Congruence & Similarity

Problem 1

$A B$ || $D E$ , $A C E$ and $B C D$ are straight lines. Given $A B=9$ cm, $A C=7.2$ cm, $C D=5.2$ cm, $D E=6$ cm. Find $B C$ .

See Solution

Problem 2

Find the scale factor of the side lengths of two similar squares with areas $16 \mathrm{~m}^{2}$ and $49 \mathrm{~m}^{2}$ .

See Solution

Problem 3

Nyatakan sama ada semua segi empat tepat adalah segi empat sama: benar atau palsu?

See Solution

Problem 4

Identify the term for two polygons that are the same shape and size.

See Solution

Problem 5

Find the base diameter of a container with height $51 \mathrm{~cm}$ and volume $3825 \pi \mathrm{cm}^{3}$ , given a similar one.

See Solution

Problem 6

An industrial container is 51 cm tall with volume $3825 \pi \mathrm{cm}^{3}$ . A similar container is 23.8 cm tall, diameter 14 cm.
(a) Find the base diameter of the industrial container.
(b) Calculate the capacity of the smaller cylindrical container.

See Solution

Problem 7

Considerați punctele coliniare $M, O, N$ și punctele $P$ și $Q$ de o parte și de alta a dreptei $M N$ . Demonstrați: a) $M Q \equiv N P$ ; b) $M P \equiv N Q$ ; c) $\triangle M P Q \equiv \triangle N Q P$ ; d) $\triangle M P N \equiv \triangle N Q M$ .

See Solution

Problem 8

If $\overline{D E} \| \overline{A C}$ , which proportion is true? A) $\frac{D A}{D E}=\frac{E C}{D E}$ B) $\frac{A C}{D E}=\frac{E C}{B E}$ C) $\frac{B E}{D E}=\frac{E C}{A C}$ D) $\frac{B E}{B C}=\frac{D E}{A C}$

See Solution

Problem 9

在 $\square A B C D$ 中, $A C, B D$ 交于点 $O$ , 点 $E, F$ 在 $A C$ 上, $A E=C F$ 。证明四边形 $E B F D$ 是平行四边形和菱形。

See Solution

Problem 10

Can Arshad construct a unique quadrilateral with sides $AB=5 \mathrm{~cm}$ , $\angle A=50^{\circ}$ , $AC=4 \mathrm{~cm}$ , $BD=5 \mathrm{~cm}$ , and $AD=6 \mathrm{~cm}$ ? Explain why or why not.

See Solution

Problem 11

Copy a figure twice, cut out, label as $A$ and $B$ , then check if side lengths and angles are the same to determine congruence.

See Solution

Problem 12

Due parallelogrammi $A B C D$ e $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ sono simili. Se $A B=30 \mathrm{~cm}$ , $B C=20 \mathrm{~cm}$ e $B^{\prime} C=16 \mathrm{~cm}$ , trova il rapporto di similitudine e $A^{\prime} B^{\prime}$ .

See Solution

Problem 13

Determine which postcard size matches the shape of a painting with dimensions 30.25 inches by 25.25 inches. Options:
1. 5 inches by 5 inches
2. 8 inches by 4 inches
3. 6.05 inches by 5.05 inches
Show your work.

See Solution

Problem 14

Which postcard matches the shape of a painting where the long side is 1.2 times the short side? Options: A (5x5), B (8x4), C (6.05x5.05).

See Solution

Problem 15

Two trees: one is 3m tall with an 18m shadow. Find the height of the other tree with a 39m shadow. Options: $12.5 \mathrm{~m}$ , $6.5 \mathrm{~m}$ , $3.25 \mathrm{~m}$ , $2.17 \mathrm{~m}$ .

See Solution

Problem 16

Identify the contrapositive: If two figures are similar, then all corresponding angles are equal.

See Solution

Problem 17

If $\triangle ABC \sim \triangle XYZ$ , complete this proportion: $\frac{BC}{AC}=\frac{?}{?}$ . Choices are A, B, C, D, or E.

See Solution

Problem 18

Find the surface area of a similar rectangular solid with width 10, given the original has dimensions 16, 20, and area 4480.

See Solution

Problem 19

If $B$ is the midpoint of $\overline{AC}$ , $D$ is the midpoint of $\overline{CE}$ , and $\overline{AB} \cong \overline{DE}$ , prove that $AE=4AB$ .

See Solution

Problem 20

Quadrilaterals $ABCD$ and $PQRS$ are scaled copies. If $AC = 6$ and $PR = 3$ , find $QS$ .

See Solution

Problem 21

Identify true statements for a scaled copy $Q$ of polygon $P$ : A) whole number sides, B) whole number angles, D) side lengths scaled by $\frac{1}{5}$ .

See Solution

Problem 22

Find the length $x$ of $\overline{TP}$ if pentagons $JKLMN$ and $PQRST$ are similar, with $QR=3.6$ , $QP=0.9$ , $RS=0.9$ , and $ST=1.8$ .

