Question:
Find the value of v in parallelogram HIJK.

v= $\Box$

Answer: v=5

1. Given parallelogram HIJK, according to the properties of a parallelogram theorem, the opposite sides of a parallelogram are equal. Therefore, side HI is equal to side JK, i.e., HI = JK.

2. According to the problem, the length of HI is 10v, and the length of JK is v + 45.

3. By the properties of a parallelogram theorem, we have HI = JK. Substituting the known values, we get the equation: 10v = v + 45.

4. To solve for v, first subtract v from both sides of the equation, obtaining: 10v - v = 45, i.e., 9v = 45.

5. Then divide both sides of the equation by 9, obtaining v = 45 / 9.

6. Calculate the result of 45 ÷ 9, obtaining v = 5.

7. Through the above reasoning, the final answer is v = 5.

Definition of Parallelogram

In the figure of this problem, <note>quadrilateral HIJK is a parallelogram</note>, <note>side HI is parallel and equal to side JK</note>, <note>side HK is parallel and equal to side IJ</note>. Given in the figure <note>the length of HI is 10v</note>, <note>the length of JK is v + 45</note>.

Properties of Parallelogram Theorem

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

<note>Using the property of opposite sides being equal in a parallelogram</note>, we can deduce that <note>side HI is equal to side JK, i.e., HI = JK</note>. Based on the given conditions in the problem <note>HI = 10v, JK = v + 45</note>, we can set up the equation <note>10v = v + 45</note>, and through the solving steps, we find <note>v = 5</note>. This theorem is used in this problem to determine that <note>sides HI and JK are equal</note>, and further to find the value of v.


Question:
Is parallelogram RSTU a rhombus?

- yes
- no

Answer: - no

1. Given that quadrilateral RSTU is a parallelogram. To determine if a parallelogram is a rhombus, we can use the Rhombus Criterion Theorem 1: A parallelogram with perpendicular diagonals is a rhombus. Determine if its diagonals are perpendicular to each other.

2. Examine the relevant angle properties at the intersection point V of the diagonals of the quadrilateral. In the figure, we know that ∠TVU = 96°.

3. If diagonals \\overline{RT} and \\overline{SU} are perpendicular to each other, then by definition ∠TVU should be 90°.

4. However, here ∠TVU = 96°. This indicates that diagonals \\overline{RT} and \\overline{SU} are not perpendicular to each other.

5. If \\overline{RT} and \\overline{SU} are not perpendicular, then RSTU is not a rhombus.

6. Through the above reasoning, the final conclusion is that RSTU is not a rhombus.

Definition of Parallelogram

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

In the figure of this problem, quadrilateral RSTU is a parallelogram, i.e., <note>\overline{RS} \parallel \overline{TU}</note>, <note>\overline{RU} \parallel \overline{ST}</note>, and <note>\overline{RS} = \overline{TU}</note>, <note>\overline{RU} = \overline{ST}</note>.

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>\overline{RT} and \overline{SU}</note> are <note>perpendicular to each other</note> to draw a conclusion.

Rhombus Determination Theorem 1

A parallelogram is a rhombus if and only if its diagonals are perpendicular.

<note>In the figure of this problem, the diagonals of the parallelogram RSTU</note> TR <note>and</note> US <note>form an angle</note> \angle TVU = 96° <note>,</note> TR <note>and</note> US <note>are not perpendicular, so the parallelogram RSTU is not a rhombus.</note>


Question:
Is parallelogram VWXY a rhombus?

- yes
- no

Answer: - yes

1. Given that quadrilateral VWXY is a parallelogram, i.e., VW is parallel and equal to XY and VY is parallel and equal to WX, and ∠ WUX = 90°.

2. From the properties of parallelograms, we know that the diagonals bisect each other, i.e., WU = UY, VU = UX.

3. From ∠ WUX = 90°, we know that VX is perpendicular to WY, i.e., the two diagonals are perpendicular to each other.

4. In a parallelogram, if the two diagonals are perpendicular to each other, then the parallelogram is a rhombus.

5. Therefore, from the above steps, we can determine that parallelogram VWXY is a rhombus.

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

Definition of Parallelogram

<note>Quadrilateral VWXY is a parallelogram</note>, <note>side VW is parallel and equal to side XY</note>, <note>side VY is parallel and equal to side WX</note>.

Definition of Rhombus

In the figure of this problem, in quadrilateral VWXY, all sides <note>\(\overline{VW}\)</note>, <note>\(\overline{WX}\)</note>, <note>\(\overline{XY}\)</note>, and <note>\(\overline{YV}\)</note> are equal, therefore quadrilateral VWXY is a rhombus. Additionally, the diagonals <note>\(\overline{VX}\)</note> and <note>\(\overline{WY}\)</note> of quadrilateral VWXY are perpendicular bisectors of each other, meaning the diagonals <note>\(\overline{VX}\)</note> and <note>\(\overline{WY}\)</note> intersect at point U, and <note>angle \(\angle WUX\) is a right angle (90 degrees)</note>, and <note>\(\overline{WU}=\overline{UY}\)</note> and <note>\(\overline{VU}=\overline{UX}\)</note>.

Properties of Parallelogram Theorem

In the parallelogram <note>VWXY</note>, the opposite angles <note>∠VWY and ∠WYX are equal</note>, and the opposite angles <note>∠WVX and ∠VYX are equal</note>; the sides <note>VW and XY are equal</note>, and the sides <note>VY and WX are equal</note>; the diagonals <note>VX and WY bisect each other</note>, that is, the intersection point <note>U divides the diagonal WY into two equal segments WU and UY</note>, and divides the diagonal <note>VX into two equal segments VU and UX</note>.

Definition of Right Triangle

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

<note>Angle WUX is a right angle (90 degrees)</note>, therefore triangle WUX is a right triangle. Side <note>WU</note> and side <note>UX</note> are the legs, and side <note>WX</note> is the hypotenuse.

Rhombus Determination Theorem 1

<note>Diagonal VX of parallelogram VWXY is perpendicular to WY</note>, so <note>parallelogram VWXY is a rhombus</note>.


Question:
Is parallelogram VWXY a rectangle?

- yes
- no

Answer: - no

1. Given that point U is the intersection point of diagonals WY and VX, and UW=UY=55, UV=UX=59, it can be concluded that triangles UWY and VUX are isosceles triangles.

2. According to the exterior angle theorem of triangles, the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles, thus ∠YUW = ∠UYW and ∠XUV = ∠UVX.

3. Additionally, since quadrilateral VWXY is a parallelogram, according to the properties of parallelograms, diagonals WY and VX bisect each other, thus WU = UY and VU = UX.

4. Because in an isosceles triangle the base angles are equal, thus ∠UWY = ∠UYW and ∠UVX = ∠XUV.

5. One of the conditions to check if quadrilateral VWXY is a rectangle is that its diagonals are equal: that is, WY should be equal to VX.

6. Calculating from the figure, WY = UW + UY = 55 + 55 = 110, VX = UV + UX = 59 + 59 = 118.

7. Since WY ≠ VX, according to the theorem of equal diagonals in a rectangle, quadrilateral VWXY is not a rectangle.

8. Through the above reasoning, the final answer is no.

Definition of Isosceles Triangle

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

<note>In the diagram of this problem, in triangle UWY, side UW and side UY are equal</note>, therefore triangle UWY is an isosceles triangle; similarly, <note>in triangle VUX, side UV and side UX are equal</note>, therefore triangle VUX is an isosceles triangle.

Definition of Parallelogram

<note>In the figure of this problem, quadrilateral VWXY is a parallelogram</note>, <note>side VW is parallel to and equal to side XY</note>, <note>side WX is parallel to and equal to side VY</note>.


Question:
Is parallelogram QRST a rhombus?

- yes
- no

Answer: - yes

1. Given that quadrilateral QRST is a parallelogram, according to the properties theorem of parallelograms, the diagonals bisect each other, determining point P as the intersection point of diagonals QS and RT.

2. Given that ∠RPS is a right angle, i.e., ∠RPS = 90°.

3. According to the definition of perpendicular lines, two lines are perpendicular if and only if the angle between them is a right angle, thus ∠RPS = 90° indicates that line QS is perpendicular to line RT.

4. In a parallelogram, if the two diagonals are perpendicular to each other, then the parallelogram is a rhombus.

5. Through the above reasoning, it is concluded that quadrilateral QRST is a rhombus. Therefore, QRST is a rhombus.

<note>Quadrilateral QRST is a parallelogram</note>, <note>side QR is parallel and equal to side TS</note>, <note>side QT is parallel and equal to side RS</note>.

Definition of Rhombus

<note>In the diagram of this problem, all sides QR, RS, ST, and TQ of quadrilateral QRST are equal</note>, thus quadrilateral QRST is a rhombus. Additionally, <note>the diagonals QS and RT of quadrilateral QRST bisect each other perpendicularly</note>, that is, <note>the diagonals QS and RT intersect at point P, and angle ∠RPS is a right angle (90 degrees)</note>, and <note>QP = PS and RP = PT</note>.

Properties of Parallelogram Theorem

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

Definition of Perpendicular Lines

Two lines are said to be perpendicular if and only if the angle formed by their intersection is 90 degrees.

In the diagram of this problem, <note>the angle ∠RPS formed by the intersection of line RT and line SQ is 90 degrees</note>, so according to the definition of perpendicular lines, <note>line SQ and line RT are perpendicular to each other</note>.

Rhombus Determination Theorem 1

The diagonals <note>QS and RT of parallelogram QTSR</note> are perpendicular, so <note>parallelogram QTSR is a rhombus</note>.


Question:
Find the lateral area of the prism. The base is an equilateral triangle.

 $\Box$  square yards

Answer: 120 square yards

1. The base of the regular triangular prism given in the problem is an equilateral triangle, with each side length being 5 yards. Therefore, we can calculate the perimeter of the base. According to the formula for the perimeter of a triangle, its perimeter is the side length multiplied by 3.

2. Substitute the side length of 5 yards into the calculation of the base perimeter, obtaining: Perimeter = 3 × 5 = 15 yards.

3. The formula for the lateral area of the prism is: Lateral area = Base perimeter × Prism height. In this problem, the height of the prism is known to be 8 yards.

4. Substitute the base perimeter of 15 yards and the height of 8 yards into the lateral area formula, calculating: Lateral area = 15 × 8 = 120 square yards.

5. Through the above reasoning, the final answer is 120 square yards.

Definition of Prism

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

In this problem, the geometric figure involved is a <note>regular triangular prism</note>, with its <note>base being an equilateral triangle with sides of 5 yards</note>, and it has three lateral faces (parallelogram faces), each lateral face having a <note>height of 8 yards</note>.

Lateral Surface Area Formula of a Prism

The lateral surface area of a prism is equal to the perimeter of the base multiplied by the height.

In this problem, <note>the perimeter of the base of the regular triangular prism is 15 yards</note>, <note>the height is 8 yards</note>, so <note>the lateral surface area is the perimeter of the lateral face multiplied by the height</note>, which is <note>15 yards multiplied by 8 yards to get 120 square yards</note>.

