1. 7.5
ESSENTIAL QUESTION:
How do you use the Triangle Proportionality Theorem and its Converse in solving missing parts?

2. 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.

4. 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

5. 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.

7. 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

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

9. Checkpoint
Find Segment Lengths and Determine Parallels 2.
ANSWER 10

10. 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

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

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

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

15. 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

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

17. Homework
Worksheet 7.5A