See Solution

Problem 23

What divides a line segment into two equal parts? A. hashmarks B. arcs C. angle bisector D. midpoint

See Solution

Problem 24

Find the length of $\overline{F G}$ if $M$ is the midpoint and $F M=12 x-4$ , $M G=5 x+10$ .

See Solution

Problem 25

If $\triangle ABC \cong \triangle DEF$ and $AB=18, BC=10, AC=25$ , what is the length of $DF$ ? A. Cannot be determined B. 18 C. 10 D. 25

See Solution

Problem 26

Which polygon is also a rectangle: kite, rhombus, trapezoid, or square?

See Solution

Problem 27

Find $W Y$ given that $W$ is the midpoint of segment $V Y$ , with $V W=9 x+7$ and $W Y=16 x-28$ .

See Solution

Problem 28

Find $W Y$ given $W$ is the midpoint of segment $VY$ , where $V W=9 x+7$ and $W Y=16 x-28$ .

See Solution

Problem 29

Find $x$ given that $Q$ is the midpoint of segment $PR$ , $QR=18$ , and $PR=5x+6$ .

See Solution

Problem 30

In $\triangle PQR \cong \triangle STR$ , find $\angle PRQ \cong$ A. $\angle RST$ B. $\angle STR$ C. $\angle SRT$ D. $\angle T$

See Solution

Problem 31

In congruent triangles $\triangle ABC$ and $\triangle STR$ , complete $\overline{BC} \cong \_$ . Options: A. $\overline{ST}$ B. $\overline{SR}$ C. $\overline{TR}$ D. $\overline{AC}$

See Solution

Problem 32

In $\triangle ABC \cong \triangle STR$ , complete $\overline{CA} \cong$ ____. Options: A. $\overline{AC}$ B. $\overline{TR}$ C. $\overline{RS}$ D. $\overline{ST}$

See Solution

Problem 33

A tree's shadow is 24 ft, and a 4-ft post casts a 6-ft shadow. What is the height of the tree?

See Solution

Problem 34

Find $x$ , $K L$ , and $J L$ given that $K$ is the midpoint of $J L$ where $J L=4x-2$ and $J K=7$ .

See Solution

Problem 35

Divide the line segment $\overline{P Q}$ into four equal parts using a compass and straightedge. Choose the correct method.

See Solution

Problem 36

Which option does NOT describe bisecting: a. Divide into 2 equal parts, b. Split in half, c. Split into 2 congruent pieces, d. Add 2 to all sides/angles?

See Solution

Problem 37

Explain why $\overline{B D} \cong \overline{B D}$ in the proof of $\angle A \simeq \angle C$ in $\triangle A B C$ .

See Solution

Problem 38

Quadrilaterals $ABCD$ and $HJKL$ are congruent. Find which angles are congruent: $\angle K \cong$ (A, D, C, or L).

See Solution

Problem 39

Quadrilateral ABCD is congruent to HJKL. Complete: $\overline{J K} \cong$ options: a. $\overline{B C}$ , b. $\overline{C B}$ , c. $\overline{H L}$ , d. $\overline{K J}$ .

See Solution

Problem 40

Pentagon ABCDE is congruent to HJKLP. Complete the congruent statements: $\overline{B A} \cong$ (a) $\overline{H P}$ , (b) $\overline{J P}$ , (c) $\overline{J H}$ , (d) $\overline{J K}$ .

See Solution

Problem 41

Pentagons ABCDE and HJKLP are congruent. Complete: $\overline{C D} \cong$ with options: a) $\overline{L P}$ , b) $\overline{A B}$ , c) $\overline{H J}$ , d) $\overline{K L}$ .

See Solution

Problem 42

Quadrilaterals ABCD and HJKL are congruent. Find $\angle K$ congruent to which angles: A, D, C, or L? Explain your reasoning.

See Solution

Problem 43

What condition makes $\triangle A E D \simeq \triangle C E B$ true? a. $\angle A E D \cong \angle C E B$ b. $\angle E A D \cong \angle E C B$ c. $\angle E D A \cong \angle E B C$ d. $\angle D E C \cong \angle B E A$

See Solution

Problem 44

Which statement proves $\triangle D E F \cong \triangle A B C$ ? a. $A B=D E$ , $B C=E F$ b. $\angle D \cong \angle A$ , $\angle B \cong \angle E$ , $\angle C \cong \angle F$ c. Rigid motions map $A$ to $D$ , $A B$ to $D E$ , $\angle B$ to $\angle E$ d. Rigid motions map $A B$ to $D E$ , $B C$ to $E F$ , $A C$ to $D F$ .

See Solution

Problem 45

The triangles $\triangle A B C$ and $\triangle D E F$ have proportional sides: $\frac{A B}{D E}=\frac{B C}{E F}=\frac{A C}{D F}$ . What is their relationship?

See Solution

Problem 46

Ariadne's shadow is 15 ft and she's 5 ft tall. If Dixon's shadow is 18 ft, find Dixon's height using similar triangles.

