangles are congruent by the SAS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △GHI≅△RQP

1. Given that line segment GH ≅ QR and HI ≅ PQ, and also given that angle ∠ H ≅ ∠ Q. According to the congruent triangles determination theorem (SAS), if two triangles have two sides and the included angle respectively equal, then these two triangles are congruent.

2. For triangle GHI and triangle PQR, GH and QR are two opposite sides of the two triangles, HI and PQ are the other two opposite sides, and ∠ H and ∠ Q are the included angles between the two opposite sides.

3. Therefore, it meets the conditions of the side-angle-side theorem: the two corresponding sides of △GHI and △PQR are respectively equal, and the included angles are also equal. Therefore, according to the congruent triangles determination theorem (SAS), it can be inferred that △GHI ≅ △PQR.

4. According to the definition of congruent triangles, the order of corresponding vertices must be consistent. Line segment GH corresponds to QR, line segment HI corresponds to PQ, so G of triangle GHI corresponds to R of triangle PQR, H corresponds to Q, I corresponds to P; the final congruent relationship obtained is: △GHI ≅ △RQP.

5. After the above reasoning, the final answer is △GHI ≅ △RQP.

Definition of Congruent Triangles

<note>Triangle GHI and triangle PQR are congruent triangles</note>, <note>the corresponding sides and corresponding angles of triangle GHI are equal to those of triangle PQR</note>, namely: <note>side GH = side QR</note> <note>side HI = side PQ</note>, and the corresponding angles are also equal: <note>angle GHI = angle RQP</note> <note>angle HIG = angle QPR</note> <note>angle IGH = angle RPQ</note>.

Triangular Congruence Theorem (SAS)

In the figure of this problem, in triangles <note>GHI</note> and <note>PQR</note>, side <note>GH is equal to side QR</note>, side <note>HI is equal to side PQ</note>, and the included angle <note>∠H is equal to the included angle ∠Q</note>. Therefore, according to the Triangular Congruence Theorem (SAS), these two triangles are congruent: <note>△GHI ≅ △PQR</note>.


Question:
Which two triangles are congruent by the SSS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △QRS≅△IJH

1. According to the markings on the triangle, determine three pairs of corresponding sides: From the markings on the figure, the markings on side QR are consistent with those on side IJ, indicating QR ≅ IJ.

2. Similarly, observing the markings on sides RS and HJ, we find that their markings are consistent, thus RS ≅ HJ.

3. Finally, the markings on sides QS and HI are consistent, leading to QS ≅ HI.

4. According to the congruence theorem for triangles (SSS), this theorem states: If the three sides of one triangle are respectively equal to the three sides of another triangle, then the two triangles are congruent.

5. In this problem, the three sides QR, RS, QS of △QRS are respectively equal to the three sides IJ, HJ, HI of △HIJ. According to the congruence theorem for triangles (SSS), these two triangles are congruent.

6. To complete the congruence correspondence, determine the corresponding angles. Since side QR corresponds to side IJ, vertex Q corresponds to vertex I.

7. At the same time, side RS corresponds to side HJ, so vertex R corresponds to vertex J.

8. Finally, side QS corresponds to side HI, so vertex S corresponds to vertex H.

9. Combining the above information, complete the triangle congruence expression: △QRS ≅ △IJH.

10. Through the above reasoning, the final answer is: Under the SSS theorem, the congruent triangles are △QRS ≅ △IJH.

Definition of Congruent Triangles

<note>Triangle QRS</note> and <note>triangle HIJ</note> are congruent triangles, meaning that the corresponding sides and angles of triangle QRS are equal to those of triangle HIJ, specifically:
Side <note>QR</note> = side <note>IJ</note>
Side <note>RS</note> = side <note>HJ</note>
Side <note>QS</note> = side <note>HI</note>,
and the corresponding angles are also equal:
Angle <note>QRS</note> = angle <note>IJH</note>
Angle <note>RSQ</note> = angle <note>JHI</note>
Angle <note>SQR</note> = angle <note>HIJ</note>.

Triangle Congruence Theorem (SSS)

In the figure of this problem, in triangles QRS and HIJ, <note>side QR is equal to side IJ</note>, <note>side RS is equal to side HJ</note>, <note>side QS is equal to side HI</note>. According to the Triangle Congruence Theorem (SSS), when the three sides of two triangles are respectively equal, the two triangles are congruent. Therefore, <note>triangle QRS is congruent to triangle HIJ</note>.


