Geometry Tutor Worksheet 10 Similar Triangles€¦ · Answer: The triangles are not similar. 5. The...

1 © MathTutorDVD.com

Geometry Tutor

Worksheet 10

Similar Triangles


Geometry Tutor - Worksheet 10 – Similar Triangles

1. Are the triangles in this figure similar? If so, give the reason for similarity and

complete the similarity statement below.

Reason: ______, ∆𝐵𝐶𝐸~∆______



Reason: ______, ∆𝑈𝑇𝑉~∆______




Reason: ______, ∆𝑈𝑇𝑆~∆______




Reason: ______, ∆𝐷𝐸𝑈~∆______

5. The triangles in below are similar. Give the reason for similarity, complete the

similarity statement, and find the missing value.

Reason: ______, ∆𝐽𝐾𝐿~∆_____, ______




Reason: ______, ∆𝑇𝑈𝑉~∆______



Reason: ______, ∆𝑇𝑈𝑉~∆______




Reason: ______, ∆𝐵𝐶𝐾~∆______

9. The triangles in below are similar. Complete the similarity statement, and find

the missing value.

∆𝑇𝑈𝑉~∆_____, ______




Reason: ______, ∆𝐹𝐺𝐻~∆______



Reason: ______, ∆𝐻𝑆𝑇~∆______



the value of 𝑥.

∆𝐷𝐸𝐹~∆______ , 𝑥 = ______



Reason: ______, ∆𝐶𝐸𝐷~∆______



the value of 𝑥.

∆𝑈𝑉𝑊~∆______ , 𝑥 = ______


similarity statement, and find the value of 𝑥.

Reason: ______, ∆𝑇𝑈𝐽~∆______ , 𝑥 = ______


16. Are the triangles in this figure similar? If so, give the reasons and complete the

similarity statement below.

Reason: ______, ∆𝑈𝑉𝑊~∆______



Reason: ______, ∆𝐸𝑅𝑆~∆______




Reason: ______, ∆𝑈𝐵𝐶~∆_____, ______



Reason: ______, ∆𝑄𝑅𝑃~∆_____, ______



the value of 𝑥.

∆𝑅𝑆𝑇~∆_____, 𝑥 = ______


Answers - Geometry Tutor - Worksheet 10 – Similar Triangles



The figure shows that the two triangles share a vertical pair of angles, so they

have one congruent angle in common, but the markings also show that there are

no other congruent corresponding angles.

Answer: The two triangles are not similar.



The figure shows that the two triangles have two pairs of congruent

corresponding angles Thus, they are similar according to the AA Triangle Similarity

Postulate.

Corresponding angles are ∠𝑈 𝑎𝑛𝑑 ∠𝐿, ∠𝑇 𝑎𝑛𝑑 ∠𝐾, ∠𝑉 𝑎𝑛𝑑 ∠𝐽.

Answer: Reason: AA, ∆𝑈𝑇𝑉~∆𝐿𝐾𝐽





have one congruent angle in common, and the markings also show that there is

one other pair of congruent corresponding angles.

Corresponding angles are ∠𝑈 and ∠𝐶, ∠𝑇 and ∠𝐵, ∠𝑇𝑆𝑈 and ∠𝐵𝑆𝐶.

Answer: Reason: AA, ∆𝑈𝑇𝑆~∆𝐶𝐵𝑆




The triangles share a pair of vertical angles which is the included angle between

two pairs of sides that we know the length of, so they have one corresponding

congruent angle. Then the ratios of the sides are

39

16 and

40

16

One ratio reduces, but the other ratio does not reduce. The result is

39

16and

5

2

The ratios are not equal.

Answer: The triangles are not similar.



The triangles have a pair of congruent angles which is the included angle between

two pairs of sides. The ratio of the corresponding sides must be equal because the

triangles are similar.

33

?=

42

28

Cross multiply to solve for the unknown value. The result is

33(28) = 42(? ); ? =33(28)

42= 22

Answer: Reason: SAS, ∆𝐽𝐾𝐿~∆𝑅𝑃𝑄, 22




The figure gives all three sides of both triangles, create ratios between the

shortest two sides, the middle two sides, and the longest two sides.

42

15?

70

25?

84

30

Simplify each ratio. The result is:

14

5=

14

5=

14

5

Notice that the ratios are all equal to each other. Thus, the triangles are similar

according to the SSS Triangle Similarity Postulate. Corresponding angles are

∠𝑇 and ∠𝑄, ∠𝑈 and ∠𝑅, ∠𝑉 and ∠𝑆.

