1. Bellwork
Clickers
Ruby is standing in her back yard and she decides to estimate the height of a tree. She stands so that the tip of her shadow coincides with the top of the tree’s shadow. Ruby is 66 inches tall. The distance from the tree to Ruby is 95 feet and the distance between the tip of the shadows and ruby is 7 feet.
What postulate or theorem can you use to show that the triangles in the diagram are similar?
About how tall is the tree, to the nearest foot?
What if? Curtis is 75 inches tall. At a different time of day, he stands so that the tip of the his shadow and the tip of the tree’s shadow coincide, as described above. His shadow is 6 feet long. How far is Curtis from the tree?

2. Bellwork Solution
Ruby is standing in her back yard and she decides to estimate the height of a tree. She stands so that the tip of her shadow coincides with the top of the tree’s shadow. Ruby is 66 inches tall. The distance from the tree to Ruby is 95 feet and the distance between the tip of the shadows and ruby is 7 feet.
What postulate or theorem can you use to show that the triangles in the diagram are similar?
About how tall is the tree, to the nearest foot?
What if? Curtis is 75 inches tall. At a different time of day, he stands so that the tip of the his shadow and the tip of the tree’s shadow coincide, as described above. His shadow is 6 feet long. How far is Curtis from the tree?

3. Use Proportionality Theorems
Section 6.6

4. Test on Wednesday

5. The Concept
Yesterday we finished our exploration of the different methodologies to prove similarity in triangles
Today we’re going to see some theorems that allow us to name proportionality within triangles and parallel lines

6. Theorems Theorem 6.4: 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
Theorem 6.5: Converse of the Triangle Proportionality Theorem
If a line divides two sides of a triangle proportionally, then it is parallel to the third side. B D E C A

7. Example
Solve for x, if DE and AC are parallel B 12 x D E 20 15 C A

8. Example
What value of x makes the lines parallel? 16 13
32.5 x

9. Example
What value of x makes the lines parallel? 6
x+3
8x-1 18

10. Example
What value of x makes the lines parallel? x 5
15x 27

11. How would we explain our answer?
In your notes
A cross brace is added to an A-Frame tent. Why is the brace not parallel to the ground?
x+3
How would we explain our answer?
15”
16”
25”
24”

12. Theorems
Theorem 6.6:
If three parallel lines intersect two transversals, then they divide the transversals proportionally A B C

13. Example
Theorem 6.6:
If three parallel lines intersect two transversals, then they divide the transversals proportionally 51 x 15 42

14. Example
What value of x makes the lines parallel? 16 x 15 20

15. Example
What value of x makes the lines parallel?
x+2 x 12 19

16. Example
What value of x makes the lines parallel?
x+2 2
x-5 4

17. Theorems
Theorem 6.7:
If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. B E A C

18. Example
Solve for x, if Ray AE bisects ABC. B 8 24 E x A 32 C

19. Example
Find x if BC=40 B x 24 E A 36 C

20. Homework
6.6
1, 2-36 even

21. HW

22. Most Important Points
Triangle Proportionality Theorems