Question:
Which two triangles are congruent by the SSS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △CDE≅△WYX

1. To solve this geometry problem, we need to find two triangles that satisfy the condition of side-side-side (SSS) congruence.

2. According to the SSS congruence theorem, if three sides of one triangle are respectively equal to three sides of another triangle, then these two triangles are congruent.

3. Look at the given figure, which includes three triangles: △CDE, △WXY, and △QRS.

4. From the marked equal sides in the figure, we can observe the following equal length relationships:

5. Segment DE ≅ Segment XY, which means in △CDE and △WXY, side DE is equal to side XY.

6. Segment CE ≅ Segment WX, which also indicates that in these two triangles, side CE is equal to side WX.

7. Segment CD ≅ Segment WY, which further confirms that in the two triangles, side CD is equal to side WY.

8. Therefore, all three sides of triangle △CDE are respectively equal to the corresponding three sides of triangle △WXY.

9. According to the SSS congruence theorem, we conclude that these two triangles △CDE and △WXY are congruent.

10. Make the corresponding vertices of the congruent triangles: since the vertices opposite the equal sides must correspond, vertex C corresponds to vertex W, vertex D corresponds to vertex Y, and vertex E corresponds to vertex X.

11. Through the above reasoning, the final answer is △CDE≅△WXY.

Definition of Congruent Triangles

<note>Triangle △CDE</note> and <note>triangle △WXY are congruent triangles</note>, the corresponding sides and angles of <note>triangle △CDE are equal to those of triangle △WXY</note>, namely:
<note>side CD = side WY</note>
<note>side DE = side XY</note>
<note>side CE = side WX</note>
At the same time, the corresponding angles are also equal:
<note>angle CDE = angle WYX</note>
<note>angle DCE = angle YWX</note>
<note>angle DEC = angle YXW</note>.

Triangle Congruence Theorem (SSS)

<note>Side CD is equal to side WY</note>, <note>side DE is equal to side XY</note>, <note>side CE is equal to side WX</note>. According to the Triangle Congruence Theorem (SSS), when the three sides of two triangles are respectively equal, the two triangles are congruent. Therefore, <note>triangle △CDE is congruent to triangle △WXY</note>.


Question:
The graph shows pentagons DEFGH and D'E'F'G'H'.

Which sequence of transformations maps DEFGH onto D'E'F'G'H'?

- a reflection across the x-axis followed by a rotation 90° clockwise around the origin
- a rotation 90° clockwise around the origin followed by a translation left 9 units
- a translation left 9 units followed by a reflection across the x-axis

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

1. First, observe the positions of pentagon DEFGH and D'E'F'G'H'. DEFGH is located in the first quadrant, while D'E'F'G'H' is located in the third quadrant, and its shape is the result of the inversion and rotation of DEFGH.

2. Since D'E'F'G'H' is the result of the inversion of DEFGH, first consider reflecting DEFGH about the x-axis. According to the definition of reflection, after reflection, the y-coordinate of each point (x, y) in DEFGH becomes -y, thus obtaining a new pentagon D''E''F''G''H'', located in the fourth quadrant.

3. Next, the new pentagon D''E''F''G''H'' needs to be rotated clockwise by 90° to match the position of pentagon D'E'F'G'H'.

4. The rule for rotating 90° is to transform each point (x, y) into (y, -x). Applying this rule, each vertex in pentagon D''E''F''G''H'' is transferred to the third quadrant, thereby matching the position of pentagon D'E'F'G'H'.

5. After the above two transformations: first reflection about the x-axis, then clockwise rotation by 90°, pentagon DEFGH precisely matches pentagon D'E'F'G'H'.

6. Therefore, the transformation sequence mapping DEFGH to D'E'F'G'H' is: first reflection about the x-axis, then clockwise rotation by 90° around the origin.

7. After the above reasoning, the final answer is reflection about the x-axis, then clockwise rotation by 90° around the origin.

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.

In the figure of this problem, <note>the pentagon DEFGH is reflected about the x-axis</note>. After the reflection, <note>each vertex (x, y) is transformed to (x, -y)</note>, resulting in a new <note>pentagon D''E''F''G''H''</note>, <note>located in the fourth quadrant</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.

<note>Pentagon D''E''F''G''H'' rotates 90° clockwise around the origin</note>, <note>each point (x, y) transforms to (y, -x)</note>, the new pentagon <note>D''E''F''G''H'' moves to the third quadrant</note>, coinciding with the position of <note>pentagon D'E'F'G'H'</note>.


