Geometry Tutor Worksheet 10 Similar Triangles
Geometry Tutor Worksheet 10 Similar Triangles
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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~∆______ 7 © MathTutorDVD.com
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 = ______ 9 © MathTutorDVD.com
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 13 © MathTutorDVD.com
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 40 and 16 16 One ratio reduces, but the other ratio does not reduce. The result is 39 5 and 16 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(? ); ? = Answer: Reason: SAS, ∆𝐽𝐾𝐿~∆𝑅𝑃𝑄, 22 15 © MathTutorDVD.com
33(28) = 22 42
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 70 84 ? ? 15 25 30 Simplify each ratio. The result is: 14 14 14 = = 5 5 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 8 and 49 28 Both ratios reduce. The result is 2 2 = 7 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 25 and 88 132 One ratio simplifies, but the other one does not. The result is 1 25 and 5 132 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) = 54 130
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 12 14 ? ? 48 72 84 Simplify each ratio. The result is: 1 1 1 = = 6 6 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 30 = 77 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 10 and 27 30 22 © MathTutorDVD.com
Both ratios simplify, with the result 1 1 = 3 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 18 = 88 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 23 © MathTutorDVD.com
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 27 = 64 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 18 ? 21 42 Simplify the ratios. The result is 3 3 = 7 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 32 44 , , and 143 88 121 Simplify the three ratios. The result is 4 4 4 = = 11 11 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 3 = 24 12 The ratios simplify to the same fraction. The result is 1 1 = 4 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 11𝑥 − 4 = 50 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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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~∆______ 7 © MathTutorDVD.com
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 = ______ 9 © MathTutorDVD.com
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 13 © MathTutorDVD.com
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 40 and 16 16 One ratio reduces, but the other ratio does not reduce. The result is 39 5 and 16 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(? ); ? = Answer: Reason: SAS, ∆𝐽𝐾𝐿~∆𝑅𝑃𝑄, 22 15 © MathTutorDVD.com
33(28) = 22 42
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 70 84 ? ? 15 25 30 Simplify each ratio. The result is: 14 14 14 = = 5 5 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 8 and 49 28 Both ratios reduce. The result is 2 2 = 7 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 25 and 88 132 One ratio simplifies, but the other one does not. The result is 1 25 and 5 132 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) = 54 130
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 12 14 ? ? 48 72 84 Simplify each ratio. The result is: 1 1 1 = = 6 6 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 30 = 77 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 10 and 27 30 22 © MathTutorDVD.com
Both ratios simplify, with the result 1 1 = 3 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 18 = 88 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 23 © MathTutorDVD.com
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 27 = 64 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 18 ? 21 42 Simplify the ratios. The result is 3 3 = 7 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 32 44 , , and 143 88 121 Simplify the three ratios. The result is 4 4 4 = = 11 11 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 3 = 24 12 The ratios simplify to the same fraction. The result is 1 1 = 4 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 11𝑥 − 4 = 50 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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