1. Splash Screen

2. Five-Minute Check (over Lesson 7–3) CCSS Then/Now New Vocabulary
Theorem 7.5: Triangle Proportionality Theorem
Example 1: Find the Length of a Side
Theorem 7.6: Converse of Triangle Proportionality Theorem
Example 2: Determine if Lines are Parallel
Theorem 7.7: Triangle Midsegment Theorem
Example 3: Use the Triangle Midsegment Theorem
Corollary 7.1: Proportional Parts of Parallel Lines
Example 4: Real-World Example: Use Proportional Segments of Transversals
Corollary 7.2: Congruent Parts of Parallel Lines
Example 5: Real-World Example: Use Congruent Segments of Transversals
Lesson Menu

3. Determine whether the triangles are similar. Justify your answer.
A. yes, SSS Similarity
B. yes, ASA Similarity
C. yes, AA Similarity
D. No, sides are not proportional.
5-Minute Check 1

4. Determine whether the triangles are similar. Justify your answer.
A. yes, SSS Similarity
B. yes, ASA Similarity
C. yes, AA Similarity
D. No, sides are not proportional.
5-Minute Check 1

5. Determine whether the triangles are similar. Justify your answer.
A. yes, AA Similarity
B. yes, SSS Similarity
C. yes, SAS Similarity
D. No, sides are not proportional.
5-Minute Check 2

6. Determine whether the triangles are similar. Justify your answer.
A. yes, AA Similarity
B. yes, SSS Similarity
C. yes, SAS Similarity
D. No, sides are not proportional.
5-Minute Check 2

7. Determine whether the triangles are similar. Justify your answer.
A. yes, AA Similarity
B. yes, SSS Similarity
C. yes, SAS Similarity
D. No, angles are not equal.
5-Minute Check 3

8. Determine whether the triangles are similar. Justify your answer.
A. yes, AA Similarity
B. yes, SSS Similarity
C. yes, SAS Similarity
D. No, angles are not equal.
5-Minute Check 3

9. Find the width of the river in the diagram.
B. 28 m
C. 24 m
D m
5-Minute Check 4

10. Find the width of the river in the diagram.
B. 28 m
C. 24 m
D m
5-Minute Check 4

11. G.SRT.4 Prove theorems about triangles.
Content Standards
G.SRT.4 Prove theorems about triangles.
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Mathematical Practices
1 Make sense of problems and persevere in solving them.
3 Construct viable arguments and critique the reasoning of others.
CCSS

12. You used proportions to solve problems between similar triangles.
Use proportional parts within triangles.
Use proportional parts with parallel lines.
Then/Now

13. midsegment of a triangle
Vocabulary

14. Concept

15. Find the Length of a Side
Example 1

16. Substitute the known measures.
Find the Length of a Side
Substitute the known measures.
Cross Products Property
Multiply.
Divide each side by 8.
Simplify.
Example 1

17. Substitute the known measures.
Find the Length of a Side
Substitute the known measures.
Cross Products Property
Multiply.
Divide each side by 8.
Simplify.
Example 1

18. A. 2.29 B
C. 12 D
Example 1

19. A. 2.29 B
C. 12 D
Example 1

20. Concept

21. In order to show that we must show that
Determine if Lines are Parallel
In order to show that we must show that
Example 2

22. Since the sides are proportional.
Determine if Lines are Parallel
Since the sides are proportional.
Answer:
Example 2

23. Since the sides are proportional.
Determine if Lines are Parallel
Since the sides are proportional.
Answer: Since the segments have proportional lengths, GH || FE.
Example 2

24. A. yes
B. no
C. cannot be determined
Example 2

25. A. yes
B. no
C. cannot be determined
Example 2

26. Concept

27. A. In the figure, DE and EF are midsegments of ΔABC. Find AB.
Use the Triangle Midsegment Theorem
A. In the figure, DE and EF are midsegments of ΔABC. Find AB.
Example 3

28. ED = AB Triangle Midsegment Theorem 1 2 5 = AB Substitution 1 2
Use the Triangle Midsegment Theorem
ED = AB Triangle Midsegment Theorem __ 1 2
5 = AB Substitution __ 1 2
10 = AB Multiply each side by 2.
Answer:
Example 3

29. ED = AB Triangle Midsegment Theorem 1 2 5 = AB Substitution 1 2
Use the Triangle Midsegment Theorem
ED = AB Triangle Midsegment Theorem __ 1 2
5 = AB Substitution __ 1 2
10 = AB Multiply each side by 2.
Answer: AB = 10
Example 3

30. B. In the figure, DE and EF are midsegments of ΔABC. Find FE.
Use the Triangle Midsegment Theorem
B. In the figure, DE and EF are midsegments of ΔABC. Find FE.
Example 3