Answer: Reason: SSS, ∆𝑇𝑈𝑉~∆𝑄𝑅𝑆




The triangles share a pair of vertical angles which is the included angle between

two pairs of sides that we know the lengths of, so they have one corresponding

congruent angle. Then the ratios of the sides are

14

49and

8

28

Both ratios reduce. The result is

2

7=

2

7

Notice that the ratios are equal. Therefore, the triangles are similar according to

the SAS Triangle Similarity Postulate.

Corresponding angles are ∠𝑇 and ∠𝑀, ∠𝑈 and ∠𝐿, ∠𝑇𝑉𝑈 and ∠𝑀𝑉𝐿.

Answer: Reason: SAS, ∆𝑇𝑈𝑉~∆𝑀𝐿𝑉




The figure shows that the two triangles share an angle, so they have one

congruent corresponding angle. Then, the upper side of the figure is 88 and the

side of the smaller triangle is 16. Putting the lengths together gives 𝐾𝑀 =

88 and 𝐾𝐿 = 132 on the lower side of the figure. The ratios of corresponding

sides of the two triangles are

16

88 and

25

132

One ratio simplifies, but the other one does not. The result is

1

5 and

25

132

The ratios are not equal, so the triangles are not similar.

Answer: The triangles are not similar.


9. The triangles in the figure below are similar. Complete the similarity statement,

and find the missing value.

The triangles are similar so set up a proportion equating the ratios of the

corresponding sides.

60

130=

?

117

Cross multiply to solve for the unknown value. The result is

60(117) = 130(? )

? =60(117)

130= 54

Corresponding angles are ∠𝑇 and ∠𝑀, ∠𝑈 and ∠𝐿, ∠𝑉 and ∠𝐾.

Answer: ∆𝑇𝑈𝑉~∆𝑀𝐿𝐾, 54




The figure gives all three sides of both triangles, create ratios between the

shortest two sides, the middle two sides, and the longest two sides.

8

48?

12

72?

14

84

Simplify each ratio. The result is:

1

6=

1

6=

1

6

Notice that the ratios are all equal to each other. Thus, the triangles are similar

according to the SSS Triangle Similarity Postulate. Corresponding angles are

∠𝐹 and ∠𝐴, ∠𝐺 and ∠𝐶, ∠𝐻 and ∠𝐵.

Answer: Reason: SSS, ∆𝐹𝐺𝐻~∆𝐴𝐶𝐵





have one congruent angle in common, and the figure also show that there is a 61°

angle in each triangle. Therefore, the triangles are similar by the AA Triangle

Similarity Postulate.

Corresponding angles are ∠𝐻 and ∠𝐺, ∠𝐻𝑆𝑇 and ∠𝐺𝑆𝐹, ∠𝑇 and ∠𝐹.

Answer: Reason: AA, ∆𝐻𝑆𝑇~∆𝐺𝑆𝐹


the value of 𝑥.




21

77=

30

11𝑥 + 11

Cross multiply to solve for 𝑥. The result is

21(11𝑥 + 11) = 77(30)

231𝑥 + 231 = 2310

231𝑥 = 2079

𝑥 = 9

Corresponding angles are ∠𝐷 and ∠𝐶, ∠𝐸 and ∠𝐵, ∠𝐹 and ∠𝐴.

Answer: ∆𝐷𝐸𝐹~∆𝐶𝐵𝐴 , 𝑥 = 9



The figure shows that the two triangles have a pair of congruent angles, so they

have one congruent corresponding angle. Then, the lengths of the sides around

that angle in each triangle are given. Set up the ratios of corresponding sides of

the two triangles which are

9

27 and

10

30


Both ratios simplify, with the result

1

3=

1

3

Notice that the simplified ratios are equal so the triangles are similar according to

the SAS Triangle Similarity Postulate.

Corresponding angles are ∠𝐶 and ∠𝑈, ∠𝐸 and ∠𝑇, ∠𝐷 and ∠𝑉.

Answer: Reason: SAS, ∆𝐶𝐸𝐷~∆𝑈𝑇𝑉


the value of 𝑥.



24

88=

18

5𝑥 + 11


24(5𝑥 + 11) = 88(18)

120𝑥 + 264 = 1584

120𝑥 = 1320

𝑥 = 11

Corresponding angles are ∠𝑈 and ∠𝑅, ∠𝑉 and ∠𝑆, ∠𝑊 and ∠𝑇.

