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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~∆______
2. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
Reason: ______, ∆𝑈𝑇𝑉~∆______
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3. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
Reason: ______, ∆𝑈𝑇𝑆~∆______
4. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
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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~∆_____, ______
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6. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
Reason: ______, ∆𝑇𝑈𝑉~∆______
7. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
Reason: ______, ∆𝑇𝑈𝑉~∆______
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8. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
Reason: ______, ∆𝐵𝐶𝐾~∆______
9. The triangles in below are similar. Complete the similarity statement, and find
the missing value.
∆𝑇𝑈𝑉~∆_____, ______
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10. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
Reason: ______, ∆𝐹𝐺𝐻~∆______
11. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
Reason: ______, ∆𝐻𝑆𝑇~∆______
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12. The triangles in below are similar. Complete the similarity statement, and find
the value of 𝑥.
∆𝐷𝐸𝐹~∆______ , 𝑥 = ______
13. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
Reason: ______, ∆𝐶𝐸𝐷~∆______
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14. The triangles in below are similar. Complete the similarity statement, and find
the value of 𝑥.
∆𝑈𝑉𝑊~∆______ , 𝑥 = ______
15. The triangles in below are similar. Give the reason for similarity, complete the
similarity statement, and find the value of 𝑥.
Reason: ______, ∆𝑇𝑈𝐽~∆______ , 𝑥 = ______
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16. Are the triangles in this figure similar? If so, give the reasons and complete the
similarity statement below.
Reason: ______, ∆𝑈𝑉𝑊~∆______
17. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
Reason: ______, ∆𝐸𝑅𝑆~∆______
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18. The triangles in below are similar. Give the reason for similarity, complete the
similarity statement, and find the missing value.
Reason: ______, ∆𝑈𝐵𝐶~∆_____, ______
19. The triangles in below are similar. Give the reason for similarity, complete the
similarity statement, and find the missing value.
Reason: ______, ∆𝑄𝑅𝑃~∆_____, ______
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20. The triangles in below are similar. Complete the similarity statement, and find
the value of 𝑥.
∆𝑅𝑆𝑇~∆_____, 𝑥 = ______
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Answers - 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.
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.
2. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
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
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3. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
The figure shows that the two triangles share a vertical pair of angles, so they
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
4. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
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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.
5. The triangles in below are similar. Give the reason for similarity, complete the
similarity statement, and find the missing value.
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
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6. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
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
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7. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
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
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8. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
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.
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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
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10. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
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
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11. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
The figure shows that the two triangles share a vertical pair of angles, so they
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
12. The triangles in below are similar. Complete the similarity statement, and find
the value of 𝑥.
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The triangles are similar so set up a proportion equating the ratios of the
corresponding sides.
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
13. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
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
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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
14. The triangles in below are similar. Complete the similarity statement, and find
the value of 𝑥.
The triangles are similar so set up a proportion equating the ratios of the
corresponding sides.
24
88=
18
5𝑥 + 11
Cross multiply to solve for 𝑥. The result is
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
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15. The triangles in below are similar. Give the reason for similarity, complete the
similarity statement, and find the value of 𝑥.
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 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
Cross multiply to solve for 𝑥. The result is
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
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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
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17. Are the triangles in this figure similar? If so, give the reason for similarity and
complete the similarity statement below.
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 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
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18. The triangles in below are similar. Give the reason for similarity, complete the
similarity statement, and find the missing value.
The figure shows that the two triangles share an angle, so they have one
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
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19. The triangles in below are similar. Give the reason for similarity, complete the
similarity statement, and find the missing value.
The figure shows that the two triangles share an angle, so they have one
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
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20. The triangles in below are similar. Complete the similarity statement, and find
the value of 𝑥.
The triangles are similar so set up a proportion equating the ratios of the
corresponding sides.
70
50=
11𝑥 − 4
60
Cross multiply to solve for 𝑥. The result is
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
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