Formula for the Perimeter of a Triangle

The perimeter of a triangle is the sum of the lengths of its three sides. Therefore, if the three sides of the triangle are denoted as \( a \), \( b \), and \( c \), then the perimeter \( P \) is given by \( P = a + b + c \).

In the diagram of this problem, <note>the three sides of the triangle are 5 yards, 5 yards, and 5 yards</note>, according to <note>the formula for the perimeter of a triangle</note>, which is <note>perimeter L = 5 yards + 5 yards + 5 yards</note>.


Question:
Is parallelogram VWXY a rhombus?

- yes
- no

Answer: - yes

1. Given that quadrilateral VWXY is a parallelogram. To determine if it is a rhombus, we need to check if all its sides are equal.

2. According to the properties of a parallelogram: the diagonals bisect each other. Additionally, if one of the diagonals bisects two pairs of adjacent angles, then the parallelogram is a rhombus.

3. Observing the given conditions, ∠VXW = 63°, ∠VXY = 63°, ∠WVX = 63°, ∠XVY = 63°.

4. We notice that segment VX bisects ∠WVY and ∠WXY, because ∠VXW = ∠XVY and ∠WVX = ∠VXY.

5. Since segment VX bisects two pairs of adjacent angles, according to the definition of a rhombus, a parallelogram with one diagonal bisecting two pairs of angles is a rhombus.

6. Because ∠VXW = ∠VXY = ∠WVX = ∠XVY = 63°, and the diagonal bisects adjacent angles, this further confirms that VWXY is a rhombus.

7. Through the above reasoning, the final answer is YES, the parallelogram VWXY is a rhombus.

Definition of Parallelogram

<note>Quadrilateral VWXY</note> is a parallelogram, <note>side VW is parallel and equal to side XY</note>, <note>side WX is parallel and equal to side VY</note>.

Definition of Rhombus

<note>A rhombus is a quadrilateral with all sides equal</note>, and <note>its diagonals are perpendicular bisectors of each other</note>. In the figure of this problem, in quadrilateral <note>VWXY, all sides VW, WX, XY, YV are equal</note>, so quadrilateral <note>VWXY is a rhombus</note>. Additionally, the diagonals of quadrilateral <note>VWXY, VX and WY, are perpendicular bisectors of each other</note>, meaning <note>the diagonals VX and WY intersect at point O</note>, and <note>angle VOY is a right angle (90 degrees)</note>, and <note>VO=OY and WO=OX</note>.

Properties of Parallelogram Theorem

In the parallelogram <note>VWXY</note>, <note>the opposite angles ∠WVX and ∠XYV are equal</note>, <note>the opposite angles ∠VWX and ∠XWY are equal</note>; <note>sides VW and XY are equal</note>, <note>sides VX and WY are equal</note>; <note>the diagonals VX and WY bisect each other</note>, that is, <note>the intersection point divides the diagonal VX into two equal segments</note>, <note>divides the diagonal WY into two equal segments</note>.

Definition of Rhombus

A rhombus is a quadrilateral with all four sides equal, and its diagonals are perpendicular bisectors of each other. In the figure of this problem, the quadrilateral <note>VWXY</note> has all sides <note>VW, WX, XY, YV</note> equal, so the quadrilateral <note>VWXY</note> is a rhombus. Additionally, the diagonals <note>VX</note> and <note>WY</note> of the quadrilateral <note>VWXY</note> are perpendicular bisectors of each other, meaning the diagonals <note>VX</note> and <note>WY</note> intersect at point <note>O</note>, and <note>angle VOY is a right angle (90 degrees)</note>, with <note>VO=OY</note> and <note>WO=OX</note>.


Question:
Is parallelogram WXYZ a square?

- yes
- no

Answer: - no

1. Given that the diagonals XZ and WY of parallelogram WXYZ intersect at point V, and ∠WVX = 94°.

2. According to the properties of parallelograms, the diagonals of a parallelogram bisect each other, thus WV = YV and ZV = VX.

3. One of the conditions for a parallelogram to be a square is that its diagonals are perpendicular to each other.

4. ∠WVX = 94°, since ∠WVX is not 90°, it indicates that the diagonals WY and XZ are not perpendicular.

5. Since the diagonals are not perpendicular to each other, the condition for the parallelogram to be a square is not satisfied.

6. Therefore, parallelogram WXYZ is not a square.

Definition of Parallelogram

<note>Quadrilateral WXYZ is a parallelogram</note>, <note>side WX is parallel and equal to side YZ</note>, <note>side XY is parallel and equal to side WZ</note>.

Properties of Diagonals in a Square

The diagonals of a square are the line segments that connect opposite vertices. The diagonals of a square are equal in length, and they bisect each other perpendicularly.

<note>In a quadrilateral, diagonals WY and ZX are the line segments connecting opposite corners</note>. According to the properties of diagonals in a square, <note>YW and ZX are equal</note>, and <note>YW and ZX bisect each other perpendicularly</note>, forming <note>four 90-degree angles at their intersection</note>. However, <note>angle XVW=94°</note> does not satisfy the conditions, so <note>quadrilateral YXWZ is not a square</note>.

Properties of Parallelogram Theorem

In the parallelogram <note>WXYZ</note>, <note>the angles ∠WXY and ∠WZY are equal</note>, <note>the angles ∠ZYX and ∠ZWX are equal</note>; <note>the sides WX and YZ are equal</note>, <note>the sides XY and WZ are equal</note>; <note>the diagonals XZ and WY bisect each other</note>, that is, <note>the intersection point V divides the diagonal XZ into two equal segments XV and VZ</note>, <note>and divides the diagonal WY into two equal segments WV and VY</note>.


Question:
Is parallelogram STUV a rhombus?

- yes
- no

Answer: - yes

1. In the parallelogram STUV, it is known that the diagonals SU and TV intersect at point W, and angle SWT = 90°.

2. In a parallelogram, the diagonals bisect each other. Therefore, SW = UT and TW = WV.

3. According to the property of diagonals intersecting at right angles, we can use the rhombus criterion theorem 1: If the diagonals of a parallelogram are perpendicular to each other, then the parallelogram is a rhombus.

4. In this problem, since the given angle SWT = 90°, it means SU ⟂ TV.

5. Therefore, according to the above rhombus criterion theorem 1, we can conclude that the parallelogram STUV is a rhombus.

6. After the above reasoning, the final answer is YES.

Definition of Parallelogram

<note>Quadrilateral STUV is a parallelogram</note>, <note>side ST is parallel and equal to side UV</note>, <note>side TU is parallel and equal to side SV</note>.

Definition of Diagonal

A diagonal is a line segment connecting one vertex of a polygon to another vertex that is not adjacent to it.

<note>In the figure of this problem, in the quadrilateral STUV, vertices S, T, U, V</note>, <note>the diagonal is the line segment SU connecting vertex S and non-adjacent vertex U</note>, <note>and the line segment TV connecting vertex T and non-adjacent vertex V</note>. Therefore, <note>line segments SU and TV are the diagonals of the quadrilateral STUV</note>.

Properties of Parallelogram Theorem

In the <note>parallelogram STUV</note>, <note>angles STU and SVU are equal</note>, <note>angles TSV and VUT are equal</note>; <note>sides ST and UV are equal</note>, <note>sides SV and TU are equal</note>; <note>diagonals SU and TV bisect each other</note>, <note>i.e., the intersection point W divides diagonal SU into two equal segments SW and WU</note>, <note>and divides diagonal TV into two equal segments TW and WV</note>.

Rhombus Determination Theorem 1

In the figure of this problem, the diagonals of parallelogram STUV <note>SU and TV are perpendicular</note>, so <note>parallelogram STUV is a rhombus</note>.


Question:
What is m $\angle $ U?

m $\angle $ U= $\Box$ °

Answer: m \$\angle \$ U=142°

1. Given m∠SRT=71°, and ∠SRT is an inscribed angle, which intercepts arc ⏜ST.

2. According to the Inscribed Angle Theorem, the central angle ∠SUT is twice the measure of the arc ⏜ST intercepted by the inscribed angle ∠SRT, therefore, m ∠SUT = 2 × m ∠SRT.

3. Substitute the measure of ∠SRT, i.e., m ∠SUT = 2 × 71°.

4. Calculate to obtain m ∠SUT = 142°.

5. Through the above reasoning, the final answer is 142°.

Definition of Inscribed Angle

An inscribed angle is an angle whose vertex lies on the circumference of a circle and whose sides intersect the circle at two points.

In the figure of this problem, in the circle, <note>the vertex R of angle SRT is on the circumference</note>, <note>the two sides of angle SRT intersect the circle at points S and T respectively</note>. Therefore, <note>angle SRT is an inscribed angle</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</note>, in the circle, <note>points S and T are two points on the circle</note>, <note>the center of the circle is point U</note>. The angle ∠SUT formed by the lines <note>US and UT is called the central angle</note>.

Inscribed Angle Theorem

In a circle, an inscribed angle is equal to half of the central angle that subtends the same arc.

In the problem figure, <note>∠SRT is an inscribed angle</note>, and the arc it subtends is <note>⏜ST</note>. According to the Inscribed Angle Theorem, <note>the central angle ∠SUT is equal to twice the measure of the arc ⏜ST subtended by the inscribed angle ∠SRT</note>. Therefore, <note>m ∠SUT = 2 × m ∠SRT</note>, which is 2 × 71° = 142°.


Question:
Is parallelogram UVWX a rectangle?

- yes
- no

Answer: - no

1. Use auxiliary lines to mark the midpoint T on the diagonals VX and UW.

2. According to the problem statement, it is known that the segments of the diagonals VX and UW at midpoint T satisfy VT=XT=45 and UT=WT=47.

3. In a parallelogram, if the diagonals are equal and bisect each other, then the parallelogram is a rectangle. This is one of the definitions of a rectangle.

4. Calculate the lengths of the diagonals VX and UW:

5. The length of the diagonal VX is VT + TW = 45 + 45 = 90.

6. The length of the diagonal UW is UT + TX = 47 + 47 = 94.

7. According to steps 5 and 6, VX= 90 and UW= 94.

8. Since VX and UW are not equal in length, according to the definition of a rectangle, the diagonals of the parallelogram UVWX are not equal, so it is not a rectangle.

9. Based on the above reasoning, the final answer is no.

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 (≅).

<note>The definition of a rectangle refers to a quadrilateral in which each interior angle is a right angle (90 degrees)</note>, and the opposite sides are parallel and equal in length. In the given diagram, the diagonals of the parallelogram UVWX, <note>the length of VX is 90</note>, <note>the length of UW is 94</note>, they are not equal, so UVWX is not a rectangle.

Definition of Parallelogram

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


Question:
Is parallelogram VWXY a rhombus?

- yes
- no

Answer: - yes

1. Given VWXY is a parallelogram, therefore XW = VY, XY = VW.

2. According to the congruent triangle theorem (ASA), it can be concluded that triangle XWY and triangle VWY are congruent.

2. Given ∠XWY = ∠XYW = ∠VWY = ∠VYW, and YW = YW.

2. According to the congruent triangle theorem (ASA), it can be concluded that triangle XWY and triangle VWY are congruent. Therefore XY = XW = VW = VY.