See Solution

Problem 47

Translate triangle $\triangle FGH$ down 2 units and left 5 units to get triangle $KLM$ . Complete the congruence statements:
$\begin{aligned} \overline{GH} \cong & \\ \angle K \cong & \\ \overline{MK} \cong & \\ H \cong & \\ \end{aligned}$

See Solution

Problem 48

What does $\sim$ signify? a. Congruency b. Similarity c. Equal measure d. Almost equal

See Solution

Problem 49

Triangle XYZ is similar to Triangle TUV.

See Solution

Problem 50

If $\triangle A F G \sim \triangle R H T$ , find $\angle T \cong$ a. $\angle A$ b. $\angle F$ c. $\angle G$ d. $\angle R$ .

See Solution

Problem 51

Sally's triangle has angles $50^{\circ}$ and $60^{\circ}$ ; Tom's has $50^{\circ}$ and $70^{\circ}$ . Are they similar?

See Solution

Problem 52

Given $\triangle \mathrm{STV} \sim \triangle \mathrm{WQX}$ , find the missing value in $\frac{S T}{T V}=\frac{W Q}{\square}$ . Choices: a. ST, b. XW, c. VS, d. $Q X$

See Solution

Problem 53

If $\triangle \mathrm{AFG} \sim \triangle \mathrm{RHT}$ , find $\angle A \cong$ which corresponds to which angle? a. $\angle \mathrm{R}$ b. $\angle \mathrm{H}$ c. $\angle T$ d. $\angle G$

See Solution

Problem 54

Triangles $A B C$ and $D E F$ have sides $A B=9$ , $B C=15$ , $D E=6$ , $E F=10$ , and $\angle B \cong \angle E$ . Which is true? a. $\angle C A B \cong \angle D E F$ b. $\frac{A B}{C B}=\frac{F E}{D E}$ c. $\triangle A B C \sim \triangle D E F$ d. $\frac{A B}{D E}=\frac{F E}{C B}$

See Solution

Problem 55

Which statement is true for triangles $A B C$ and $D E F$ with $A B=9, B C=15, D E=6, E F=10$ , and $\angle B \cong \angle E$ ? a. $\angle C A B \cong \angle D E F$ b. $\frac{A B}{C B}=\frac{F E}{D E}$ c. $\triangle A B C \cong \triangle D E F$ d. $\frac{A B}{D E}=\frac{F E}{C B}$

See Solution

Problem 56

Are triangles $\triangle ABC$ and $\triangle DEF$ similar given $\angle A=30^{\circ}$ , $\angle D=30^{\circ}$ , $\angle F=38^{\circ}$ ? Justify.

See Solution

Problem 57

Determine what information is needed to show $\triangle L M N$ can be mapped onto $\triangle O P Q$ . Options: a. $O Q=6$ , b. $M N=9$ , c. $\angle L M N \cong \angle Q O P$ , d. $\angle N L M \cong \angle Q O P$ .

See Solution

Problem 58

Pentagon ABCDE is similar to Pentagon RYMNT. Find the missing side: $\frac{A B}{B C}=\frac{R Y}{\square}$ . Options: A E, ER, TN, YM.

See Solution

Problem 59

If $\triangle \mathrm{AFG} \sim \triangle \mathrm{DRH}$ , find the missing value in $\frac{D R}{A F}=\frac{D H}{\square}$ . Choices: HD, AH, AG.

See Solution

Problem 60

Find the value of $x$ given $P M=2 x-5$ , $P N=6 x$ , and $M N=5 x+4$ .

See Solution

Problem 61

If $\overline{P Q} \cong \overline{Q R}$ with $P Q=3x-8$ , $Q R=2x$ , and $R S=1.5x+4$ , is $P S=24$ true or false?

See Solution

Problem 62

Pentagon $\mathrm{ABCDE}$ is similar to Pentagon RYMNT. Find $\frac{N T}{D E}=\frac{R T}{\square}$ using TN, AE, YM, ER.

See Solution

Problem 63

Why are congruent triangles common in architecture? Use math terms and give two complete sentences for full credit.

See Solution

Problem 64

Explain why SSA does not prove triangle congruence in your own words. Use math vocabulary and three complete sentences.

See Solution

Problem 65

Find $y$ given $R S=6x+5$ , $S T=8x-1$ , and $T U=\pi y+13$ with midpoints $S$ and $T$ .

See Solution

Problem 66

Find $x$ and the length of $PQ$ given $PQ=2x+1$ and $QR=5x-44$ , with $Q$ as the midpoint of $\overline{PR}$ .

See Solution

Problem 67

If $AB=49$ and $BC=22$ , find the length $AC$ .

See Solution

Problem 68

Show how to use rigid motions (translations, reflections, rotations) to prove $\triangle ABC \cong \triangle XYZ$ given $\angle A \cong \angle X$ and $\angle B \cong \angle Y$ .

See Solution

Problem 69

Given $AEC$ is a line and $\triangle ABC \cong \triangle DEC$ . Find $AB$ , $CD$ , $\angle BCA$ , and $y$ using given lengths and angles.