Question:
The graph shows triangles VWX and V'W'X'.

Which sequence of transformations maps VWX onto V'W'X'?

- a rotation 180° around the origin followed by a translation right 1 unit and up 2 units
- a reflection across the x-axis followed by a rotation 90° counterclockwise around the origin
- a translation left 1 unit and down 6 units followed by a reflection across the y-axis

Answer: - a translation left 1 unit and down 6 units followed by a reflection across the y-axis

1. Given the coordinates of point V are (-8, 9), translate it left by 1 unit to get the coordinates of V'' (-9, 9), then translate it down by 6 units to get the coordinates of point V''' (-9, 3).

2. Given the coordinates of point W are (-7, -2), translate it left by 1 unit to get the coordinates of W'' (-8, -2), then translate it down by 6 units to get the coordinates of point W''' (-8, -8).

3. Given the coordinates of point X are (-3, 4), translate it left by 1 unit to get the coordinates of X'' (-4, 4), then translate it down by 6 units to get the coordinates of point X''' (-4, -2).

4. After the above translation operations, we get triangle V'''W'''X''' with coordinates V'''(-9, 3), W'''(-8, -8), X'''(-4, -2).

5. Now reflect triangle V'''W'''X''' across the y-axis: after reflection, point V''' maps to point V', with new coordinates (9, 3).

6. Similarly, point W''' maps to point W', with new coordinates (8, -8).

7. Finally, point X''' maps to point X', with new coordinates (4, -2).

8. After translation and reflection operations, the final triangle V'W'X' has vertex coordinates V'(9, 3), W'(8, -8), X'(4, -2), which match the target triangle V'W'X' in the problem.

9. Based on the above reasoning, the final answer is a translation left 1 unit and down 6 units followed by a reflection across the y-axis.

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.

<note>Point V(-8, 9)</note> is translated to <note>Point V'''(-9, 3)</note>, <note>Point W(-7, -2)</note> is translated to <note>Point W'''(-8, -8)</note>, <note>Point X(-3, 4)</note> is translated to <note>Point X'''(-4, -2)</note>. These points are moved specific distances according to the rules in the definition, completing the translation in the specified direction.

Reflection Transformation

<note>Figure V'''W'''X''' undergoes reflection transformation over the y-axis</note>, <note>point V''' becomes point V'</note>, <note>point W''' becomes point W'</note>, <note>point X''' becomes point X'</note>. <note>The reflection axis y-axis is the reference line for the figure's flip</note>. After the reflection transformation, <note>the coordinates of point V''' (-9, 3) become the coordinates of point V' (9, 3)</note>, <note>the coordinates of point W''' (-8, -8) become the coordinates of point W' (8, -8)</note>, <note>the coordinates of point X''' (-4, -2) become the coordinates of point X' (4, -2)</note>. Each point's coordinates are transformed to their corresponding coordinates symmetrical over the reflection axis y-axis.

Translation Invariance Theorem

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

In this problem diagram, <note>triangle VWX</note> is transformed into <note>V'''W'''X'''</note> through <note>translation</note>, its <note>shape and size remain unchanged</note>, and the <note>distances between the vertices remain unchanged</note>.


Question:
The graph shows triangles VWX and V'W'X'.

Which sequence of transformations maps VWX onto V'W'X'?

- a reflection across the x-axis followed by a rotation 90° counterclockwise around the origin
- a rotation 180° around the origin followed by a translation right 1 unit and up 2 units
- a translation left 1 unit and down 6 units followed by a reflection across the y-axis

Answer: - a translation left 1 unit and down 6 units followed by a reflection across the y-axis

1. First, consider the vertex coordinates of triangle VWX: V(-8, 9), W(-7, -2), X(-3, 4). The goal is to transform them into the vertex coordinates of V'W'X': V'(9, 3), W'(8, -8), X'(4, -2).

2. First, apply the translation formula to shift triangle VWX left 1 unit and down 6 units. The new vertex coordinates after the transformation are: V''(-9, 3), W''(-8, -8), X''(-4, -2).

3. Next, perform a reflection transformation of V''W''X'' across the y-axis. The coordinates after the reflection are: V'''(9, 3), W'''(8, -8), X'''(4, -2).

