1. Geometry 8.4 Proportionality Theorems
8.4 Warmup
Explain why the triangles are similar. Then find the value of x 4 18 x 15 6 20 x x
Hint: Use Pyth. Thm.
January 18, 2019
Geometry 8.4 Proportionality Theorems

2. Geometry 8.6 Proportions and Similar Triangles
8.2 Practice A
January 18, 2019
Geometry 8.6 Proportions and Similar Triangles

3. Geometry 8.6 Proportions and Similar Triangles
8.2 Practice A
January 18, 2019
Geometry 8.6 Proportions and Similar Triangles

4. Geometry 8.6 Proportions and Similar Triangles
8.3 Practice A
In Exercises 5 and 6, show that the triangles are similar and write a similarity statement. Explain your reasoning.
January 18, 2019
Geometry 8.6 Proportions and Similar Triangles

5. 8.4 Proportionality Theorems
Geometry
8.4 Proportionality Theorems

6. Geometry 8.4 Proportionality Theorems
8.4 Essential Question
What proportionality relationships exist in a triangle intersected by an angle bisector or by a line parallel to one sides?
January 18, 2019
Geometry 8.4 Proportionality Theorems

7. Geometry 8.4 Proportionality Theorems
Goals
Use proportionality theorems to calculate segment lengths.
Solve problems using these theorems.
January 18, 2019
Geometry 8.4 Proportionality Theorems

8. Theorem 8.6 Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.
𝐼𝑓 𝐷𝐸 // 𝐴𝐵 , 𝑡ℎ𝑒𝑛 𝐴𝐷 𝐵𝐸 = 𝐷𝐶 𝐸𝐶
𝑜𝑟 𝐴𝐷 𝐷𝐶 = 𝐵𝐸 𝐸𝐶 A B C D E
January 18, 2019
Geometry 8.4 Proportionality Theorems

9. Geometry 8.4 Proportionality Theorems
Proof
CDE ~ CAB.
Why?
AA~ C t s D E v r A B
January 18, 2019
Geometry 8.4 Proportionality Theorems

10. Geometry 8.4 Proportionality Theorems
Proof
CDE ~ CAB.
Recall the Addition Property of Proportions: C t s
t + v D E
r + s v r A B
January 18, 2019
Geometry 8.4 Proportionality Theorems

11. Theorem 8.6 Simplified 𝐼𝑓 𝐷𝐸 // 𝐴𝐵 , 𝑡ℎ𝑒𝑛 𝑎 𝑏 = 𝑐 𝑑 𝑜𝑟 𝑎 𝑐 = 𝑏 𝑑 b a c
𝐼𝑓 𝐷𝐸 // 𝐴𝐵 , 𝑡ℎ𝑒𝑛 𝑎 𝑏 = 𝑐 𝑑 𝑜𝑟 𝑎 𝑐 = 𝑏 𝑑
January 18, 2019
Geometry 8.4 Proportionality Theorems

12. Geometry 8.4 Proportionality Theorems
Example 1 Find a. 8 a 3 9
January 18, 2019
Geometry 8.4 Proportionality Theorems

13. Geometry 8.4 Proportionality Theorems
Example 2 You do it. Find x. 7 5 x 12
January 18, 2019
Geometry 8.4 Proportionality Theorems

14. Geometry 8.4 Proportionality Theorems
We did this before…
Do you remember where?
January 18, 2019
Geometry 8.4 Proportionality Theorems

15. Triangle Midsegment Theorem
A midsegment of a triangle is parallel to the third side and ½ its length. R S
But this was a special case where the parallel line divides the two sides in half.
January 18, 2019
Geometry 8.4 Proportionality Theorems

16. Geometry 8.4 Proportionality Theorems
Example 3 Find x. x
14 – x 6 10
January 18, 2019
Geometry 8.4 Proportionality Theorems

17. Geometry 8.4 Proportionality Theoremsx
14 – x 6 10 16 14
January 18, 2019
Geometry 8.4 Proportionality Theorems

18. Theorem 8.7 Conv. of Δ Propor. Thm.
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
𝐼𝑓 𝐴𝐷 𝐵𝐸 = 𝐷𝐶 𝐸𝐶 ,𝑡ℎ𝑒𝑛 𝐷𝐸 // 𝐴𝐵 A B C D E
January 18, 2019
Geometry 8.4 Proportionality Theorems