See Solution

Problem 70

In square $ABCD$ , points $E$ on $AB$ and $F$ on $BC$ satisfy $AF = DE$ . Show: (a) $\triangle ABF \cong \triangle DAE$ ; (b) Is $AF$ perpendicular to $DE$ ? Why? (c) Prove $\triangle ABF \sim \triangle AGE$ .

See Solution

Problem 71

Prove that $\triangle ABC \sim \triangle DEC$ given $\angle BAC = \angle CDE$ . Find if $\triangle ABC$ is right-angled and the area of $ABDE$ .

See Solution

Problem 72

In rectangle $ABCD$ , with $AB = 20$ and $BC = 15$ :
(a) Find $AC$ . (b) Prove $\triangle ABC \sim \triangle CMD$ . (c) Find $MC$ . (d) Find ratio $AM : MC$ .

See Solution

Problem 73

Find $R S$ if $S$ is the midpoint of $\overline{R T}$ , $R S=5 x+17$ , and $S T=8 x-31$ .

See Solution

Problem 74

Find $D E$ given that $B$ is the midpoint of $A C$ , $A C=C D$ , $A B=3 x+4$ , $A C=11 x-17$ , and $C E=49$ .

See Solution

Problem 75

Find $A C$ if line $y$ bisects $A C$ , where $A B=4-5 x$ and $B C=2 x+25$ . Solve: $4-5 x=2 x+25$

See Solution

Problem 76

Find the length of $A C$ if line $y$ bisects $A C$ , $A B=4-5 x$ , and $B C=2 x+25$ .

See Solution

Problem 77

A man is $2 \mathrm{~cm}$ tall in a photo and $1.8 \mathrm{~m}$ tall in reality. Find the scale factor.

See Solution

Problem 78

After securing the string at point A and setting its length over half of $AB$ , what should you do next?

See Solution

Problem 79

Identify the property: If $\angle Q \cong \angle S$ and $\angle S \cong \angle P$ , then $\angle Q \cong \angle P$ .

See Solution

Problem 80

Find the missing side length of two similar triangles with sides 36, 36, 18 and 24, 48.

See Solution

Problem 81

Polygon $A$ has sides $2.5$ , $2.5$ , $1.5$ , angles $53^\circ$ , $82^\circ$ . Polygon $B$ has one side $5$ .
a. Find the scale factor from $A$ to $B$ .
b. Calculate the unknown side lengths in $B$ .
c. Find the unknown angles in $A$ .

See Solution

Problem 82

Is Jackson correct to conclude that triangles are similar from $\frac{3}{6}=\frac{2}{4}$ ? Explain your reasoning.

See Solution

Problem 83

Is this shape a square? Choose the correct reason: A. Perpendicular sides, equal length. B. Parallel sides, equal length. C. Opposite sides not parallel. D. Sides not congruent.

See Solution

Problem 84

Identify corresponding points and sides in two $H$ -shaped polygons. Find the scale factor for the smaller copy.

See Solution

Problem 85

Find the model wingspan if the actual wingspan is 211 feet and the scale is 1 in: $40 \mathrm{ft}$ .

See Solution

Problem 86

Use the distance formula to check if $\overline{AB}$ and $\overline{CD}$ are congruent: $\sqrt{(-6-1)^{2}+(1--1)^{2}}$ and $\sqrt{(15-4)^{2}+(-4-3)^{2}}$ . Are they congruent?

See Solution

Problem 87

Check if line segments $\overline{AB}$ and $\overline{CD}$ are congruent using points $A(1,-1)$ , $B(-6,1)$ , $C(4,3)$ , $D(5,-4)$ . Use distance formula: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .

See Solution

Problem 88

Find $y$ if point $S$ is the midpoint of $\overline{R T}$ with $R S=6y+3$ and $S T=3y+9$ . Also find $R S$ and $R T$ .

See Solution

Problem 89

Find $K J$ if $K$ is the midpoint of $\overline{H J}$ , $H K=x+6$ , and $H J=4 x-6$ . A. 15 B. 9 C. 4 D. 10

See Solution

Problem 90

Prove that $\triangle A D C \cong \triangle E B C$ using reasons for statements 2, 4, and 5. Choose from I-V.

See Solution

Problem 91

Find the scale factor from Polygon A (sides 2.5, 2.5, 1.5) to Polygon B (side 5) using the formula: scale factor = $\frac{\text{side in B}}{\text{corresponding side in A}}$ .

See Solution

Problem 92

Classify the shape: opposite sides are parallel, all sides congruent, vertices not necessarily right angles. Options: Square, Rectangle, Rhombus, Parallelogram.

See Solution

Problem 93

Quadrilateral A has sides $6, 9, 9, 12$ . Quadrilateral B is a scaled copy with shortest side 2. Find the perimeter of B.

See Solution

Problem 94

Name the segment in polygon $P Q R S$ that corresponds to segment $A D$ from polygon $A B C D$ .