4. Check if the obtained vertex coordinates of triangle V'''W'''X''' match the vertex coordinates of V'W'X': it is found that the coordinates of V'''W'''X''' perfectly match the coordinates of V'W'X', hence the transformation sequence is: first translate left 1 unit and down 6 units, followed by a reflection across the y-axis.

5. Based on the above reasoning, the final answer is: a translation left 1 unit and down 6 units followed by a reflection across the y-axis.

Definition of Translation

After <note>triangle VWX is translated 1 unit to the left and 6 units down</note>, <note>the vertex coordinates change from V(-8, 9), W(-7, -2), X(-3, 4) to V''(-9, 3), W''(-8, -8), X''(-4, -2)</note>.

Reflection Transformation

In the figure of this problem, triangle <note>V''W''X'' is reflected over the y-axis</note>, and the reflection point of vertex <note>V''(-9, 3) is V'''(9, 3)</note>, the reflection point of vertex <note>W''(-8, -8) is W'''(8, -8)</note>, and the reflection point of vertex <note>X''(-4, -2) is X'''(4, -2)</note>. These reflection points are all on the opposite side of the y-axis, and the shape and size of the figure remain unchanged.

Translation Formula

If a point \( x(x, y) \) is translated horizontally by \( a \) units and vertically by \( b \) units, then the coordinates of the translated point \( x' \) are \( (x + a, y + b) \). Additionally, when translating to the left, \( a \) should be replaced with its opposite sign, and when translating downward, \( b \) should be replaced with its opposite sign.

In this problem, <note>points V(-8,9), W(-7, -2), X(-3, 4)</note> are translated 1 unit to the left horizontally and 6 units down vertically to get <note>points V''(-8-1, 9-6), W''(-7-1, -2-6), X''(-3-1, 4-6)</note>.


Question:
The graph shows triangles FGH and F'G'H'.

Which sequence of transformations maps FGH onto F'G'H'?

- a rotation 90° clockwise around the origin followed by a reflection across the y-axis
- a rotation 90° counterclockwise around the origin followed by a reflection across the y-axis
- a rotation 180° around the origin followed by a translation right 3 units and up 3 units

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

1. First, analyze the coordinates of the triangles FGH and F'G'H'.

2. The vertex coordinates of triangle FGH are: F(-4, 6), G(0, 3), H(-4, 3).

3. The vertex coordinates of triangle F'G'H' are: F'(6, -4), G'(3, 0), H'(3, -4).

4. First, perform a 90° counterclockwise rotation transformation centered at the origin, which can be derived from the formula (x, y) -> (-y, x).

5. Substitute the vertices of triangle FGH into the rotation formula:

6. It can be seen that the rotated triangle F''G''H'': F''(-6, -4), G''(-3, 0), H''(-3, -4) still has a distance from F'G'H', requiring further reflection to map.

7. Continue with the reflection transformation about the y-axis, which will transform the point (x, y) to (-x, y).

8. Substitute the vertices of triangle F''G''H'' into the reflection formula:

9. The final coordinates of triangle F'''G'''H''' are F'''(6, -4), G'''(3, 0), H'''(3, -4), which match the coordinates of F'G'H', therefore this sequence of transformations can map FGH to F'G'H'.

10. Through the above reasoning, the final answer is a 90° counterclockwise rotation followed by a reflection transformation about the y-axis.

Rotation Transformation

In the figure of this problem, <note>triangle FGH is rotated 90° counterclockwise around the origin</note>, its vertex coordinates change from <note>F(-4, 6), G(0, 3), and H(-4, 3) to F''(-6, -4), G''(-3, 0), H''(-3, -4)</note>. The specific formula is <note>(x, y) -> (-y, x)</note>.

Reflection Transformation

<note>Triangle F''G''H''</note> undergoes <note>reflection transformation across the y-axis</note>, and its <note>vertex coordinates</note> change from <note>F''(-6, -4)</note>, <note>G''(-3, 0)</note>, <note>H''(-3, -4)</note> to <note>F'''(6, -4)</note>, <note>G'''(3, 0)</note>, <note>H'''(3, -4)</note>. The specific formula is (x, y) -> <note>(-x, y)</note>.


Question:
The graph shows triangles RST and R'S'T'.

Which sequence of transformations maps RST onto R'S'T'?

