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7.5 ESSENTIAL QUESTION: How do you use the Triangle Proportionality Theorem and its Converse in solving missing parts?

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Theorem 7.4 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.

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Example 1 Find the value of x. SOLUTION CD DB = CE EA 4 8 x 12 =

Find Segment Lengths Find the value of x. SOLUTION CD DB = CE EA Triangle Proportionality Theorem 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 = 8x Divide each side by 8. 6 = x Simplify. 4

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Theorem 7.5 Converse of the Triangle Proportionality Theorem

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

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Given the diagram, determine whether MN is parallel to GH.

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 1 ANSWER Because ≠ 3 1 8 , MN is not parallel to GH. 7

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Find the value of the variable.

Checkpoint Find Segment Lengths and Determine Parallels Find the value of the variable. 1. ANSWER 8

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Checkpoint Find Segment Lengths and Determine Parallels 2. ANSWER 10

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Given the diagram, determine whether QR is parallel to ST. Explain.

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

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VOCABULARY A midsegment of a triangle: a segment that connects the midpoints of two sides of a triangle.

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VOCABULARY A midsegment of a triangle: a segment that connects the midpoints of two sides of a triangle.

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The Midsegment Theorem

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long.

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Example 4 Find the length of QS. SOLUTION

Use the Midsegment Theorem Find the length of QS. 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. 1 2 QS = PT = (10) = 5 ANSWER The length of QS is 5. 15

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Find the value of the variable.

Checkpoint Use the Midsegment Theorem Find the value of the variable. 5. ANSWER 8 6. ANSWER 28

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Homework Worksheet 7.5A

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