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Proportions and Similar Triangles Section 8.6: Proportions and Similar Triangles

Theorem 8.4: 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. If TU ║ QS, then

Theorem 8.5: Converse of the Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. If , then TU ║ QS.

Theorem 8.6 If three parallel lines intersect two transversals, then they divide the transversals proportionally. If r ║ s and s║ t and l and m intersect, r, s, and t, then .

Theorem 8.7 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. If CD bisects ACB, then .

Example 1: In the diagram, UY is parallel to VX, UV = 3, UW = 18, and XW = 16. What is the length of YX? U 3 18 V W 16 Y X YX = 3.2

Example 2: Given the diagram, determine whether PQ is parallel to TR Example 2: Given the diagram, determine whether PQ is parallel to TR. Q P 9.75 9 T R 24 26 S Yes If PQ is parallel to TR then the sides lengths would be in proportion.

HOMEWORK (Day 1) pg. 502; 11 – 20

Example 3: In the diagram 1  2  3, and PQ = 9, QR = 15, and ST = 11. What is the length of TU?

Example 4: In the diagram , LN = 15, LK = 3 and KN = 17 Example 4: In the diagram , LN = 15, LK = 3 and KN = 17. Use the given side lengths to find the length of MN. L M N K MN = 12.75

Example 5: FJ || GI. Find the values of the variables Example 5: FJ || GI. Find the values of the variables. F 2 G 8 9 y H x I 12 J x = 9.6, y = 7.2

HOMEWORK (Day 2) pg. 503; 21, 23, 25