See Solution

Problem 95

Dos terrenos tienen igual área: uno cuadrado y otro rectangular. Si el lado del cuadrado se relaciona con el menor del rectangular como 3 a 2, ¿cuál es la relación de sus perímetros? A) $17 / 18$ B) $15 / 16$ C) $12 / 13$ D) $13 / 14$ E) $49 / 50$

See Solution

Problem 96

Ariadne's shadow is 15 ft, she's 5 ft tall. Dixon's shadow is 18 ft. How tall is Dixon? $15 \mathrm{ft}$ Dixon is feet tall.

See Solution

Problem 97

Find the missing value of $a$ if $M$ is the midpoint of $\overline{F G}$ with $F G=14 a+1$ and $F M=4.5$ .

See Solution

Problem 98

Quadrilateral A has sides 3, 6, 6, 9. Quadrilateral B scales A to shortest side 2. Does perimeter decrease by 4? (A) No (B) Yes. Explain.

See Solution

Problem 99

Figures R, S, and T are scaled copies. Find scale factors: T to S, S to R, R to T, and T to R.

See Solution

Problem 100

What fraction of Polygon B's area is Polygon A's area if B is a scaled copy of A with a scale factor of 5?

See Solution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Congruence & Similarity

Problem 1

ABA BAB || DED EDE, ACEA C EACE and BCDB C DBCD are straight lines. Given AB=9A B=9AB=9 cm, AC=7.2A C=7.2AC=7.2 cm, CD=5.2C D=5.2CD=5.2 cm, DE=6D E=6DE=6 cm. Find BCB CBC.

Problem 2

Find the scale factor of the side lengths of two similar squares with areas 16 m216 \mathrm{~m}^{2}16 m2 and 49 m249 \mathrm{~m}^{2}49 m2.

Problem 3

Nyatakan sama ada semua segi empat tepat adalah segi empat sama: benar atau palsu?

Problem 4

Identify the term for two polygons that are the same shape and size.

Problem 5

Find the base diameter of a container with height 51 cm51 \mathrm{~cm}51 cm and volume 3825πcm33825 \pi \mathrm{cm}^{3}3825πcm3, given a similar one.

Problem 6

An industrial container is 51 cm tall with volume 3825πcm33825 \pi \mathrm{cm}^{3}3825πcm3. A similar container is 23.8 cm tall, diameter 14 cm.(a) Find the base diameter of the industrial container. (b) Calculate the capacity of the smaller cylindrical container.

Problem 7

Problem 8

Problem 9

在 □ABCD\square A B C D□ABCD 中, AC,BDA C, B DAC,BD 交于点 OOO, 点 E,FE, FE,F 在 ACA CAC 上, AE=CFA E=C FAE=CF。证明四边形 EBFDE B F DEBFD 是平行四边形和菱形。

Problem 10

Can Arshad construct a unique quadrilateral with sides AB=5 cmAB=5 \mathrm{~cm}AB=5 cm, ∠A=50∘\angle A=50^{\circ}∠A=50∘, AC=4 cmAC=4 \mathrm{~cm}AC=4 cm, BD=5 cmBD=5 \mathrm{~cm}BD=5 cm, and AD=6 cmAD=6 \mathrm{~cm}AD=6 cm? Explain why or why not.

Problem 11

Copy a figure twice, cut out, label as AAA and BBB, then check if side lengths and angles are the same to determine congruence.

Problem 12

Problem 13

Determine which postcard size matches the shape of a painting with dimensions 30.25 inches by 25.25 inches. Options: 1. 5 inches by 5 inches2. 8 inches by 4 inches3. 6.05 inches by 5.05 inchesShow your work.

Problem 14

Which postcard matches the shape of a painting where the long side is 1.2 times the short side? Options: A (5x5), B (8x4), C (6.05x5.05).

Problem 15

Two trees: one is 3m tall with an 18m shadow. Find the height of the other tree with a 39m shadow. Options: 12.5 m12.5 \mathrm{~m}12.5 m, 6.5 m6.5 \mathrm{~m}6.5 m, 3.25 m3.25 \mathrm{~m}3.25 m, 2.17 m2.17 \mathrm{~m}2.17 m.

Problem 16

Identify the contrapositive: If two figures are similar, then all corresponding angles are equal.

Problem 17

If △ABC∼△XYZ\triangle ABC \sim \triangle XYZ△ABC∼△XYZ, complete this proportion: BCAC=??\frac{BC}{AC}=\frac{?}{?}ACBC​=??​. Choices are A, B, C, D, or E.

Problem 18

Find the surface area of a similar rectangular solid with width 10, given the original has dimensions 16, 20, and area 4480.

Problem 19

If BBB is the midpoint of AC‾\overline{AC}AC, DDD is the midpoint of CE‾\overline{CE}CE, and AB‾≅DE‾\overline{AB} \cong \overline{DE}AB≅DE, prove that AE=4ABAE=4ABAE=4AB.

Problem 20

Quadrilaterals ABCDABCDABCD and PQRSPQRSPQRS are scaled copies. If AC=6AC = 6AC=6 and PR=3PR = 3PR=3, find QSQSQS.