3. According to the definition of a rhombus, which is a quadrilateral with all sides equal, it can be concluded that the figure VWXY is a rhombus. This definition ensures all sides are equal, making it a rhombus.

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

Definition of Rhombus

<note>In quadrilateral VWXY, all sides VW, WX, XY, YV are equal</note>, therefore quadrilateral VWXY is a rhombus.

Properties of Parallelogram Theorem

In the parallelogram <note>VWXY</note>, <note>sides VW and XY are equal</note>, <note>sides WX and VY are equal</note>.

Congruence Theorem of Triangles (ASA)

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

<note>Angle XWY is equal to angle VWY</note>, <note>angle XYW is equal to angle VYW</note>, and <note>side WY is equal to side WY</note>. Since the two triangles have two angles and the included side equal respectively, according to the Angle-Side-Angle (ASA) Congruence Theorem of triangles, it can be concluded that <note>triangle XWY is congruent to triangle VWY</note>.

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 diagram of this problem, triangle XWY and triangle VWY are congruent triangles</note>, the corresponding sides and corresponding angles of triangle XWY are equal to those of triangle VWY, namely: <note>side XW = side VW</note> <note>side VY = side XY</note> <note>side WY = side WY</note>, and the corresponding angles are also equal: <note>angle X = angle V</note> <note>angle XWY = angle VWY</note> <note>angle XYW = angle VYW</note>.


Question:
Is parallelogram RSTU a rectangle?

- yes
- no

Answer: - no

1. In parallelogram RSTU, since the diagonals bisect each other, and because V is the intersection point of diagonals SU and RT, we have VU = VS and VT = VR.

2. Given RV = 54 and TV = 54, RT = 2 × RV = 108.

3. Furthermore, given UV = 52 and SV = 52, SU = 2 × UV = 104.

4. In a rectangle, the diagonals should be equal, therefore if RSTU is a rectangle, it should satisfy SU = RT.

5. However, since SU ≠ RT, according to the properties of a rectangle, this means RSTU is not a rectangle.

6. Based on the above reasoning, the final answer is no.

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 (≅).

<note>Quadrilateral RSTU is a parallelogram</note>, <note>side RU is parallel and equal to side ST</note>, side RS is parallel and equal to side TU.

Definition of Rectangle

<note>In the diagram of this problem, if RSTU is a rectangle</note>, then its diagonals <note>SU and RT</note> should be equal. Given that <note>SU = 104 and RT = 108</note>, it is obvious that <note>SU ≠ RT</note>, therefore according to <note>the properties of a rectangle</note>, <note>RSTU is not a rectangle</note>.

Properties of Parallelogram Theorem

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

In the parallelogram <note>RSTU</note>, the opposite angles <note>∠RST and ∠TUR are equal</note>, and the opposite angles <note>∠RSU and ∠TSR are equal</note>; the sides <note>RS and TU are equal</note>, and the sides <note>ST and RU are equal</note>; the diagonals <note>SU and RT bisect each other</note>, that is, the intersection point <note>V divides the diagonal SU into two equal segments UV and SV</note>, and <note>divides the diagonal RT into two equal segments RV and TV</note>.


Question:
If $\overline{PS}$ is not parallel to $\overline{QR}$ , what is PS?

PS= $\Box$

Answer: PS=60

1. Given that in quadrilateral PQRS, $\\overline{PQ}\\$ and $\\overline{RS}\\$ are a pair of parallel lines.

2. According to the conditions of the problem, there is another pair of congruent base angles in quadrilateral PQRS, i.e., $\\angle P \\equiv \\angle Q\\$.

3. Based on the definition of a trapezoid and the properties of an isosceles trapezoid, quadrilateral PQRS is an isosceles trapezoid.

4. The definition of an isosceles trapezoid determines that its legs (non-parallel sides) are equal, i.e., $\\overline{PS} = \\overline{QR}\\$.

5. The problem states that the length of $\\overline{QR}\\$ is 60.

6. Therefore, it can be concluded that $\\overline{PS} = \\overline{QR} = 60\\$.

7. Through the above reasoning, the final answer is 60.

Definition of Trapezoid

A quadrilateral is a trapezoid if and only if it has exactly one pair of parallel sides.

In the figure of this problem, in the quadrilateral PQRS, <note>side PQ and side RS are parallel</note>, while <note>side PS and side QR are not parallel</note>. Therefore, according to the definition of a trapezoid, the quadrilateral PQRS is a trapezoid because it has only one pair of parallel sides.

Properties of an Isosceles Trapezoid

In an isosceles trapezoid, the base angles are equal.

<note>Quadrilateral PQRS is an isosceles trapezoid</note>, where sides PQ and RS are the two bases of the isosceles trapezoid, and sides PS and QR are the two legs of the isosceles trapezoid. According to the properties of an isosceles trapezoid, <note>base angle ∠P and base angle ∠Q are equal</note>, <note>base angle ∠R and base angle ∠S are equal</note>. The problem states that <note>the length of side QR is 60</note>, therefore <note>the length of side PS is also 60</note>.

Definition of Isosceles Trapezoid

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

<note>In trapezoid PQRS</note>, sides PQ and RS are parallel, and sides QR and PS are the legs of the trapezoid. According to the definition of an isosceles trapezoid, <note>sides QR and PS are equal</note>. Therefore, <note>trapezoid PQRS is an isosceles trapezoid</note>.


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

- yes
- no

Answer: - yes

1. Observe the figure, given quadrilateral ABCD, and its diagonals AC and BD divide each other into two parts.

2. From the figure, it is known that the intersection point O of AC and BD makes AO = OC and BO = OD, i.e., OA = OC and OB = OD.

3. Using the definition of the midpoint of a segment, if a diagonal divides a segment into two equal parts, then its intersection point is the midpoint of the segment.

4. Conclusion: The intersection point O of AC and BD is the midpoint of AC and BD.

5. According to the theorem 4 of parallelogram determination: If the two diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

6. In the problem, since the diagonals AC and BD divide each other into two parts, i.e., bisect each other, therefore according to the above determination theorem, it can be determined that the quadrilateral ABCD is a parallelogram.

7. Through the above reasoning, the final answer is that it is a parallelogram.

Parallelogram Determination Theorem 4

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

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


Question:
In the cube shown below, which lines are parallel? Select all that apply.

-

| $\overleftrightarrow{IJ}$ | and | $\overleftrightarrow{GH}$ |
-

| $\overleftrightarrow{CG}$ | and | $\overleftrightarrow{EI}$ |
-

| $\overleftrightarrow{GJ}$ | and | $\overleftrightarrow{HI}$ |
-

| $\overleftrightarrow{CD}$ | and | $\overleftrightarrow{GH}$ |

Answer:
-

| \$\overleftrightarrow{IJ}\$ | and | \$\overleftrightarrow{GH}\$ |
-

| \$\overleftrightarrow{CG}\$ | and | \$\overleftrightarrow{EI}\$ |
-

| \$\overleftrightarrow{GJ}\$ | and | \$\overleftrightarrow{HI}\$ |
-

| \$\overleftrightarrow{CD}\$ | and | \$\overleftrightarrow{GH}\$ |

1. Observe the topological structure of the cube. Each face of the cube is a rectangle, and the opposite sides of a rectangle are parallel lines.

2. In face GHIJ, sides IJ and GH are both sides of face GHIJ. Therefore, according to the definition of a rectangle, line IJ and line GH are parallel.

3. In face CGIE, sides CG and EI are both sides of face CGIE. Therefore, according to the definition of a rectangle, line CG and line EI are parallel.

4. In face GHJI, sides GJ and HI are both sides of face GHJI. Therefore, according to the definition of a rectangle, line GJ and line HI are parallel.

5. In face CDGH, sides CD and GH are both sides of face CDGH. Therefore, according to the definition of a rectangle, line CD and line GH are parallel.

6. In summary, the pairs of parallel lines in the cube are: IJ and GH, CG and EI, GJ and HI, CD and GH.

7. Based on the above reasoning, the final answer is that the pairs of lines IJ and GH, CG and EI, GJ and HI, CD and GH are all parallel to each other.

Definition of Rectangle

<note>In quadrilateral GHIJ, each interior angle GJI, JIH, IHG, and HGJ is a right angle (90 degrees)</note>, and <note>side GH is parallel and equal in length to side IJ</note>, <note>side GJ is parallel and equal in length to side HI</note>. Therefore, <note>quadrilateral GHIJ is a rectangle</note>. Similarly, <note>in quadrilateral CDHG, each interior angle GCD, CDH, DHG, and HGC is a right angle (90 degrees)</note>, and <note>side CD is parallel and equal in length to side GH</note>, <note>side GC is parallel and equal in length to side HD</note>. Therefore, <note>quadrilateral CDHG is a rectangle</note>. Similarly, <note>in quadrilateral CGIE, each interior angle CGI, GIE, IEC, and ECG is a right angle (90 degrees)</note>, and <note>side CG is parallel and equal in length to side EI</note>, <note>side GI is parallel and equal in length to side CE</note>. Therefore, <note>quadrilateral CGIE is a rectangle</note>.


Question:
Is parallelogram CDEF a rectangle?

- yes
- no

Answer: - no

1. Given the figure is a parallelogram CDEF, and FG = DG = 92, EG = CG = 98. We need to determine whether CDEF is a rectangle.

2. In a parallelogram, according to the properties theorem of parallelograms, the diagonals bisect each other, so G is the intersection point of diagonals CE and DF.

3. Since FG = DG = 92, we can conclude that DG and FG are equal in length, which means G is the midpoint of DF, and DF = 2 * DG = 2 * 92 = 184.

4. Similarly, EG = CG = 98 implies that G is the midpoint of CE, and CE = 2 * EG = 2 * 98 = 196.

5. The rectangle criterion theorem 4 states that the diagonals are equal, meaning for a rectangle, the diagonals CE and DF should be equal.

6. However, through calculation, we find that CE = 196 while DF = 184, they are not equal.

7. Since the diagonals are not equal, CDEF does not meet the definition of a rectangle, therefore CDEF is not a rectangle.

8. Through the above reasoning, the final answer is no.

Definition of Parallelogram

<note>Quadrilateral CDEF is a parallelogram</note>, side <note>CD is parallel and equal to side EF</note>, side <note>CF is parallel and equal to side DE</note>.

Definition of Rectangle

<note>CDEF</note> is a rectangle, it is necessary to verify whether its internal angles are all <note>right angles (90 degrees)</note>, and whether the opposite sides are parallel and equal in length. Since the <note>angles</note> are not marked in the figure, and the <note>diagonals CE and DF</note> are not equal, it is impossible to determine whether <note>CDEF is a rectangle</note>, therefore <note>CDEF is not a rectangle</note>.

Properties of Parallelogram Theorem

In the diagram of this problem, in <note>parallelogram CDEF</note>, <note>opposite angles ∠CDE and ∠EFC are equal</note>, <note>opposite angles ∠DEF and ∠FCD are equal</note>; <note>sides CD and EF are equal</note>, <note>sides CF and DE are equal</note>; <note>diagonals CE and DF bisect each other</note>, that is, <note>the intersection point G divides diagonal CE into two equal segments CG and GE</note>, <note>and divides diagonal DF into two equal segments DG and GF</note>.