- a reflection across the x-axis followed by a rotation 90° clockwise around the origin
- a rotation 180° around the origin followed by a translation left 3 units and up 4 units
- a translation right 3 units and up 10 units followed by a reflection across the y-axis

Answer: - a translation right 3 units and up 10 units followed by a reflection across the y-axis

1. First, observe the figure and record the coordinates of the three vertices of triangle RST and the three vertices of triangle R'S'T'.

2. Calculate the series of geometric transformations from figure RST to R'S'T'.

3. First, translate RST: translate RST 3 units to the right (this keeps the y-coordinate of the origin unchanged), then translate 10 units up to align it with triangle R'S'T' in the same region, obtaining triangle R''T''S'', where vertex R moves from (-5, -3) to R''(-2, 7), vertex S moves from (-9, -5) to S''(-6, 5), and vertex T moves from (-8, -2) to T''(-6, 8).

4. Check if the translated points correspond to the coordinates of R'S'T' through reflection transformation.

5. Reflect the translated triangle about the y-axis and verify if the triangle coincides with R'S'T'.

6. Calculate the coordinates obtained after transformation and compare them with the coordinates of R'S'T' to determine the correctness of the transformation. After reflection, vertex R'' moves from (-2, 7) to R'(2, 7), vertex S'' moves from (-6, 5) to S'(6, 5), and vertex T'' moves from (-6, 8) to T'(6, 8). These points match the positions of the vertices of triangle R'S'T'.

7. Confirm that the first step of the transformation is translation, and the second step is reflection about the y-axis to correspond with R'S'T'.

8. Based on the above reasoning, it is finally determined that the transformation from RST to R'S'T' is first translating 3 units to the right, then 10 units up, and then reflecting about the y-axis.

Definition of Translation

<note>In the figure of this problem</note>, <note>triangle RST</note> is transformed into <note>triangle R''T''S''</note> by <note>translating 3 units to the right and 10 units up</note>. For example, <note>vertex R moves from (-5, -3) to (-2, 7)</note>, <note>vertex S moves from (-9, -5) to (-6, 5)</note>, and <note>vertex T moves from (-8, -2) to (-6, 8)</note>.

Reflection Transformation

<note>The translated triangle R''S''T'' is reflected about the y-axis</note>, resulting in <note>triangle R'S'T'</note>. After reflection, vertex <note>R'' moves from (-2, 7) to (2, 7)</note>, vertex <note>S'' moves from (-6, 5) to (6, 5)</note>, vertex <note>T'' moves from (-6, 8) to (6, 8)</note>. These points coincide with <note>the positions of the vertices of triangle R'S'T'</note>.


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

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

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

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

1. First, observe the positions of triangles UVW and U'V'W', and find that UVW is in the third quadrant while U'V'W' is in the second quadrant.

2. To map UVW to U'V'W', first consider the reflection across the y-axis. The reflection transformation states that if a point (x, y) is reflected across the y-axis, it becomes (-x, y).

3. Apply the reflection: reflecting triangle UVW across the y-axis results in triangle U''V''W''. The vertex coordinates change from U(-2, -6) to U''(2, -6), V(-5, -8) to V''(5, -8), and W(-5, -6) to W''(5, -6).

4. Next, perform the translation. According to the problem, we need to translate left 9 units and up 10 units. The translation theorem states that if a shape is translated by (p, q), its point (x, y) becomes (x+p, y+q).

5. Therefore, we translate each vertex of triangle U''V''W''. U''(2, -6) translates to U'(2-9, -6+10)=U'(-7, 4); V''(5, -8) translates to V'(5-9, -8+10)=V'(-4, 2); W''(5, -6) translates to W'(5-9, -6+10)=W'(-4, 4).

6. Through the above reflection and translation, triangle UVW is accurately mapped to U'V'W', and U''V''W'' translates to U'V'W'.

7. Based on the above reasoning, the final answer is: a reflection across the y-axis followed by a translation left 9 units and up 10 units.

Definition of Translation

<note>Triangle U''V''W'' is translated 9 units to the left and 10 units up to obtain triangle U'V'W'</note>. According to the definition of translation, <note>point U''(2, -6) is translated to U'(-7, 4)</note>, <note>point V''(5, -8) is translated to V'(-4, 2)</note>, <note>point W''(5, -6) is translated to W'(-4, 4)</note>.

