1. Bell Ringer

2. Proportions and Similar Triangles

3. Example 1 Find Segment Lengths Find the value of x. 4 8 x 12 = Substitute 4 for CD, 8 for DB, x for CE, and 12 for EA. 4 · 12 = 8 · x Cross product property 48 = 8x Multiply. 48 8 = 8x8x 8 Divide each side by 8. SOLUTION CD DB = CE EA Triangle Proportionality Theorem 6 = x Simplify.

4. Example 2 Find Segment Lengths Find the value of y. 3 9 y 20 – y = Substitute 3 for PQ, 9 for QR, y for PT, and (20 – y) for TS. 3(20 – y) = 9 · y Cross product property 60 – 3y = 9y Distributive property PQ QR = PT TS Triangle Proportionality Theorem SOLUTION You know that PS = 20 and PT = y. By the Segment Addition Postulate, TS = 20 – y.

5. Example 2 Find Segment Lengths 60 12 = 12y 12 Divide each side by 12. 60 – 3y + 3y = 9y + 3y Add 3y to each side. 60 = 12y Simplify. 5 = y Simplify.

6. Example 3 Determine Parallels Given the diagram, determine whether MN is parallel to GH. SOLUTION Find and simplify the ratios of the two sides divided by MN. LM MG = 56 21 = 8 3 LN NH = 48 16 = 3 1 ANSWER Because ≠ 3 1 8 3, MN is not parallel to GH.

7. Now You Try Find Segment Lengths and Determine Parallels Find the value of the variable. 1. 2. ANSWER 8 10

8. Checkpoint Find Segment Lengths and Determine Parallels 4. 3. Given the diagram, determine whether QR is parallel to ST. Explain. ANSWER Converse of the Triangle Proportionality Theorem. = 6 12 4 8 Yes; || so QR ST by the ≠ 17 23 15 21 no; ANSWER Now You Try

9. A midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle.

10. Example 4 Use the Midsegment Theorem Find the length of QS. 1 2 QS = PT = (10) = 5 1 2 ANSWER The length of QS is 5. SOLUTION From the marks on the diagram, you know S is the midpoint of RT, and Q is the midpoint of RP. Therefore, QS is a midsegment of  PRT. Use the Midsegment Theorem to write the following equation.

11. Checkpoint Use the Midsegment Theorem Find the value of the variable. ANSWER 8 24 5. 6. ANSWER 28 7. Use the Midsegment Theorem to find the perimeter of  ABC. Now You Try

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13. Complete Page 390 #s 2-36 even only