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?

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?

Use Proportionality Theorems Section 6.6

Test on Wednesday

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

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

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

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

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

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

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”

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

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

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

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

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

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

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

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

Homework 6.6 1, 2-36 even

HW

Most Important Points Triangle Proportionality Theorems