Problem 21

Identify true statements for a scaled copy QQQ of polygon PPP: A) whole number sides, B) whole number angles, D) side lengths scaled by 15\frac{1}{5}51​.

Problem 22

Find the length xxx of TP‾\overline{TP}TP if pentagons JKLMNJKLMNJKLMN and PQRSTPQRSTPQRST are similar, with QR=3.6QR=3.6QR=3.6, QP=0.9QP=0.9QP=0.9, RS=0.9RS=0.9RS=0.9, and ST=1.8ST=1.8ST=1.8.

Problem 23

What divides a line segment into two equal parts? A. hashmarks B. arcs C. angle bisector D. midpoint

Problem 24

Find the length of FG‾\overline{F G}FG if MMM is the midpoint and FM=12x−4F M=12 x-4FM=12x−4, MG=5x+10M G=5 x+10MG=5x+10.

Problem 25

If △ABC≅△DEF\triangle ABC \cong \triangle DEF△ABC≅△DEF and AB=18,BC=10,AC=25AB=18, BC=10, AC=25AB=18,BC=10,AC=25, what is the length of DFDFDF? A. Cannot be determined B. 18 C. 10 D. 25

Problem 26

Which polygon is also a rectangle: kite, rhombus, trapezoid, or square?

Problem 27

Find WYW YWY given that WWW is the midpoint of segment VYV YVY, with VW=9x+7V W=9 x+7VW=9x+7 and WY=16x−28W Y=16 x-28WY=16x−28.

Problem 28

Find WYW YWY given WWW is the midpoint of segment VYVYVY, where VW=9x+7V W=9 x+7VW=9x+7 and WY=16x−28W Y=16 x-28WY=16x−28.

Problem 29

Find xxx given that QQQ is the midpoint of segment PRPRPR, QR=18QR=18QR=18, and PR=5x+6PR=5x+6PR=5x+6.

Problem 30

In △PQR≅△STR\triangle PQR \cong \triangle STR△PQR≅△STR, find ∠PRQ≅\angle PRQ \cong∠PRQ≅ A. ∠RST\angle RST∠RST B. ∠STR\angle STR∠STR C. ∠SRT\angle SRT∠SRT D. ∠T\angle T∠T

Problem 31

In congruent triangles △ABC\triangle ABC△ABC and △STR\triangle STR△STR, complete BC‾≅_\overline{BC} \cong \_BC≅_. Options: A. ST‾\overline{ST}ST B. SR‾\overline{SR}SR C. TR‾\overline{TR}TR D. AC‾\overline{AC}AC

Problem 32

In △ABC≅△STR\triangle ABC \cong \triangle STR△ABC≅△STR, complete CA‾≅\overline{CA} \congCA≅ ____. Options: A. AC‾\overline{AC}AC B. TR‾\overline{TR}TR C. RS‾\overline{RS}RS D. ST‾\overline{ST}ST

Problem 33

A tree's shadow is 24 ft, and a 4-ft post casts a 6-ft shadow. What is the height of the tree?

Problem 34

Find xxx, KLK LKL, and JLJ LJL given that KKK is the midpoint of JLJ LJL where JL=4x−2J L=4x-2JL=4x−2 and JK=7J K=7JK=7.

Problem 35

Divide the line segment PQ‾\overline{P Q}PQ​ into four equal parts using a compass and straightedge. Choose the correct method.

Problem 36

Which option does NOT describe bisecting: a. Divide into 2 equal parts, b. Split in half, c. Split into 2 congruent pieces, d. Add 2 to all sides/angles?

Problem 37

Explain why BD‾≅BD‾\overline{B D} \cong \overline{B D}BD≅BD in the proof of ∠A≃∠C\angle A \simeq \angle C∠A≃∠C in △ABC\triangle A B C△ABC.

Problem 38

Quadrilaterals ABCDABCDABCD and HJKLHJKLHJKL are congruent. Find which angles are congruent: ∠K≅\angle K \cong∠K≅ (A, D, C, or L).

Problem 39

Quadrilateral ABCD is congruent to HJKL. Complete: JK‾≅\overline{J K} \congJK≅ options: a. BC‾\overline{B C}BC, b. CB‾\overline{C B}CB, c. HL‾\overline{H L}HL, d. KJ‾\overline{K J}KJ.

Problem 40

Pentagon ABCDE is congruent to HJKLP. Complete the congruent statements: BA‾≅\overline{B A} \congBA≅ (a) HP‾\overline{H P}HP, (b) JP‾\overline{J P}JP, (c) JH‾\overline{J H}JH, (d) JK‾\overline{J K}JK.

Problem 41

Pentagons ABCDE and HJKLP are congruent. Complete: CD‾≅\overline{C D} \congCD≅ with options: a) LP‾\overline{L P}LP, b) AB‾\overline{A B}AB, c) HJ‾\overline{H J}HJ, d) KL‾\overline{K L}KL.

