8.4 Warmup. Explain why the triangles are similar. Then find the value of x Hint: Use Pyth. Thm.

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1 8.4 Warmup Explain why the triangles are similar. Then find the value of x x x x Hint: Use Pyth. Thm. 1

2 Geometry 8.4 Proportionality Theorems

3 8.4 Essential Question What proportionality relationships exist in a triangle intersected by an angle bisector or by a line parallel to one of the sides? 3

4 Goals Use proportionality theorems to calculate segment lengths. Solve problems using these theorems. 4

5 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. D C E If DE AB then AD DC BE EC A B 5

6 Proof CDE ~ CAB. Why? C AA~ s D t E r v A B 6

7 Proof CDE ~ CAB. r s t v s t r v s t s t r v r + s A Recall the Addition Property of Proportions: a b c d a c b d b d r s D C t E t + v v B 7

8 Theorem 8.6 Simplified a c a c b d or b d a b c d 8

9 Theorem 8.6 Simplified D a c E b d A B If DE // AB, then a b = c d or a c = b d 9

10 Example 1 Find a. a a 24 a a

11 Example 2 You do it. Find x x 5x 84 x x 11

12 We did this before Do you remember where? 12

13 Triangle Midsegment Theorem A midsegment of a triangle is parallel to the third side and ½ its length. R C S CR RA CS SB A RS 1 2 AB B But this was a special case where the parallel line divides the two sides in half. 13

14 Example 3 Find x. x 6 14 x 10 10x6(14 x) 10x84 6x 16x 84 x x x

15 or x x 6 16x x x 10 15

16 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. D C AD DC If then DE AB. BE EC E A B 16

17 Your Turn Problems Identify what theorem(s) apply. Write the proportion and solve. 17

18 Example 4 Is PQ RS? YES 8 12? or P 12 Q R S 18

19 Your Turn 4 Is EB DC? A E B D C Yes: Sides are proportional

20 Example 5 Is AB CD? YES 6? A B C D 20

21 Your Turn 5 Solve for x. A E B x 8 D C x 8 15x 80 x

22 Theorem 8.8 Three Parallel Lines Thm If three (or more) parallel lines intersect two transversals, they divide the transversals proportionally. a c a c b d b d 22

23 Example 6 AB AM A BC MN BC MN B M DE OP C N AB AM CE NP E D O P AD CE AO NP 23

24 Example 7 Solve for x x x + 2 Set up the proportion and solve x x 2 14x10x20 4x 20 x 5 24

25 Example 7 Check it x x

26 Example 8 You do it. Solve for x. 24 x x x 5 x x100 24x 100 4x x 25 26

27 Example 9 In the diagram, 1, 2, and 3 are all congruent, GF = 120 yards, DE = 150 yards, and CD = 300 yards. Find the distance HF between Main Street and South Main Street. 27

28 Theorem 8.9 Triangle Angle Bisector Thm. 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 c = b d a Theorem Demo b d c a c = b d 28

29 Example 10 Solve for x. 6 8 x x 24 x 4 x 3 29

30 Example 11 Solve for x. 8? x What is this value in terms of x? 30

31 Example 11 Solve for x x x x 8 x 14 14(20 x) 8x x8x x Now, write the proportion and solve x

32 Example 12 You do it. Solve for a. a 24 a a a

33 Your Turn. Solve for a. a 8 a a a

34 Your Turn. Solve for x. 9 x x x x 2 36 x x 6 34

35 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 other side in proportion to the other two sides. 35

36 Homework 36

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