19. Geometry 8.4 Proportionality Theorems
Your Turn Problems
Identify what theorem(s) apply.
Write the proportion and solve.
January 18, 2019
Geometry 8.4 Proportionality Theorems

20. Geometry 8.4 Proportionality Theorems
Example 4
Is PQ || RS ?
YES 8 6 P Q 9 12 R S
January 18, 2019
Geometry 8.4 Proportionality Theorems

21. Geometry 8.4 Proportionality Theorems
Your Turn Is EB || DC? A B C D E 10 12 4
4.8
Yes: Sides are divided proportionally.
January 18, 2019
Geometry 8.4 Proportionality Theorems

22. Geometry 8.4 Proportionality Theorems
Example 5
Is AB || CD ?
YES 6 9 24 16 A B 10 15 C D
January 18, 2019
Geometry 8.4 Proportionality Theorems

23. Geometry 8.4 Proportionality Theorems
Your Turn Solve for x. A B C D E 10 15 x 8
January 18, 2019
Geometry 8.4 Proportionality Theorems

24. Theorem 8.8 Three Parallel Lines Thm
If three (or more) parallel lines intersect two transversals, they divide the transversals proportionally. a c d b
January 18, 2019
Geometry 8.4 Proportionality Theorems

25. Geometry 8.4 Proportionality Theorems
Example 6 A B C D E M N O P
January 18, 2019
Geometry 8.4 Proportionality Theorems

26. Geometry 8.4 Proportionality Theorems
Example 7 Solve for x. 10 14 x
x + 2
Set up the proportion and solve.
January 18, 2019
Geometry 8.4 Proportionality Theorems

27. Geometry 8.4 Proportionality Theorems
Example 7 Check it. 10 14 x
x + 2 5 7
January 18, 2019
Geometry 8.4 Proportionality Theorems

28. Example 8 You do it. Solve for x.24 20
January 18, 2019
Geometry 8.4 Proportionality Theorems

29. Geometry 8.4 Proportionality Theorems
Example 9
In the diagram, ∠1, ∠2, and ∠3 are all congruent, GF = 120 yards, DE = 150 yards, and CD = 300 yards. Find HF, the distance between Main Street and South Main Street.
January 18, 2019
Geometry 8.4 Proportionality Theorems

30. Theorem 8.9 Triangle Angle Bisector Thm.
Theorem Demo
If a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the other sides. a b c d
January 18, 2019
Geometry 8.4 Proportionality Theorems

31. Geometry 8.4 Proportionality Theorems
Example 10 Solve for x. 6 8 x 4
January 18, 2019
Geometry 8.4 Proportionality Theorems

32. Geometry 8.4 Proportionality Theorems
Example 11 Solve for x. 20 14 8 ? x
What is this value in terms of x?
January 18, 2019
Geometry 8.4 Proportionality Theorems

33. Geometry 8.4 Proportionality Theorems
Example 11 Solve for x. 20 14 8
20 – x x
Now, write the proportion and solve.
January 18, 2019
Geometry 8.4 Proportionality Theorems

34. Example 12 You do it. Solve for a.
44 12 a 10 24
January 18, 2019
Geometry 8.4 Proportionality Theorems

35. Geometry 8.4 Proportionality Theorems
Your Turn. Solve for a. a 8 4 6
January 18, 2019
Geometry 8.4 Proportionality Theorems

36. Geometry 8.4 Proportionality Theorems
Your Turn. Solve for x. x 9 4
January 18, 2019
Geometry 8.4 Proportionality Theorems

37. Geometry 8.4 Proportionality Theorems
Summary
If a line parallel to a side of a triangle intersects the other two sides, it divides them proportionally.
If a ray bisects an angle of a triangle, it divides the opposite side in proportion to the other two sides.
January 18, 2019
Geometry 8.4 Proportionality Theorems

38. Geometry 8.4 Proportionality Theorems
Homework
January 18, 2019
Geometry 8.4 Proportionality Theorems