$A B$ || $D E$ , $A C E$ and $B C D$ are straight lines. Given $A B=9$ cm, $A C=7.2$ cm, $C D=5.2$ cm, $D E=6$ cm. Find $B C$ .

Find the scale factor of the side lengths of two similar squares with areas $16 \mathrm{~m}^{2}$ and $49 \mathrm{~m}^{2}$ .

Find the base diameter of a container with height $51 \mathrm{~cm}$ and volume $3825 \pi \mathrm{cm}^{3}$ , given a similar one.

An industrial container is 51 cm tall with volume $3825 \pi \mathrm{cm}^{3}$ . A similar container is 23.8 cm tall, diameter 14 cm.
(a) Find the base diameter of the industrial container.
(b) Calculate the capacity of the smaller cylindrical container.

在 $\square A B C D$ 中, $A C, B D$ 交于点 $O$ , 点 $E, F$ 在 $A C$ 上, $A E=C F$ 。证明四边形 $E B F D$ 是平行四边形和菱形。

Can Arshad construct a unique quadrilateral with sides $AB=5 \mathrm{~cm}$ , $\angle A=50^{\circ}$ , $AC=4 \mathrm{~cm}$ , $BD=5 \mathrm{~cm}$ , and $AD=6 \mathrm{~cm}$ ? Explain why or why not.

Copy a figure twice, cut out, label as $A$ and $B$ , then check if side lengths and angles are the same to determine congruence.

Determine which postcard size matches the shape of a painting with dimensions 30.25 inches by 25.25 inches. Options:
1. 5 inches by 5 inches
2. 8 inches by 4 inches
3. 6.05 inches by 5.05 inches
Show your work.

Two trees: one is 3m tall with an 18m shadow. Find the height of the other tree with a 39m shadow. Options: $12.5 \mathrm{~m}$ , $6.5 \mathrm{~m}$ , $3.25 \mathrm{~m}$ , $2.17 \mathrm{~m}$ .

If $\triangle ABC \sim \triangle XYZ$ , complete this proportion: $\frac{BC}{AC}=\frac{?}{?}$ . Choices are A, B, C, D, or E.

If $B$ is the midpoint of $\overline{AC}$ , $D$ is the midpoint of $\overline{CE}$ , and $\overline{AB} \cong \overline{DE}$ , prove that $AE=4AB$ .

Quadrilaterals $ABCD$ and $PQRS$ are scaled copies. If $AC = 6$ and $PR = 3$ , find $QS$ .

Identify true statements for a scaled copy $Q$ of polygon $P$ : A) whole number sides, B) whole number angles, D) side lengths scaled by $\frac{1}{5}$ .

Find the length $x$ of $\overline{TP}$ if pentagons $JKLMN$ and $PQRST$ are similar, with $QR=3.6$ , $QP=0.9$ , $RS=0.9$ , and $ST=1.8$ .

Find the length of $\overline{F G}$ if $M$ is the midpoint and $F M=12 x-4$ , $M G=5 x+10$ .

If $\triangle ABC \cong \triangle DEF$ and $AB=18, BC=10, AC=25$ , what is the length of $DF$ ? A. Cannot be determined B. 18 C. 10 D. 25

Find $W Y$ given that $W$ is the midpoint of segment $V Y$ , with $V W=9 x+7$ and $W Y=16 x-28$ .

Find $W Y$ given $W$ is the midpoint of segment $VY$ , where $V W=9 x+7$ and $W Y=16 x-28$ .

Find $x$ given that $Q$ is the midpoint of segment $PR$ , $QR=18$ , and $PR=5x+6$ .

In $\triangle PQR \cong \triangle STR$ , find $\angle PRQ \cong$ A. $\angle RST$ B. $\angle STR$ C. $\angle SRT$ D. $\angle T$

In congruent triangles $\triangle ABC$ and $\triangle STR$ , complete $\overline{BC} \cong \_$ . Options: A. $\overline{ST}$ B. $\overline{SR}$ C. $\overline{TR}$ D. $\overline{AC}$

In $\triangle ABC \cong \triangle STR$ , complete $\overline{CA} \cong$ ____. Options: A. $\overline{AC}$ B. $\overline{TR}$ C. $\overline{RS}$ D. $\overline{ST}$

Find $x$ , $K L$ , and $J L$ given that $K$ is the midpoint of $J L$ where $J L=4x-2$ and $J K=7$ .

Divide the line segment $\overline{P Q}$ into four equal parts using a compass and straightedge. Choose the correct method.

Explain why $\overline{B D} \cong \overline{B D}$ in the proof of $\angle A \simeq \angle C$ in $\triangle A B C$ .

Quadrilaterals $ABCD$ and $HJKL$ are congruent. Find which angles are congruent: $\angle K \cong$ (A, D, C, or L).

Quadrilateral ABCD is congruent to HJKL. Complete: $\overline{J K} \cong$ options: a. $\overline{B C}$ , b. $\overline{C B}$ , c. $\overline{H L}$ , d. $\overline{K J}$ .