Reflection Transformation

In the figure of this problem, <note>triangle UVW</note> undergoes <note>reflection transformation over the y-axis</note>, resulting in <note>point U becoming point U''</note>, <note>point V becoming point V''</note>, and <note>point W becoming point W''</note>. <note>The y-axis is the reference line for the reflection of triangle UVW</note>. After the reflection transformation, <note>the coordinates of point U (-2, -6) become the coordinates of point U'' (2, -6)</note>, <note>the coordinates of point V (-5, y2) become the coordinates of point V'' (5, -8)</note>, and <note>the coordinates of point W (-5, -6) become the coordinates of point W'' (5, -6)</note>. Each point's coordinates are transformed into the coordinates of its corresponding point reflected over the y-axis.

Translation Invariance Theorem

In the figure of this problem, <note>triangle U''V''W''</note> is translated to obtain <note>triangle U'V'W'</note>, <note>the shape and size of triangle U''V''W'' and U'V'W' remain unchanged</note>, which conforms to the Translation Invariance Theorem.


Question:
Which rule explains why these triangles are congruent?

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

Answer: - These triangles cannot be proven congruent.

1. Given \\overline{UV} ≅ \\overline{UY}, according to the markings in the problem.

2. Observing the figure, \\angle VUW and \\angle XUY are vertical angles, according to the definition of vertical angles, we have: \\angle VUW ≅ \\angle XUY.

3. Based on the above conditions, there is one pair of corresponding segments equal and one pair of corresponding angles equal.

4. Since there is no criterion based on 'AS' in the determination of triangles, at least one method is needed to obtain three pairs of corresponding parts equal.

5. Through the above reasoning, it is not possible to determine that triangles \\triangle VUW and \\triangle XUY are congruent based on the given conditions, the information provided in the problem is insufficient to draw a conclusion.

Definition of Vertical Angles

<note>Two intersecting lines VU and XY intersect at point U</note>, forming four angles: <note>angle VUW, angle XUY, angle WUY, and angle VUX</note>. According to the definition of vertical angles, <note>angle VUW and angle XUY are vertical angles, angle WUY and angle VUX are vertical angles</note>. Since the angles of vertical angles are equal, <note>angle VUW = angle XUY, angle WUY = angle VUX</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>Triangles WUV and XUY</note> are congruent if they satisfy any of the following conditions:
(1. Side-Side-Side (SSS): The sides of <note>triangle WUV</note>, <note>WU, UV, WV</note>, are respectively equal to the sides of <note>triangle XUY</note>, <note>UX, UY, YX</note>, i.e., <note>WU=UX, UV=UY, WV=YX</note>.
(2. Side-Angle-Side (SAS): The sides of <note>triangle WUV</note>, <note>WU, UV and their included angle WUV</note>, are respectively equal to the sides of <note>triangle XUY</note>, <note>UX, UY and their included angle XUY</note>, i.e., <note>WU=UX, UV=UY, angle WUV=angle XUY</note>.
(3. Angle-Side-Angle (ASA): The angles of <note>triangle WUV</note>, <note>WUV, WVU and their included side UV</note>, are respectively equal to the angles of <note>triangle XUY</note>, <note>XUY, XYU and their included side UY</note>, i.e., <note>angle WUV=angle XUY, angle WVU=angle XYU, UV=


Question:
Which two triangles are congruent by the SAS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △RST≅△DEC

1. In triangle △RST, sides ST and RT are known.

2. In triangle △CDE, sides CE and CD are known.

3. According to the figure in the problem, side ST ≅ side CE, side RT ≅ side CD.

4. According to the figure in the problem, ∠RTS and ∠DCE are included angles. ∠RTS ≅ ∠DCE.

5. △RST and △DEC satisfy the congruent triangles theorem (SAS), the correspondence of sides, angles, and sides (two sides and the included angle are equal), so △RST ≅ △DEC.

6. The vertices of △RST correspond to the vertices of △DEC as follows: ∠T corresponds to ∠C, ∠R corresponds to ∠D, ∠S corresponds to ∠E.

7. Therefore, based on the above correspondence, △RST ≅ △DEC.

Definition of Triangle

A geometric figure is called a triangle if it is formed by three non-collinear points and the line segments connecting them.

In the figure of this problem, <note>triangle △RST</note> is a geometric figure composed of three non-collinear points <note>R, S, T</note> and their connecting line segments <note>RS, ST, TR</note>. <note>Points R, S, T</note> are the three vertices of the triangle, and <note>line segments RS, ST, TR</note> are the three sides of the triangle; <note>triangle △DEC</note> is a geometric figure composed of three non-collinear points <note>D, E, C</note> and their connecting line segments <note>DE, EC, CD</note>. <note>Points D, E, C</note> are the three vertices of the triangle, and <note>line segments DE, EC, CD</note> are the three sides of the triangle.