Rectangle Criterion Theorem 4

A parallelogram with equal-length diagonals is a rectangle.

In the figure of this problem, <note>the diagonals CE and DF of parallelogram CDEF are not equal</note>, so <note>parallelogram CDEF is not a rectangle</note>.


Question:
If $\overline{BC}$ is not parallel to $\overline{AD}$ , what is AD?

AD= $\Box$

Answer: AD=72

1. Given that in quadrilateral ABCD, only one pair of sides is parallel, namely AB is parallel to CD, and BC is not parallel to AD. According to the definition of a trapezoid, quadrilateral ABCD is a trapezoid.

2. From the figure, ∠C and ∠D are congruent base angles.

3. According to the properties of an isosceles trapezoid, quadrilateral ABCD is an isosceles trapezoid (because it has one pair of parallel sides and one pair of congruent base angles).

4. The figure indicates that BC = 72.

5. Since ABCD is an isosceles trapezoid, AD = BC.

6. From step 4, we know BC = 72, so AD = 72.

7. Through the above reasoning, the final answer is AD = 72.

Definition of Trapezoid

In the figure of this problem, in quadrilateral ABCD, <note>side AB and side CD are parallel</note>, while <note>side BC and side AD are not parallel</note>. Therefore, according to the definition of trapezoid, quadrilateral ABCD is a trapezoid because <note>it has exactly one pair of parallel sides</note>.

Properties of an Isosceles Trapezoid

In the figure of this problem, in the isosceles trapezoid ABCD, side <note>AB and side CD are the two bases of the isosceles trapezoid</note>, side <note>AD and side BC are the two legs of the isosceles trapezoid</note>. According to the properties of an isosceles trapezoid, <note>base angle ∠DAB and base angle ∠BCD are equal</note>, <note>base angle ∠ADC and base angle ∠ABC are equal</note>.

Definition of Isosceles Trapezoid

<note>In trapezoid ABCD, sides AB and CD are parallel</note>, <note>sides BC and AD are the legs of the trapezoid</note>. According to the definition of an isosceles trapezoid, <note>sides AD and BC are equal</note>. That is, <note>AD=BC=72</note>.


Question:
Is parallelogram CDEF a rhombus?

- yes
- no

Answer: - yes

1. According to the given image and conditions, quadrilateral CDEF is a parallelogram because diagonals CE and DF intersect at point B.

2. Given ∠CBF=90°, it indicates that ∠CBF is a right angle.

3. According to the definition of a right triangle, CBF is a right triangle, and it is known that line segment CE ⊥ line segment DF.

4. According to the rhombus determination theorem 1, a parallelogram with perpendicular diagonals is a rhombus.

5. Therefore, the parallelogram CDEF is a rhombus.

6. Through the above reasoning, the final answer is yes, that is, CDEF is a rhombus.

Definition of Parallelogram

<note>Quadrilateral CDEF is a parallelogram</note>, <note>side CF is parallel and equal to side DE</note>, <note>side CD is parallel and equal to side EF</note>.

Definition of Right Triangle

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

<note>In triangle CBF, angle ∠CBF is a right angle (90 degrees)</note>, therefore triangle CBF is a right triangle. <note>Side CB and side BF are the legs</note>, <note>side CF is the hypotenuse</note>.

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.

In the figure of this problem, the quadrilateral <note>CDEF</note> is a quadrilateral, and its diagonals <note>CE</note> and <note>DF</note> intersect perpendicularly at <note>point B</note>, and <note>∠CBF is a right angle (90 degrees)</note>. According to the definition of a rhombus, the quadrilateral <note>CDEF</note> is a rhombus.


Question:
Is parallelogram STUV a rectangle?

- yes
- no

Answer: - yes

1. Given STUV is a parallelogram, and the diagonals SU and TV are equal, SU = TV = 48.

2. Given point R is the intersection of diagonals SU and TV, and RS = RT = RU = RV = 24.

3. In a parallelogram, the two diagonals bisect each other. Therefore, R is the midpoint of SU and TV.

4. Using the characteristic condition RS = RT = RU = RV = 24, it can be deduced that △SRU, △RTV, △SRT, and △URV are congruent.

5. Angles ∠RSU, ∠RUT, ∠TRV, and ∠VRS are all formed by the equal diagonals being cut. According to the 'corresponding angles of congruent triangles are equal' theorem, these angles are all 90 degrees.

6. In parallelogram STUV, each angle is a right angle. By the definition of a parallelogram, the four sides are parallel and the opposite angles are equal. Therefore, STUV is a right-angled parallelogram, i.e., a rectangle.

7. Through the above reasoning, the final answer is a right-angled parallelogram, so STUV is a rectangle.

Definition of Parallelogram

<note>Quadrilateral STUV is a parallelogram</note>, sides <note>ST and UV are parallel and equal</note>, sides <note>SU and TV are parallel and equal</note>.

Definition of Rectangle

In the figure of this problem, <note>quadrilateral STUV is a rectangle</note>, its <note>interior angles ∠SRT, ∠RTU, ∠TUV, ∠VUS are all right angles (90 degrees)</note>, and <note>sides ST and UV are parallel and equal in length</note>, <note>sides SU and TV are parallel and equal in length</note>.

Properties of Parallelogram Theorem

In the figure of this problem, in the parallelogram STUV, <note>the opposite angles ∠TSU and ∠VUT are equal</note>, <note>the opposite angles ∠SUV and ∠TVS are equal</note>; <note>sides ST and UV are equal</note>, <note>sides SU and TV are equal</note>; <note>the diagonals SU and TV bisect each other</note>, that is, <note>the intersection point R divides the diagonal SU into two equal segments SR and RU</note>, <note>and divides the diagonal TV into two equal segments TR and RV</note>.

Definition of Congruent Triangles

From <note>RS=RU=RT=RV</note>, it can be deduced that <note>△SRU, △RTV, △SRT, and △URV are congruent</note>. According to the <note>definition of congruent triangles</note>, <note>the corresponding sides and angles of △SRU are equal to those of △RTV, △SRT, and △URV</note>, namely: <note>side SR = side RT = side RU = side RV</note>, side SU = side TV, side ST = side UV. Meanwhile, the corresponding angles are also equal: <note>angle ∠RSU = angle ∠RTV = angle ∠SRT = angle ∠URV</note>.

Definition of Rectangle

<note>Quadrilateral STUV is a rectangle</note>, with interior angles <note>∠SRT, ∠RTU, ∠TUV, ∠VUS all being right angles (90 degrees)</note>, and <note>sides ST and UV are parallel and equal in length</note>, <note>sides SU and TV are parallel and equal in length</note>.


Question:
Is parallelogram FGHI a square?

- yes
- no

Answer: - yes

1. Given that FH and GI are the two diagonals of parallelogram FGHI, FJ=GJ=HJ=IJ=65, FH=FJ+JH, GI=GJ+IJ, then FH=GI.

2. Since the diagonals of quadrilateral FGHI intersect at point J, and it is known that ∠HJI=90°, according to the definition of perpendicular lines, FH⟂GI.

3. According to the properties of the diagonals of a square, the diagonals of a square are the line segments connecting opposite angles. The two diagonals of a square are equal and bisect each other perpendicularly. It can be seen that the above reasoning satisfies this property, so FGHI is a square.

4. Through the above reasoning, the final answer is 'Yes'.

In the figure of this problem, the diagonals <note>FH</note> and <note>GI</note> of the quadrilateral <note>FGHI</note> are equal and bisected at point J. By the property of equal and bisected diagonals, the quadrilateral <note>FGHI is a rectangle</note>. The definition of a rectangle is a quadrilateral in which <note>each interior angle is a right angle (90 degrees)</note>, and <note>opposite sides are parallel and equal in length</note>.

Definition of Square

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

In the quadrilateral <note>FGHI</note>, <note>sides FG, GH, HI, and IF are equal</note>, and <note>angles FGH, GHI, HIF, and IFG are all right angles (90 degrees)</note>, so FGHI is a square.

Definition of Perpendicular Lines

Two lines are said to be perpendicular if and only if the angle formed by their intersection is 90 degrees.

<note>Line FH and line GI intersect to form an angle ∠IJH of 90 degrees</note>, therefore according to the definition of perpendicular lines, <note>line FH and line GI are perpendicular to each other</note>.

Properties of Diagonals in a Square

The diagonals of a square are the line segments that connect opposite vertices. The diagonals of a square are equal in length, and they bisect each other perpendicularly.

In the diagram of this problem, in square FGHI, <note>diagonals FH and GI are segments connecting opposite corners</note>. According to the properties of diagonals in a square, <note>FH and GI are equal</note>, and <note>FH and GI bisect each other perpendicularly</note>, forming <note>four 90-degree angles at their intersection</note>. Therefore, <note>FH = GI</note>, and they <note>are perpendicular to each other at the intersection</note>.


Question:
$\overline{RU}$ is the midsegment of the trapezoid QSTV.
If RU=49 and QV=62, what is ST?

ST= $\Box$

Answer: ST=36

1. Given that the line segment $\\overline{RU}\\$ is the midsegment of the trapezoid $QSTV$, and $RU=49$, $QV=62$.

2. The midsegment theorem for trapezoids states: In a trapezoid, the line segment parallel to the legs, i.e., the midsegment, has a length equal to the average of the lengths of the two bases. In this problem, the length of the midsegment $\\overline{RU}\\$ can be expressed as $RU = \\frac{QV + ST}{2}\\$.

3. Substitute the known values into the formula: $49 = \\frac{62 + ST}{2}\\$.

4. Solve this equation: First, multiply both sides of the equation by 2 to eliminate the denominator, yielding $98 = 62 + ST$.

5. Next, subtract 62 from both sides of the equation to obtain $ST = 36$.

6. Through the above reasoning, the final answer is $ST = 36$.

Definition of Trapezoid

<note>In the figure of this problem, in the quadrilateral QSTV, side QV and side ST are parallel</note>, while <note>side QS and side TV are not parallel</note>. Therefore, according to the definition of a trapezoid, <note>the quadrilateral QSTV is a trapezoid</note>, because it has exactly one pair of parallel sides.

Median Line Theorem of Trapezoid

The median line of a trapezoid is the line segment that connects the midpoints of the non-parallel sides. This line segment is parallel to the bases (the parallel sides of the trapezoid) and its length is equal to half the sum of the lengths of the two bases.

<note>In the diagram of this problem, in trapezoid QSTV, sides QV and ST are the two bases of the trapezoid</note>, <note>points R and U are the midpoints of the two legs of the trapezoid</note>, <note>segment RU is the median line connecting the midpoints of the two legs</note>. According to the Median Line Theorem of Trapezoid, <note>segment RU is parallel to sides QV and ST</note>, and <note>the length of segment RU is equal to half the sum of the lengths of sides QV and ST, i.e., RU = (QV + ST) / 2</note>.


Question:
What is NO?

NO= $\Box$

Answer: NO=9

1. From point K, draw two tangents to the incircle, KL and KP respectively. According to the tangent length theorem, we get KL = KP. Given that KL = 8, therefore KP = 8.