Pentagon ABCDE is congruent to HJKLP. Complete the congruent statements: $\overline{B A} \cong$ (a) $\overline{H P}$ , (b) $\overline{J P}$ , (c) $\overline{J H}$ , (d) $\overline{J K}$ .

Pentagons ABCDE and HJKLP are congruent. Complete: $\overline{C D} \cong$ with options: a) $\overline{L P}$ , b) $\overline{A B}$ , c) $\overline{H J}$ , d) $\overline{K L}$ .

Quadrilaterals ABCD and HJKL are congruent. Find $\angle K$ congruent to which angles: A, D, C, or L? Explain your reasoning.

The triangles $\triangle A B C$ and $\triangle D E F$ have proportional sides: $\frac{A B}{D E}=\frac{B C}{E F}=\frac{A C}{D F}$ . What is their relationship?

What does $\sim$ signify? a. Congruency b. Similarity c. Equal measure d. Almost equal

If $\triangle A F G \sim \triangle R H T$ , find $\angle T \cong$ a. $\angle A$ b. $\angle F$ c. $\angle G$ d. $\angle R$ .

Sally's triangle has angles $50^{\circ}$ and $60^{\circ}$ ; Tom's has $50^{\circ}$ and $70^{\circ}$ . Are they similar?

Given $\triangle \mathrm{STV} \sim \triangle \mathrm{WQX}$ , find the missing value in $\frac{S T}{T V}=\frac{W Q}{\square}$ . Choices: a. ST, b. XW, c. VS, d. $Q X$

If $\triangle \mathrm{AFG} \sim \triangle \mathrm{RHT}$ , find $\angle A \cong$ which corresponds to which angle? a. $\angle \mathrm{R}$ b. $\angle \mathrm{H}$ c. $\angle T$ d. $\angle G$

Are triangles $\triangle ABC$ and $\triangle DEF$ similar given $\angle A=30^{\circ}$ , $\angle D=30^{\circ}$ , $\angle F=38^{\circ}$ ? Justify.

Determine what information is needed to show $\triangle L M N$ can be mapped onto $\triangle O P Q$ . Options: a. $O Q=6$ , b. $M N=9$ , c. $\angle L M N \cong \angle Q O P$ , d. $\angle N L M \cong \angle Q O P$ .

Pentagon ABCDE is similar to Pentagon RYMNT. Find the missing side: $\frac{A B}{B C}=\frac{R Y}{\square}$ . Options: A E, ER, TN, YM.

If $\triangle \mathrm{AFG} \sim \triangle \mathrm{DRH}$ , find the missing value in $\frac{D R}{A F}=\frac{D H}{\square}$ . Choices: HD, AH, AG.

Find the value of $x$ given $P M=2 x-5$ , $P N=6 x$ , and $M N=5 x+4$ .

If $\overline{P Q} \cong \overline{Q R}$ with $P Q=3x-8$ , $Q R=2x$ , and $R S=1.5x+4$ , is $P S=24$ true or false?

Pentagon $\mathrm{ABCDE}$ is similar to Pentagon RYMNT. Find $\frac{N T}{D E}=\frac{R T}{\square}$ using TN, AE, YM, ER.

Find $y$ given $R S=6x+5$ , $S T=8x-1$ , and $T U=\pi y+13$ with midpoints $S$ and $T$ .

Find $x$ and the length of $PQ$ given $PQ=2x+1$ and $QR=5x-44$ , with $Q$ as the midpoint of $\overline{PR}$ .

If $AB=49$ and $BC=22$ , find the length $AC$ .

Show how to use rigid motions (translations, reflections, rotations) to prove $\triangle ABC \cong \triangle XYZ$ given $\angle A \cong \angle X$ and $\angle B \cong \angle Y$ .

Given $AEC$ is a line and $\triangle ABC \cong \triangle DEC$ . Find $AB$ , $CD$ , $\angle BCA$ , and $y$ using given lengths and angles.

In square $ABCD$ , points $E$ on $AB$ and $F$ on $BC$ satisfy $AF = DE$ . Show: (a) $\triangle ABF \cong \triangle DAE$ ; (b) Is $AF$ perpendicular to $DE$ ? Why? (c) Prove $\triangle ABF \sim \triangle AGE$ .

Prove that $\triangle ABC \sim \triangle DEC$ given $\angle BAC = \angle CDE$ . Find if $\triangle ABC$ is right-angled and the area of $ABDE$ .

In rectangle $ABCD$ , with $AB = 20$ and $BC = 15$ :
(a) Find $AC$ . (b) Prove $\triangle ABC \sim \triangle CMD$ . (c) Find $MC$ . (d) Find ratio $AM : MC$ .

Find $R S$ if $S$ is the midpoint of $\overline{R T}$ , $R S=5 x+17$ , and $S T=8 x-31$ .

Find $D E$ given that $B$ is the midpoint of $A C$ , $A C=C D$ , $A B=3 x+4$ , $A C=11 x-17$ , and $C E=49$ .