7 5 ESSENTIAL QUESTION l How do you

Theorem 7. 4 Triangle Proportionality Theorem l If a line parallel to one side

Example 1 Find Segment Lengths Find the value of x. SOLUTION CD CE =

Theorem 7. 5 Converse of the Triangle Proportionality Theorem l If a line divides

Example 3 Determine Parallels Given the diagram, determine whether MN is parallel to GH.

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

Checkpoint Find Segment Lengths and Determine Parallels 2. ANSWER 10

Checkpoint Find Segment Lengths and Determine Parallels Given the diagram, determine whether QR is

VOCABULARY l A midsegment of a triangle: a segment that connects the midpoints of

The Midsegment Theorem l The segment connecting the midpoints of two sides of a

Example 4 Use the Midsegment Theorem Find the length of QS. SOLUTION From the

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

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7. 5 ESSENTIAL QUESTION: l How do you use the Triangle Proportionality Theorem and its Converse in solving missing parts?

Theorem 7. 4 Triangle Proportionality Theorem l If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

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

Theorem 7. 5 Converse of the Triangle Proportionality Theorem l If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

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 56 8 = = MG 21 3 ANSWER LN 48 3 = = NH 16 1 8 3 Because ≠ , MN is not parallel to GH. 3 1

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

Checkpoint Find Segment Lengths and Determine Parallels 2. ANSWER 10

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

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

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

Example 4 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. QS = 1 1 PT = (10) = 5 2 2 ANSWER The length of QS is 5.

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

Homework l Worksheet 7. 5 A