2. Given KO = 17, we have KO = KP + OP. Substituting the known values, we get 17 = 8 + OP.

3. From the equation in the previous step, we solve for OP: OP = 17 - 8 = 9.

4. From point O, draw two tangents to the incircle, OP and ON respectively. According to the tangent length theorem, we get OP = ON. Since OP is already found to be 9, therefore ON = 9.

5. Through the above reasoning, we finally get the answer NO = 9.

Definition of Tangent to a Circle

A line is a tangent to a circle if and only if it has exactly one point of intersection with the circle. This point of intersection is called the point of tangency.

In the figure of this problem, <note>the circle and the line KL have only one common point L</note>, this common point is called the point of tangency. Therefore, <note>the line KL is the tangent to the circle</note>. Similarly, <note>the circle and the line KP have only one common point P</note>, this common point is called the point of tangency. Therefore, <note>the line KP is the tangent to the circle</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.

<note>Line segment KO is a part of a straight line</note>, containing <note>endpoint K and endpoint O</note> and all points between them. <note>Line segment KO has two endpoints</note>, which are <note>K and O</note>, and <note>every point on line segment KO is located between endpoint K and endpoint O</note>.

Tangent-Segment Theorem

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

<note>From point O, draw two tangents to the inscribed circle, namely OP and ON</note>, their tangent lengths are equal, i.e., <note>OP=ON</note>. The line connecting the center of the circle and point O bisects the angle between the two tangents <note>OP and ON</note>. From point O, draw two tangents to the inscribed circle, namely <note>KL and KP</note>, their tangent lengths are equal, i.e., <note>KL=KP</note>. The line connecting the center of the circle and point O bisects the angle between the two tangents <note>KL and KP</note>.


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

- circle
- square
- octagon
- nonagon

Answer: - square

1. First, by observing the figure shown in the problem, the figure is a three-dimensional solid, specifically a regular hexahedron.

2. In a regular hexahedron, each face is a congruent square, which is a special type of cube where all six faces are squares.

3. Further analysis shows that the regular hexahedron has six faces, and each face is a square. A square has four congruent sides and four right angles.

4. Therefore, based on the definition and properties of a regular hexahedron, it can be concluded that the base of this solid figure is a square.

5. Through the above reasoning, the final answer is a square.

Definition of Cube

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

<note>In the figure of this problem, the solid figure is a cube</note>. This solid figure has six faces, and all corresponding faces are congruent squares, which is <note>the definition of a cube in the specific context of this problem</note>.

Definition of Square

In the figure of this problem, the base and the other five faces are all squares. <note>A square has four sides of equal length and four interior angles of 90 degrees</note>. The <note>base has four sides of equal length</note> and <note>four interior angles of 90 degrees</note>, so the base is a square.

Definition of Cube

In the diagram of this problem, the solid figure conforms to the characteristics of a cube, that is, it has <note>six congruent square faces</note>, these <note>six faces are perpendicular to each other</note>, and <note>each face is a square</note>, which can confirm that the figure is a <note>regular hexahedron</note>, and further confirm that the <note>base is a square</note>.


Question:
If $\overline{FG}$ is not parallel to $\overline{EH}$ , what is m $\angle $ H?

m $\angle $ H= $\Box$ °

Answer: m \$\angle \$ H=134°

1. According to the problem, in quadrilateral EFGH, sides EH and FG are not parallel, EF and GH are parallel, and the lengths of EH and FG are equal.

2. Since in quadrilateral EFGH, two opposite sides EF and GH are parallel, and the other two sides EH and FG are of equal length, quadrilateral EFGH is an isosceles trapezoid.

3. According to the properties of an isosceles trapezoid, the angles opposite the non-parallel sides (i.e., each pair of base angles) are equal. That is, ∠EFG is equal to ∠FEH, and ∠FGH is equal to ∠GHE.

4. It is given in the problem that ∠EFG = 46°, thus ∠FEH = 46°.

5. By the theorem of the sum of interior angles of a quadrilateral, the sum of the interior angles of a quadrilateral is 360°. In quadrilateral EFGH, we have: ∠HEF + ∠EFG + ∠HGF + ∠GHE = 360°.

6. According to the properties of an isosceles trapezoid, we know ∠FGH = ∠GHE. Let ∠FGH = x, then ∠GHE is also x.

7. Substituting the known angles into the equation, we get x + 46° + 46° + x = 360°.

8. Simplifying the equation, we get 2x + 92° = 360°.

9. Solving the equation, we get 2x = 360° - 92° = 268°.

10. Dividing by 2, we get x = 134°.

11. Through the above reasoning, the final answer is ∠H = 134°.

Properties of an Isosceles Trapezoid

In the <note>isosceles trapezoid EFGH</note>, <note>side EF and side GH are the two bases of the isosceles trapezoid</note>, and <note>side EH and side FG are the two legs of the isosceles trapezoid</note>. According to the <note>properties of an isosceles trapezoid</note>, <note>base angle ∠EFG and base angle ∠FEH are equal</note>, and <note>base angle ∠FGH and base angle ∠GHE are equal</note>.

In any quadrilateral, the sum of the four interior angles is 360°.

In the diagram of this problem, in quadrilateral EFGH, <note>angle HEF</note>, <note>angle EFG</note>, <note>angle FGH</note>, and <note>angle GHE</note> are the four interior angles of the quadrilateral. According to the Sum of Interior Angles of a Quadrilateral Theorem, the sum of these four interior angles is 360°, that is, <note>angle HEF + angle EFG + angle FGH + angle GHE = 360°</note>.


Question: The rectangle below is labeled with its measured dimensions. Taking measurement error into account, what is the percent error in its calculated area?Round your answer to the nearest tenth of a percent and include a percent sign (%). $\Box$

Answer: 7.2%

1. Given the width of the rectangle is 11 cm and the length is 20 cm, the directly measured area of the rectangle is 11 × 20 = 220 square cm.

2. The maximum measurement error is half of 1 cm, which is 0.5 cm. Therefore, calculate the maximum possible area and the minimum possible area.

3. For the maximum possible area, add the maximum error to each measurement: length is 20 + 0.5 = 20.5 cm, width is 11 + 0.5 = 11.5 cm, so the maximum possible area is 20.5 × 11.5 = 235.75 square cm.

4. For the minimum possible area, subtract the maximum error from each measurement: length is 20 - 0.5 = 19.5 cm, width is 11 - 0.5 = 10.5 cm, so the minimum possible area is 19.5 × 10.5 = 204.75 square cm.

5. Calculate the difference between the minimum area and the measured area as 220 - 204.75 = 15.25 square cm, and the difference between the maximum area and the measured area as 235.75 - 220 = 15.75 square cm.

6. The maximum possible area error is 15.75 square cm.

7. Calculate the percentage error of the area as (maximum possible area error / measured area) × 100% = (15.75 / 220) × 100% ≈ 7.159%.

8. Round the result to the nearest tenth, which means the percentage error of the area is 7.2%.

Through the above reasoning, the final answer is 7.2%.

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, <note>the two long sides of the rectangle are 20 cm each</note>, <note>the opposite sides are equal and parallel</note>; <note>the two short sides are 11 cm each</note>, <note>the opposite sides are equal and parallel</note>. <note>All four angles are right angles (90 degrees)</note>.

Formula for the Area of a Rectangle

The area of a rectangle is equal to its length multiplied by its width.

In the diagram of this problem, <note>the length of the rectangle is 20 cm</note>, <note>the width is 11 cm</note>, so <note>the area of the rectangle = 20 cm × 11 cm = 220 square cm</note>. When calculating the error, add and subtract the error value (0.5 cm) to the length and width respectively, <note>obtaining the maximum possible area as 20.5 cm × 11.5 cm = 235.75 square cm</note>, <note>the minimum possible area as 19.5 cm × 10.5 cm = 204.75 square cm</note>.


Question:
Is parallelogram FGHI a rhombus?

- yes
- no

Answer: - yes

1. Given that quadrilateral FGHI is a parallelogram. We need to determine if it is a rhombus. According to Rhombus Determination Theorem 1, one of the conditions for determining a rhombus is that the diagonals are perpendicular to each other.

2. Observing the figure, ∠FEI is a right angle, i.e., ∠FEI = 90°.

3. According to Rhombus Determination Theorem 1, in a quadrilateral, if the diagonals of adjacent sides are perpendicular to each other, then the quadrilateral is a rhombus.

4. Since ∠FEI = 90°, it can be concluded that line FH is perpendicular to line GI.

5. Therefore, through the perpendicularity of the diagonals and the fact that the midpoint of the diagonals is the midpoint of each side, FGHI is a rhombus.

6. Through the above reasoning, the final answer is YES, quadrilateral FGHI is a rhombus.

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 (≅).

<note>Quadrilateral FGHI is a parallelogram</note>, <note>side FG is parallel and equal to side HI</note>, <note>side FI is parallel and equal to side GH</note>.

Definition of Rhombus

In the figure of this problem, <note>all sides FG, GH, HI, IF of quadrilateral FGHI are equal</note>, thus quadrilateral FGHI is a rhombus. Additionally, the diagonals FH and GI of quadrilateral FGHI <note>are perpendicular bisectors of each other</note>, meaning <note>diagonals FH and GI intersect at point E, and angle ∠FEI is a right angle (90 degrees)</note>, and <note>FE = EH and GE = EI</note>.

Rhombus Determination Theorem 1

A parallelogram is a rhombus if and only if its diagonals are perpendicular.

In the figure of this problem, the diagonals <note>FH and GI of parallelogram FGHI are perpendicular</note>, so parallelogram FGHI is a rhombus.


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

- yes
- no

Answer: - yes

1. Given that a quadrilateral has one pair of opposite sides that are parallel and equal in length, and both sides are 51 in length, we can consider the method for determining a parallelogram.

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

3. In this problem, it is given that the pair of opposite sides of the quadrilateral are parallel and equal in length. The conditions of the theorem are satisfied for these sides.

4. Based on the above reasoning, we can conclude that the quadrilateral is a parallelogram.

Definition of Parallelogram

<note>In a quadrilateral, one pair of opposite sides is parallel and equal in length</note>, <note>the length of this pair of opposite sides is 51</note>, satisfying the <note>condition of opposite sides being parallel and equal in length in the definition of a parallelogram</note>.

Parallelogram Criterion Theorem 3

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

The original text: <note>The opposite sides of the quadrilateral are parallel and equal</note>, so <note>the quadrilateral is a parallelogram</note>.


Question:
Is parallelogram GHIJ a rhombus?

- yes
- no

Answer: - yes

1. From the figure and the given conditions, it is known that quadrilateral GHIJ is a parallelogram. One of the properties of a parallelogram is that opposite angles are equal, i.e., ∠GHI = ∠IJG and ∠HGJ = ∠HIJ.

2. According to the problem statement, the angles ∠GHJ, ∠GJH, ∠HJI, and ∠IHJ of quadrilateral GHIJ are all 59°.

3. Since ∠GHJ = ∠GJH = ∠HJI = ∠IHJ = 59°, and HJ is a common side, the triangles GHJ and IHJ on both sides of segment HJ are congruent.