Answer: ∆𝑈𝑉𝑊~∆𝑅𝑆𝑇 , 𝑥 = 11



similarity statement, and find the value of 𝑥.


congruent corresponding angle. Then, the upper side of the figure is 64 and the

side of the smaller triangle is 4𝑥 − 4, and the left side shows that 𝑈𝐿 = 72 and

𝑈𝑇 = 27. Putting the lengths together the ratios of corresponding sides of the

two triangles gives the proportion

4𝑥 − 4

64=

27

72


72(4𝑥 − 4) = 64(27)

288𝑥 − 288 = 1728

288𝑥 = 2016

𝑥 = 7

Corresponding angles are ∠𝑈𝑇𝐽 and ∠𝐿, ∠𝑈 and ∠𝑈, ∠𝑈𝐽𝑇 and ∠𝐾.

Answer: Reason: SAS, ∆𝑇𝑈𝐽~∆𝐿𝑈𝐾 , 𝑥 = 7


16. Are the triangles in this figure similar? If so, give the reasons and complete the

similarity statement below.

The triangles have a pair of congruent angles, 76°, which is the included angle

between two pairs of sides. The ratio of the corresponding sides must be equal

for the triangles to be similar.

9

21 ?

18

42

Simplify the ratios. The result is

3

7=

3

7

The simplified ratios are equal, so the triangles are similar according to the SAS

Triangle Similarity Postulate.

Corresponding angles are ∠𝑈 and ∠𝐺, ∠𝑉 and ∠𝐻, ∠𝑊 and ∠𝐹.

Answer: Reason: SAS, ∆𝑈𝑉𝑊~∆𝐺𝐻𝐹





congruent corresponding angle. Then, the upper side of the figure is 143 and part

of the side of the larger triangle is 91, so 𝐸𝑆 = 52. On the left side, part of the

side of the larger triangle is 56, so 𝐸𝑅 = 32. Putting the lengths together the

ratios of corresponding sides of the two triangles gives the proportion

52

143,32

88, and

44

121

Simplify the three ratios. The result is

4

11=

4

11=

4

11

All three ratios are the same, so the triangles are similar according to the SSS

Triangle Similarity Postulate. We could also use the SAS Triangle Similarity

Postulate.

Corresponding angles are ∠𝐸 and ∠𝐸, ∠𝐸𝑅𝑆 and ∠𝐷, ∠𝐸𝑆𝑅 and ∠𝐹.

Answer: SSS or SAS, ∆𝐸𝑅𝑆~∆𝐸𝐷𝐹





congruent corresponding angle. Then, the right side of the figure shows 𝑈𝑇 = 12

so 𝐶𝑇 = 9. Putting the lengths together the ratios of corresponding sides of the

two triangles gives the proportion

6

24=

3

12

The ratios simplify to the same fraction. The result is

1

4=

1

4

Corresponding angles are ∠𝑈 and ∠𝑈, ∠𝑇 and ∠𝑈𝐶𝐵, ∠𝑆 and ∠𝑈𝐵𝐶 .

Answer: Reason: SAS, ∆𝑈𝑇𝑆~∆𝑈𝐶𝐵 , 9





congruent corresponding angle. Then, the upper side of the figure shows that

𝑄𝑊 = 9 and 𝑄𝑅 = 18. The lower side shows that 𝑄𝑃 = 22 and 𝑄𝑉 is unknown.

Putting the lengths together the ratios of corresponding sides of the two triangles

gives the proportion

9

18=

?

22

Cross multiply to solve for the unknown. The result is

9(22) = 18(? )

198 = 18(? )

? =198

18

? = 11

Corresponding angles are ∠𝑄 and ∠𝑄, ∠𝑅 and ∠𝑄𝑊𝑉, ∠𝑃 and ∠𝑄𝑉𝑊. The

triangles are similar according to the SAS Triangle Similarity Postulate. Since all

three sides are given, we could also use SSS Triangle Similarity Postulate.

Answer: Reason: SAS or SSS, ∆𝑄𝑅𝑃~∆𝑄𝑊𝑉 , 11



the value of 𝑥.



70

50=

11𝑥 − 4

60


50(11𝑥 − 4) = 70(60)

550𝑥 − 200 = 4200

550𝑥 = 4400

𝑥 = 8

Corresponding angles are ∠𝑅 and ∠𝐷, ∠𝑆 and ∠𝐵, ∠𝑇 and ∠𝐶.

Answer: ∆𝑅𝑆𝑇~∆𝐷𝐵𝐶 , 𝑥 = 8

Geometry Tutor Worksheet 10 Similar Triangles€¦ · Answer: The triangles are not similar. 5. The...

Documents

Transcript of Geometry Tutor Worksheet 10 Similar Triangles€¦ · Answer: The triangles are not similar. 5. The...