Definition of Congruent Triangles

<note>Triangle RST and triangle DEC are congruent triangles</note>, the corresponding sides and angles of triangle RST are equal to those of triangle DEC, namely: <note>side ST = side CE</note>, <note>side RT = side CD</note>, and the corresponding angles are also equal: <note>angle RST = angle DEC</note>.

Triangular Congruence Theorem (SAS)

In the figure of this problem, triangles <note>△RST</note> and <note>△DEC</note>, <note>side ST is equal to side CE</note>, <note>side RT is equal to side CD</note>, and <note>angle ∠RTS is equal to angle ∠DCE</note>. Therefore, according to the Triangular Congruence Theorem (SAS), these two triangles are congruent.


Question:
Which two triangles are congruent by the SAS Theorem? Complete the congruence statement.

△ $\Box$ ≅△ $\Box$

Answer: △PQR≅△VXW

1. In the provided figure, first observe the corresponding elements of △PQR and △VWX.

2. Given that segment QR ≅ segment WX and segment PR ≅ segment VW, and ∠PRQ ≅ ∠VWX. According to the problem statement, these are two sides of the triangles and the included angle.

3. According to the congruent triangles criterion (SAS), two triangles are congruent if and only if two sides and the included angle are respectively congruent. The congruence theorem can be applied to these triangles.

4. In the figure, observe that ∠PRQ and ∠VWX are the included angles between the two known congruent sides.

5. Therefore, according to the congruent triangles criterion (SAS), triangles △PQR and △VWX are congruent.

6. To write the congruence statement, match the corresponding vertices. Since ∠R ≅ ∠W, vertex R corresponds to vertex W. Since the opposite side of P corresponds to the opposite side of V, vertex P corresponds to vertex V. Similarly, vertex Q corresponds to vertex X.

7. Through the above reasoning, the final answer is △PQR ≅ △VXW.

Definition of Congruent Triangles

<note>△PQR≅△VWX</note> means that triangles PQR and VWX each have three corresponding sides and three corresponding angles that are equal respectively. The specific congruence relationships are: <note>PQ≅VX</note>, <note>PR≅VW</note>, <note>QR≅WX</note>; <note>∠P≅∠V</note>, <note>∠Q≅∠X</note>, <note>∠R≅∠W</note>.

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 figure of this problem, <note>angle PRQ is a geometric figure formed by two rays PR and QR</note>, these two rays <note>have a common endpoint R</note>. This <note>common endpoint R is called the vertex of angle PRQ</note>, and <note>rays PR and QR are called the sides of angle PRQ</note>. Similarly, <note>angle VWX is a geometric figure formed by two rays VW and WX</note>, these two rays <note>have a common endpoint W</note>. This <note>common endpoint W is called the vertex of angle VWX</note>, and <note>rays VW and WX are called the sides of angle VWX</note>.

Definition of Line Segment

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

Original: <note>The line segment PQ is a part of a straight line, including endpoint P and endpoint Q and all points between them</note>. The line segment PQ has two endpoints, which are <note>P and Q</note>, and every point on the line segment PQ lies between endpoint P and endpoint Q. <note>The line segment VX is a part of a straight line, including endpoint V and endpoint X and all points between them</note>. The line segment VX has two endpoints, which are <note>V and X</note>, and every point on the line segment VX lies between endpoint V and endpoint X. <note>The line segment PR is a part of a straight line, including endpoint P and endpoint R and all points between them</note>. The line segment PR has two endpoints, which are <note>P and R</note>, and every point on the line segment PR lies between endpoint P and endpoint R. <note>The line segment VW is a part of a straight line, including endpoint V and endpoint W and all points between them</note>. The line segment VW has two endpoints, which are <note>V and W</note>, and every point on the line segment VW lies between endpoint V and endpoint W.

Triangular Congruence Theorem (SAS)

In the figure of this problem, in <note>triangle PQR and triangle VWX</note>, <note>side RQ is equal to side WX</note>, <note>side PR is equal to side VW</note>, and <note>included angle ∠PRQ is equal to included angle ∠VWX</note>. Therefore, according to the <note>Triangular Congruence Theorem (SAS)</note>, these two triangles are congruent.