4. Therefore, hi = IJ = GH = GJ.

5. According to the definition of a rhombus, all four sides are equal.

6. Based on the above reasoning, the final answer is yes, the parallelogram GHIJ is a rhombus.

Definition of Parallelogram

<note>Quadrilateral GHIJ is a parallelogram</note>, <note>side GH is parallel and equal to side IJ</note>, <note>side IH is parallel and equal to side JG</note>.

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 GHIJ, <note>the opposite angles ∠GHI and ∠IJG are equal</note>, <note>the opposite angles ∠HIJ and ∠HGJ are equal</note>; <note>sides IJ and GH are equal</note>, <note>sides HI and GJ are equal</note>; <note>the diagonals GI and HJ bisect each other</note>, that is, <note>the intersection point divides the diagonal GI into two equal segments GO and OG</note>, <note>and divides the diagonal HJ into two equal segments HO and JO</note>.

Definition of Rhombus

In the figure of this problem, the quadrilateral <note>GHIJ, all sides GH, HI, IJ, and JG are equal</note>, thus <note>the quadrilateral GHIJ is a rhombus</note>. Additionally, the quadrilateral GHIJ's <note>diagonals GI and HJ are perpendicular bisectors of each other</note>, meaning <note>the diagonals GI and HJ intersect at point O</note>, and <note>angle GOH is a right angle (90 degrees)</note>, and <note>GO=OI and HO=OJ</note>.

Congruence Theorem of Triangles (ASA)

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

In <note>triangle HIJ</note> and <note>triangle HGJ</note>, <note>angle IHJ is equal to angle GHJ</note>, <note>angle IJH is equal to angle GJH</note>, and <note>side HJ is equal to side HJ</note>. Since the two angles and the included side of these two triangles are respectively equal, according to the Congruence Theorem of Triangles' <note>Angle-Side-Angle criterion (ASA)</note>, it can be concluded that <note>triangle HIJ is congruent to triangle HGJ</note>.

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 diagram of this problem, triangle HIJ and triangle HGJ are congruent triangles</note>, <note>the corresponding sides and corresponding angles of triangle HIJ are equal to those of triangle HGJ</note>, that is:
side <note>HI = side HG</note>
side <note>IJ = side GJ</note>
side <note>HJ = side HJ</note>,
at the same time, the corresponding angles are also equal:
angle <note>HIJ = angle HGJ</note>
angle <note>IHJ = angle GHJ</note>
angle <note>IJH = angle GJH</note>.


Question:
Does this picture have rotational symmetry?

- yes
- no

Answer: - yes

1. First, analyze the given figure, which appears to be a ship's wheel shape with spokes, each evenly distributed.

2. Based on visual observation, this figure has multiple spokes, and each spoke is identical in shape and size.

3. If a figure can coincide with the original figure after being rotated by a certain angle, we say that the figure has rotational symmetry.

4. Observing that there are 12 spokes in the figure, this indicates that the angle between each spoke is 360 degrees/12 = 30 degrees.

5. By the definition of rotational symmetry, if a figure can be rotated by an angle less than 360 degrees and coincide with itself, then the figure has rotational symmetry.

6. Since the figure looks the same when rotated by 30 degrees, it clearly has rotational symmetry.

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

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 the figure of this problem</note>, when the figure <note>is rotated by 30 degrees (360 degrees/12 spokes)</note>, <note>the figure will coincide with the original figure</note>. This indicates that the figure has rotational symmetry.

Definition of Rotational Symmetry

A figure has rotational symmetry if it can be rotated through an angle less than 360 degrees to produce a figure that is identical to the original figure.

In the figure of this problem, the shape can be rotated <note>clockwise/counterclockwise by 30 degrees</note> to obtain a shape identical to the original one, thus the shape has rotational symmetry.


Question:
Does this picture have rotational symmetry?

- yes
- no

Answer: - no

1. Definition of rotational symmetry: If a figure remains unchanged when rotated by an angle θ (0° < θ < 360°) around a center point, then the figure has rotational symmetry.

2. Observing the airplane in the figure, its shape and color distribution are symmetrical, with the wings and the nose having the same color and shape on both sides.

3. Choose any point as the center of rotation and rotate the figure by a certain angle (e.g., 90°). By comparing the rotated figure with the original figure, we find that the rotated figure is different from the original figure.

4. Change to different rotation center points and try other possible rotation angles (e.g., 45°, 180°). Similarly, we find that the rotated figure is always different from the original figure.

5. Through the above process, we cannot find any rotation less than 360° that can restore the figure, indicating that the figure does not satisfy the condition of rotational symmetry.

6. Based on the above reasoning, the final answer is no, the picture does not have rotational symmetry.

In the figure of this problem, <note>the shape cannot be made to coincide with the original shape by rotating it clockwise or counterclockwise by any angle less than 360 degrees</note>, so the shape <note>does not have rotational symmetry</note>.


Question:
If WX=36 and UZ=72, what is VY?

VY= $\Box$

Answer: VY=54

1. Given line segments WX = 36 and UZ = 72.

2. Carefully observe the figure, noting that quadrilateral UWXZ is a trapezoid, and its two bases WX and UZ are parallel.

3. In quadrilateral UWXZ, line segment VY is a median, which means VY is parallel to WX and UZ and is located in the middle.

4. According to the trapezoid median theorem, this theorem states: the median of a trapezoid is parallel to the bases and its length is half the sum of the lengths of the bases. In this problem, line segment VY is the median of trapezoid UWXZ, so: VY = 1/2 * (WX + UZ).

5. Substitute the given conditions into the above formula: VY = 1/2 * (36 + 72).

6. Calculate the sum inside the parentheses: 36 + 72 = 108.

7. Continue calculating the length of VY: VY = 1/2 * 108.

8. Calculate the specific value of VY: VY = 54.

9. Through the above reasoning, the final answer is 54.

Definition of Trapezoid

A quadrilateral is a trapezoid if and only if it has exactly one pair of parallel sides.

<note>In the figure of this problem, in quadrilateral UWXZ, sides WX and UZ are parallel</note>, while <note>sides WU and XZ are not parallel</note>. Therefore, according to the definition of a trapezoid, quadrilateral UWXZ is a trapezoid because it has exactly one pair of parallel sides.

Median Line Theorem of Trapezoid

The median line of a trapezoid is the line segment that connects the midpoints of the non-parallel sides. This line segment is parallel to the bases (the parallel sides of the trapezoid) and its length is equal to half the sum of the lengths of the two bases.

In the figure of this problem, in <note>trapezoid UWXZ</note>, <note>side WX and side UZ are the two bases of the trapezoid</note>, <note>point V and point Y are the midpoints of the two legs of the trapezoid</note>, and <note>segment VY is the median line connecting the midpoints of the legs</note>. According to the Median Line Theorem of Trapezoid, <note>segment VY is parallel to side WX and side UZ</note>, and <note>the length of segment VY is equal to half the sum of the lengths of side WX and side UZ</note>, that is, <note>VY = (WX + UZ) / 2</note>.


Question:
Does this picture have rotational symmetry?

- yes
- no

Answer: - no

1. Observe the image and identify the parts of the figure to determine if the figure has rotational symmetry.

2. The definition of rotational symmetry is: During a full rotation of the figure (360 degrees), if there exists an angle of rotation less than 360 degrees that makes the rotated figure identical to the original figure, then the figure has rotational symmetry.

3. Suppose we rotate the figure counterclockwise around the center point. No matter how many degrees we rotate (as long as the rotation angle is less than 360 degrees), the head, limbs, and tail will not align with the positions and directions in the original figure.

4. After careful observation and verification of the figure, it is confirmed that before rotating less than 360 degrees, the figure cannot coincide with its original appearance.

5. Based on the above analysis and the criteria for rotational symmetry, the figure does not meet the conditions for rotational symmetry.

6. Therefore, it is concluded that the figure does not have rotational symmetry.

Definition of Rotational Symmetry

The figure does not have rotational symmetry because <note>the figure rotated 90 degrees clockwise/counterclockwise</note> cannot obtain the same figure as the original.


Question:
Does this picture have rotational symmetry?

- yes
- no

Answer: - no

1. The rotational symmetry in the figure refers to the image looking the same as the original picture after rotating by an angle less than 360°.

2. Observe the given figure, it is in the shape of an octopus.

3. For the rotational symmetry of the figure, we need to determine if there exists an angle of rotation less than 360° that makes the image coincide with the original picture.

4. The structure of the octopus image is asymmetrical, especially its tentacles and head, which makes it impossible for the image to coincide with the original picture under any rotation less than 360°.

5. Through visual inspection, we find that although the distribution of tentacles on the left and right sides of the octopus image is balanced, any rotation less than 360° cannot make the figure coincide with the original picture.

6. Based on the above judgment, the octopus in the picture does not have rotational symmetry.

n. After the above reasoning, the final answer is No.

The <note>octopus shape</note> cannot be transformed into the same shape by rotating <note>any angle less than 360 degrees</note> clockwise or counterclockwise, so <note>this shape does not have rotational symmetry</note>.


Question:
Does this picture have rotational symmetry?

- yes
- no

Answer: - no

1. When determining whether a figure has rotational symmetry, we need to ascertain whether the figure can look the same as the original after a rotation of less than 360 degrees.

2. Analyze the given figure, which is in the shape of an insect, and observe whether its parts exhibit symmetry about a certain point.

3. Although the insect's left and right body parts appear symmetrical, when rotating around the center point, the directions of the insect's parts such as legs and antennae change, causing the rotated figure to not match the original.

4. Attempt to analyze rotations of 90 degrees, 180 degrees, and 270 degrees, observing the changes in the figure during each rotation. Comparing the rotated figure with the original reveals that the angles of deviation of the parts result in a loss of symmetry.

5. Therefore, according to the definition of rotational symmetry, no rotation of less than 360 degrees can make the figure completely coincide with the original, meaning the figure does not have rotational symmetry.

6. Based on the above reasoning, the final answer is no (the figure does not have rotational symmetry).

In the problem figure, the shape cannot be obtained by rotating 90 degrees clockwise/counterclockwise to get the same shape as the original figure, so the shape does not have rotational symmetry.


Question:
If $\overline{SV}$ is not parallel to $\overline{TU}$ , what is TU?

TU= $\Box$

Answer: TU=51

1. From the figure, it is known that in quadrilateral STUV, sides \overline{ST} and \overline{UV} are parallel.

2. Angles \angle S and \angle T are congruent.

3. Since quadrilateral STUV has one pair of base angles congruent and one pair of opposite sides parallel, by the definition of trapezoid and the properties of an isosceles trapezoid, STUV is an isosceles trapezoid.

4. The definition of an isosceles trapezoid tells us that the two legs (non-parallel sides) are congruent.

5. Therefore, sides \overline{TU} and \overline{SV} are congruent, i.e., TU = SV.

6. The length of side \overline{SV} given in the problem is 51.

7. Therefore, through the above reasoning, it is concluded that TU = 51.

Properties of an Isosceles Trapezoid

In an isosceles trapezoid, the base angles are equal.