31. FE = BC Triangle Midsegment Theorem
Use the Triangle Midsegment Theorem __ 1 2
FE = BC Triangle Midsegment Theorem
FE = (18) Substitution __ 1 2
FE = 9 Simplify.
Answer:
Example 3

32. FE = BC Triangle Midsegment Theorem
Use the Triangle Midsegment Theorem __ 1 2
FE = BC Triangle Midsegment Theorem
FE = (18) Substitution __ 1 2
FE = 9 Simplify.
Answer: FE = 9
Example 3

33. C. In the figure, DE and EF are midsegments of ΔABC. Find mAFE.
Use the Triangle Midsegment Theorem
C. In the figure, DE and EF are midsegments of ΔABC. Find mAFE.
Example 3

34. By the Triangle Midsegment Theorem, AB || ED.
Use the Triangle Midsegment Theorem
By the Triangle Midsegment Theorem, AB || ED.
AFE  FED Alternate Interior Angles Theorem
mAFE = mFED Definition of congruence
mAFE = 87 Substitution
Answer:
Example 3

35. By the Triangle Midsegment Theorem, AB || ED.
Use the Triangle Midsegment Theorem
By the Triangle Midsegment Theorem, AB || ED.
AFE  FED Alternate Interior Angles Theorem
mAFE = mFED Definition of congruence
mAFE = 87 Substitution
Answer: mAFE = 87
Example 3

36. A. In the figure, DE and DF are midsegments of ΔABC. Find BC.
Example 3

37. A. In the figure, DE and DF are midsegments of ΔABC. Find BC.
Example 3

38. B. In the figure, DE and DF are midsegments of ΔABC. Find DE.
Example 3

39. B. In the figure, DE and DF are midsegments of ΔABC. Find DE.
Example 3

40. C. In the figure, DE and DF are midsegments of ΔABC. Find mAFD.
Example 3

41. C. In the figure, DE and DF are midsegments of ΔABC. Find mAFD.
Example 3

42. Concept

43. Use Proportional Segments of Transversals
MAPS In the figure, Larch, Maple, and Nuthatch Streets are all parallel. The figure shows the distances in between city blocks. Find x.
Example 4

44. Triangle Proportionality Theorem
Use Proportional Segments of Transversals
Notice that the streets form a triangle that is cut by parallel lines. So you can use the Triangle Proportionality Theorem.
Triangle Proportionality Theorem
Cross Products Property
Multiply.
Divide each side by 13.
Answer:
Example 4

45. Triangle Proportionality Theorem
Use Proportional Segments of Transversals
Notice that the streets form a triangle that is cut by parallel lines. So you can use the Triangle Proportionality Theorem.
Triangle Proportionality Theorem
Cross Products Property
Multiply.
Divide each side by 13.
Answer: x = 32
Example 4

46. In the figure, Davis, Broad, and Main Streets are all parallel
In the figure, Davis, Broad, and Main Streets are all parallel. The figure shows the distances in between city blocks. Find x.
A. 4
B. 5
C. 6
D. 7
Example 4

47. In the figure, Davis, Broad, and Main Streets are all parallel
In the figure, Davis, Broad, and Main Streets are all parallel. The figure shows the distances in between city blocks. Find x.
A. 4
B. 5
C. 6
D. 7
Example 4

48. Concept

49. 2x – 7 = 5 Subtract x from each side. 2x = 12 Add 7 to each side.
Use Congruent Segments of Transversals
ALGEBRA Find x and y.
To find x:
3x – 7 = x + 5 Given
2x – 7 = 5 Subtract x from each side.
2x = 12 Add 7 to each side.
x = 6 Divide each side by 2.
Example 5

50. Use Congruent Segments of Transversals
To find y:
The segments with lengths 9y – 2 and 6y are congruent since parallel lines that cut off congruent segments on one transversal cut off congruent segments on every transversal.
Example 5

51. 9y – 2 = 6y + 4 Definition of congruence
Use Congruent Segments of Transversals
9y – 2 = 6y + 4 Definition of congruence
3y – 2 = 4 Subtract 6y from each side.
3y = 6 Add 2 to each side.
y = 2 Divide each side by 3.
Answer:
Example 5

52. 9y – 2 = 6y + 4 Definition of congruence
Use Congruent Segments of Transversals
9y – 2 = 6y + 4 Definition of congruence
3y – 2 = 4 Subtract 6y from each side.
3y = 6 Add 2 to each side.
y = 2 Divide each side by 3.
Answer: x = 6; y = 2
Example 5

53. Find a and b.
A ;
B. 1; 2
C. 11;
D. 7; 3 __ 2 3
Example 5

54. Find a and b.
A ;
B. 1; 2
C. 11;
D. 7; 3 __ 2 3
Example 5

55. End of the Lesson