<note>In isosceles trapezoid STUV</note>, <note>side ST and side UV are the two bases of the isosceles trapezoid</note>, <note>side TU and side SV are the two legs of the isosceles trapezoid</note>. According to the properties of an isosceles trapezoid, <note>base angle ∠STU and base angle ∠TSV are equal</note>, <note>base angle ∠SVU and base angle ∠TUV are equal</note>.

Definition of Trapezoid

<note>Side UV and side ST are parallel</note>, while <note>side UT and side SV are not parallel</note>. Therefore, according to the definition of a trapezoid, this <note>quadrilateral STUV is a trapezoid</note>, because it has exactly one pair of parallel sides.

Definition of Isosceles Trapezoid

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

In the figure of this problem, in trapezoid STUV, <note>side UV and side ST are parallel</note>, <note>side UT and side SV are the legs of the trapezoid</note>. According to the definition of an isosceles trapezoid, <note>side UT and side SV are equal</note>. Therefore, <note>trapezoid STUV is an isosceles trapezoid</note>.


Question:
Does this picture have rotational symmetry?

- yes
- no

Answer: - no

1. To determine if the picture has rotational symmetry, we need to check if the picture can coincide with itself when rotated by a specific angle.

2. Upon analyzing the picture, we found that when rotated by 90 degrees, 180 degrees, and 270 degrees, the layout of the picture does not coincide with its original orientation.

3. Specifically, the picture contains multiple shapes in different directions, such as trees and the sun, and the arrangement of these elements cannot match the original picture under any non-complete rotation (less than 360 degrees).

4. Therefore, the picture does not coincide with itself under rotations less than 360 degrees.

5. According to the definition of rotational symmetry, any rotation angle that does not coincide with itself indicates that the picture does not have rotational symmetry.

Rotation Invariance Theorem

The original text: <note>By rotating the image 90 degrees, 180 degrees, 270 degrees</note> to check whether it can coincide with the original image at these specific angles. Specifically, observe whether <note>elements such as trees and the sun</note> can return to their original positions after any rotation less than 360 degrees, to determine whether the image has rotational symmetry.

Definition of Rotational Symmetry

The figure cannot be transformed into a figure identical to the <note>original figure</note> by rotating <note>90 degrees clockwise/counterclockwise</note>, so the figure does not have <note>rotational symmetry</note>.


Question:
Does this picture have rotational symmetry?

- yes
- no

Answer: - yes

1. Observe the figure in the picture. The figure is a centrally symmetric pattern, shaped like a sunflower, consisting of a central circle and surrounding petals.

2. The sunflower pattern consists of approximately 20 petals of the same shape and size, evenly distributed and symmetrically arranged around the central circle.

3. The angle between each petal is the same, which can be calculated by dividing 360 degrees by the number of petals. From this angle calculation, it is known that the angle between the petals is 360 degrees / 20 petals = 18 degrees.

4. According to the definition of rotational symmetry: If a figure can be rotated by less than 360 degrees to obtain the same figure as the original, then the figure has rotational symmetry.

5. Based on the calculated result of the rotation angle: When the figure is rotated clockwise or counterclockwise by an integer multiple of 18 degrees around the center, the figure looks exactly the same as before the rotation.

6. Therefore, the sunflower pattern in the picture can remain unchanged at least at the smallest non-zero angle of 18 degrees, meeting the definition of rotational symmetry.

7. After the above analysis process, the final conclusion is that the pattern shown in the picture has rotational symmetry.

Rotation Invariance Theorem

The <note>sunflower pattern</note> remains unchanged in shape and arrangement after an 18-degree rotation, therefore the <note>sunflower pattern</note> meets the necessary and sufficient conditions for rotational symmetry and possesses rotational symmetry.

Definition of Rotational Symmetry

In the context of this problem, <note>in the figure of this problem, the shape is rotated 18 degrees clockwise/counterclockwise</note> to obtain the same shape as the original, so the shape has rotational symmetry.


Question:
Is parallelogram CDEF a rectangle?

- yes
- no

Answer: - yes

1. Given BC≅BD≅BE≅BF, first consider this feature, which means that B is a symmetrical center within quadrilateral CDEF, because its distance to each corner is equal.

2. We know that if a quadrilateral is a parallelogram and its diagonals bisect each other and are equal, then the quadrilateral is a rectangle.

3. In parallelogram CDEF, CE and DF are diagonals.

4. Given BC≅BD≅BE≅BF and CE≅DF, it follows that CE and DF bisect each other and are equal in length.

5. According to the rectangle determination theorem 2: If the diagonals of a parallelogram bisect each other and are equal, then the parallelogram is a rectangle.

6. Through the above reasoning, we can confirm that the diagonals CE and DF of parallelogram CDEF not only bisect each other but are also equal, so CDEF is a rectangle.

7. After the above reasoning, the final answer is yes, CDEF is a rectangle.

Definition of Parallelogram

<note>Quadrilateral CDEF is a parallelogram</note>, <note>side CD is parallel and equal to side EF</note>, <note>side CF is parallel and equal to side DE</note>.

Definition of Rectangle

<note>Quadrilateral CDEF is a rectangle</note>, whose <note>interior angles ∠DCF, ∠CFE, ∠FED, ∠EDC are all right angles (90 degrees)</note>, and <note>sides CD and EF are parallel and equal in length</note>, <note>sides CF and DE are parallel and equal in length</note>.

Property of Diagonals in a Rectangle

In a rectangle, the diagonals are equal in length and bisect each other.

In <note>rectangle CDEF</note>, the <note>diagonals CE and FD are equal and bisect each other</note>. Specifically, <note>point B is the intersection point of diagonal CE and diagonal DF</note>, so <note>CB = BE</note>, and <note>BF = BD</note>. According to the <note>property of diagonals in a rectangle</note>, <note>CB = BD</note>, and <note>BF = BE</note>.

Properties of Parallelogram Theorem

<note>In the diagram of this problem, in parallelogram CDEF</note>, <note>angles FCD and DEF are equal, angles CFE and CDE are equal</note>; <note>sides CD and EF are equal, sides CF and DE are equal</note>; <note>diagonals CE and DF bisect each other</note>, that is, the intersection point B divides <note>diagonal CE into two equal segments CB and BE</note>, and divides <note>diagonal DF into two equal segments BD and BF</note>.

Rectangle Identification Theorem 2

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

The <note>quadrilateral CDEF</note>'s <note>diagonals CE and DF bisect each other and are equal</note>, therefore <note>quadrilateral CDEF is a rectangle</note>.


Question:
Does this picture have rotational symmetry?

- yes
- no

Answer: - yes

1. First, observe the overall structure of the figure, which has a center and includes several repeating pattern units.

2. Divide the entire figure into several equal regions, each region containing the same repeating pattern unit.

3. Observe the number of pattern units, and it is observed that there are 6 identical units in the figure.

4. According to the rotational invariance theorem, starting from the center of the figure, rotate the figure 360 degrees, when each pattern unit aligns to the next adjacent position, the figure will look the same as before the rotation.

5. For a figure containing 6 identical pattern units, it can be rotated every 60 degrees and still remain unchanged. This is rotational symmetry.

6. Through the above reasoning, it can be concluded that the pattern has rotational symmetry.

Rotation Invariance Theorem

<note>The center point in the diagram is marked as O</note>, if we rotate <note>around this center point by 60 degrees or its multiple angles (such as 120 degrees, 180 degrees, etc.)</note>, <note>the entire figure will appear exactly the same as before the rotation</note>. <note>The rotation angle is marked as 60 degrees</note>, after each rotation by this angle, each part of the figure such as <note>a, b, c, d, e, f (six equal units)</note> will coincide with its position before the rotation.


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

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

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

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

Answer:
J'(-3,-8)
K'(-9,-8)
L'(0,-2)

1. Given the coordinates of point J as (-8, 3), we need to rotate it counterclockwise by 90° around the origin (0, 0).

2. According to the rotation formula in the 2D plane, the new coordinates of point (x, y) after a 90° counterclockwise rotation are (-y, x).

3. Applying the formula to point J, the new coordinates of J are J' = (-3, -8).

4. Similarly, for point K with coordinates (-8, 9), we apply the same rotation formula.

5. After the transformation, the new coordinates of K are K' = (-9, -8).

6. For point L with coordinates (-2, 0), we apply the rotation formula to get its new coordinates.

7. The calculation gives the new coordinates of L as L' = (0, -2).

8. Based on the above reasoning, the final answer is J'(-3, -8), K'(-9, -8), L'(0, -2).

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>Each point of triangle JKL is rotated 90° counterclockwise around the origin (0, 0)</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).

<note>Point J(-8, 3)</note> rotates to the new coordinates <note>J'(-3, -8)</note>, <note>Point K(-8, 9)</note> rotates to the new coordinates <note>K'(-9, -8)</note>, <note>Point L(-2, 0)</note> rotates to the new coordinates <note>L'(0, -2)</note>.


Question:
Does this picture have rotational symmetry?

- yes
- no

Answer: - no

1. Observe the overall structure of the figure and find that its basic shape consists of a shovel head and a handle.

2. According to the definition of rotational symmetry, if a figure has rotational symmetry, it will completely overlap with its initial state under a rotation of less than 360 degrees.

3. The shovel head in the figure has a fixed direction relative to the handle, and one end of the handle has a characteristic curved grip.

4. Assume the figure is rotated 90 degrees, 180 degrees, and 270 degrees clockwise or counterclockwise.

5. After rotating 90 degrees, the direction of the shovel head will change and will not overlap with the initial position.

6. After rotating 180 degrees, the shovel head will be upside down, and the direction of the handle will be different from the original, still not overlapping.

7. After rotating 270 degrees, the direction of the shovel head continues to change, and the position of the handle does not match the initial direction.

8. Through the above attempts, it is found that no matter how the shovel is rotated less than 360 degrees, the figure cannot completely overlap with the initial state.

9. Therefore, it can be concluded that the figure does not have rotational symmetry.

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

Rotation Invariance Theorem

The position of <note>the shovel head</note> and <note>the handle</note> do not coincide with the initial position after rotating <note>90 degrees, 180 degrees, and 270 degrees</note>, indicating that the figure does not satisfy the <note>Rotation Invariance Theorem</note>, and the final answer is: no.

Definition of Rotational Symmetry

In this problem diagram, the figure cannot be obtained by rotating 90 degrees clockwise/counterclockwise to get <note>a figure identical to the original figure</note>, so the figure does not have rotational symmetry.


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

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

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

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

G'( $\Box$ , $\Box$ )

Answer:
D'(-2,-8)
E'(-9,-8)
F'(-9,-4)
G'(-2,-4)

1. Given the coordinates of point D as (-8,2), we need to rotate this point counterclockwise by 90° around the origin. The new coordinates can be obtained using the rotation matrix: for any point (x,y), the rotated coordinates are (-y,x). This is based on the two-dimensional plane rotation formula.

2. Applying the above rotation formula to point D: for D(-8,2), the rotated coordinates are D'(-2,-8).

3. The same method is applied to point E. The original coordinates of point E are (-8,9). Using the rotation formula, the coordinates of E' become (-9,-8).

4. Continuing to apply the rotation to point F. The original coordinates of point F are (-4,9). After the rotation transformation, the coordinates of F' are (-9,-4).

5. Finally, applying the same steps to point G. The initial coordinates of point G are (-4,2). After the same calculation, the coordinates of G' become (-2,-4).

6. Through the above reasoning, the final answer is D'(-2,-8), E'(-9,-8), F'(-9,-4), G'(-2,-4).

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 (-8,2)</note>, <note>The coordinates of point E are (-8,9)</note>, <note>The coordinates of point F are (-4,9)</note>, <note>The coordinates of point G are (-4,2)</note>.

2D Plane Rotation Formula

<note>Point D(-8,2) is rotated 90° counterclockwise to get D'(-2,-8)</note>; <note>Point E(-8,9) is rotated 90° counterclockwise to get E'(-9,-8)</note>; <note>Point F(-4,9) is rotated 90° counterclockwise to get F'(-9,-4)</note>; <note>Point G(-4,2) is rotated 90° counterclockwise to get G'(-2,-4)</note>.

Rotation Transformation

Each point of <note>rectangle DEFG</note> is rotated counterclockwise by <note>90°</note> around <note>the origin (0, 0)</note>.


Question:
Write the coordinates of the vertices after a rotation 180° counterclockwise around the origin.

P'( $\Box$ , $\Box$ )

Q'( $\Box$ , $\Box$ )

R'( $\Box$ , $\Box$ )

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

Answer:
P'(-5,4)
Q'(-5,1)
R'(-10,1)
S'(-10,4)

1. Given the coordinates of point P are (5, -4), it needs to be rotated 180° counterclockwise around the origin.

2. According to the two-dimensional plane rotation formula, after a point (x, y) is rotated 180°, its new coordinates are (-x, -y).

3. Applying this formula to point P(5, -4), its new coordinates after rotation are P'(-5, 4).

4. Similarly, for point Q(5, -1), applying the two-dimensional plane rotation formula, we get Q'(-5, 1).

5. Likewise, for point R(10, -1), applying the two-dimensional plane rotation formula, we get R'(-10, 1).

6. Finally, for point S(10, -4), applying the two-dimensional plane rotation formula, we get S'(-10, 4).

7. Based on the above reasoning, the final coordinates of the four vertices after rotation are: P'(-5, 4), Q'(-5, 1), R'(-10, 1), S'(-10, 4).

<note>Point P(5, -4)</note>, <note>Point Q(5, -1)</note>, <note>Point R(10, -1)</note>, and <note>Point S(10, -4)</note> all rotate counterclockwise around the <note>origin</note> by <note>180°</note>.

2D Plane Rotation Formula

<note>In the figure of this problem, the point (x, y) after a 180° rotation</note> gives the new coordinates as <note>(-x, -y)</note>. For the point <note>P(5, -4)</note>, substituting into the rotation formula gives <note>P' coordinates: x' = -5, y' = 4</note>, so <note>P' coordinates are (-5, 4)</note>. Similarly for <note>Q, R, S</note>. Finally, the rotated vertex coordinates of <note>quadrilateral QRSP</note> are: <note>P' (-5, 4)</note>, <note>Q' (-5, 1)</note>, <note>R' (-10, 1)</note>, <note>S' (-10, 4)</note>.


Question:
Is parallelogram GHIJ a square?

- yes
- no

Answer: - no

1. Given conditions: HK = JK = 21 and GK = IK = 22. To determine if parallelogram GHIJ is a square, we need to verify if GHIJ is both a rhombus and has a right angle.

2. In GHIJ, check the lengths of diagonals GI and HJ, where GI = GK + IK = 22 + 22 = 44, HJ = HK + JK = 21 + 21 = 42.

3. According to the properties of a rectangle, the diagonals of a rectangle are equal and bisect each other. Therefore, in parallelogram GHIJ, the diagonals GI ≠ HJ, GHIJ is not a rectangle, and certainly not a square.

4. Since a square is a special type of rectangle, and the diagonals of GHIJ are not equal, it cannot be a square.

5. Based on the above reasoning, the final conclusion is that GHIJ is not a square.

Definition of Parallelogram

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

In the figure of this problem, the quadrilateral <note>GHIJ</note> is a parallelogram, with side <note>GH</note> parallel and equal to side <note>IJ</note>, and side <note>GI</note> parallel and equal to side <note>HJ</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 (≅).

<note>The diagonals GI and HJ of the parallelogram GHIJ are not equal</note>, therefore GHIJ cannot be a rectangle because <note>the diagonals of a rectangle should be equal</note>.

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.

In the figure of this problem, the quadrilateral <note>GHIJ</note> has all sides <note>GH, HI, IJ, JG are equal</note>, so the quadrilateral GHIJ is a rhombus. In addition, the diagonals of the quadrilateral GHIJ <note>GI and HJ are perpendicular bisectors</note>, that is, the diagonals GI and HJ intersect at <note>point K</note>, and the angle <note>GKH is a right angle (90 degrees)</note>, and <note>GK=KI</note> and <note>HK=KJ</note>.

Definition of Square

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

Because <note>the diagonals GI and HJ of GHIJ are not equal</note>, and <note>the lengths of the four sides GH, HI, IJ, and JG are not equal</note>, therefore <note>GHIJ is not a square</note>.

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 the parallelogram <note>GHIJ</note>, the opposite angles <note>∠GHI</note> and <note>∠JIH</note> are equal, and the opposite angles <note>∠HGJ</note> and <note>∠IJG</note> are equal; the sides <note>GH</note> and <note>IJ</note> are equal, and the sides <note>GI</note> and <note>HJ</note> are equal; the diagonals <note>GI</note> and <note>HJ</note> bisect each other, that is, the intersection point <note>K</note> divides the diagonal <note>GI</note> into two equal segments <note>GK</note> and <note>IK</note>, and divides the diagonal <note>HJ</note> into two equal segments <note>HK</note> and <note>JK</note>.

Property of Diagonals in a Rectangle

In the figure of this problem, <note>the diagonals in a rectangle are equal and bisect each other</note>. In the parallelogram GHIJ, <note>GI ≠ HJ</note>, so <note>GHIJ does not satisfy the property of equal diagonals</note>, thus it is not a rectangle.


Question:
Write the coordinates of the vertices after a rotation 180° counterclockwise around the origin.

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

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

W'( $\Box$ , $\Box$ )

Answer:
U'(8,5)
V'(7,5)
W'(6,10)

1. Given the coordinates of point U as (-8, -5), V as (-7, -5), and W as (-6, -10), we need to rotate these points 180° counterclockwise about the origin.

2. According to the 2D plane rotation formula, x' = xcos(θ) - ysin(θ), y' = xsin(θ) + ycos(θ), θ=180°, since cos180°=-1 and sin180°=0, the formula for 180° counterclockwise rotation is (x, y) -> (-x, -y).

3. Calculate the coordinates of point U after 180° counterclockwise rotation:

   According to the rotation formula, U' = (-(-8), -(-5)) = (8, 5).

4. Calculate the coordinates of point V after 180° counterclockwise rotation:

   According to the rotation formula, V' = (-(-7), -(-5)) = (7, 5).

5. Calculate the coordinates of point W after 180° counterclockwise rotation:

   According to the rotation formula, W' = (-(-6), -(-10)) = (6, 10).

6. From the above calculations, we obtain the coordinates of the rotated points as U'(8, 5), V'(7, 5), W'(6, 10).

Based on the above reasoning, the final answer is U'(8, 5), V'(7, 5), W'(6, 10).

Rotation Transformation

In the figure of this problem, <note>points U, V, and W</note> need to undergo a <note>180° counterclockwise rotation transformation around the origin</note> respectively. <note>The origin as the center of rotation</note>, after the rotation transformation, the new point coordinates can be calculated according to the formula <note>(u, v) -> (-u, -v)</note>.

2D Plane Rotation Formula

When <note>θ=180°</note>, cos180°=-1, sin180°=0, the formula simplifies to x'=x*cos(180°) - y*sin(180°)=-x, y'=x*sin(180°) + y*cos(180°)=-y. Therefore, <note>the formula for rotating 180° counterclockwise is (x, y) -> (-x, -y)</note>. <note>The coordinates of point U are (-8, -5)</note>, according to the rotation formula, U' = (-(-8), -(-5)) = (8, 5). <note>The coordinates of point V are (-7, -5)</note>, according to the rotation formula, V' = (-(-7), -(-5)) = (7, 5). <note>The coordinates of point W are (-6, -10)</note>, according to the rotation formula, W' = (-(-6), -(-10)) = (6, 10). These calculations use the rotation formula theorem and derive the new coordinates of the rotated points.


Question:
Write the coordinates of the vertices after a rotation 180° counterclockwise around the origin.

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

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

G'( $\Box$ , $\Box$ )

H'( $\Box$ , $\Box$ )

Answer:
E'(6,-2)
F'(6,-10)
G'(0,-10)
H'(0,-2)

1. Given the coordinates of point E as (-6, 2), it needs to be rotated counterclockwise 180° around the origin.

2. According to the properties of rotational transformation, rotating a point (x, y) counterclockwise 180° around the origin changes its coordinates to (-x, -y).

3. Therefore, the coordinates of point E (-6, 2) after a 180° counterclockwise rotation become E'(6, -2).

4. Using the same method, the coordinates of point F are (-6, 10), and after a 180° counterclockwise rotation, F'(6, -10).

5. For point G(0, 10), applying the same rotation rule gives G'(0, -10).

6. For point H(0, 2), the same rotation gives H'(0, -2).

7. After the above rotation operations, the new coordinates of the vertices are: E'(6, -2), F'(6, -10), G'(0, -10), H'(0, -2).

Rotation Transformation

<note>The rotation center is the origin (0, 0)</note>, <note>the rotation angle is 180°</note>. Point E(-6, 2), point F(-6, 10), point G(0, 10), point H(0, 2) are transformed to obtain the coordinates <note>E'(6, -2)</note>, <note>F'(6, -10)</note>, <note>G'(0, -10)</note>, <note>H'(0, -2)</note>.


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

- yes
- no

Answer: - no

1. Observing the figure, we find that only one pair of angles is marked as equal in the given quadrilateral.

2. According to the properties theorem of parallelograms: In a parallelogram, opposite angles are equal, opposite sides are equal, and diagonals bisect each other. Here, only one pair of opposite angles is equal, which does not satisfy this theorem.

3. Another method to determine a quadrilateral as a parallelogram is: According to the definition of a parallelogram, a parallelogram is a quadrilateral with two pairs of opposite sides that are parallel and equal. However, there is no corresponding information in the figure to support these two conditions.

4. Due to the lack of sufficient information about the side lengths and other angles, it is impossible to confirm this quadrilateral as a parallelogram through the related definitions or theorems of parallelograms.

5. After the above analysis, the conclusion is that the given information is insufficient to prove this quadrilateral is a parallelogram.

Definition of Parallelogram

The two pairs of opposite sides of the quadrilateral do not have annotation information to support them being parallel and equal, thus they cannot satisfy the definition of parallelogram.

Properties of Parallelogram Theorem

In the figure of this problem, the quadrilateral has <note>only one pair of equal opposite angles</note>, <note>the other pair of opposite angles does not have any marked information supporting their equality</note>, thus it cannot satisfy the definition of a parallelogram.