1. 7.5 Proportions In Triangles
Objective: Use Side-Splitter theorem & Triangle-Angle-Bisector Theorem to calculate segment lengths.

2. Theorems 7.4 Side-Splitter Theorem
If a line parallel to one side of a triangle intersects the other two sides, then it divides the two side proportionally.
If TU ║ QS, then RT RU = TQ US

3. Ex. 1: Finding the length of a segment
In the diagram AB ║ ED, BD = 8, DC = 4, and AE = 12. What is the length of EC?

4. Step: DC EC BD AE 4 EC 8 12 4(12) 8 6 = EC Reason Side-Splitter Thm.
4(12) 8
6 = EC
Reason
Side-Splitter Thm.
Substitute
Multiply each side by 12.
Simplify. = = EC =

5. Ex. 2: Determining Parallels
Given the diagram, determine whether MN ║ GH. LM 56 8 = = MG 21 3 LN 48 3 = = NH 16 1 8 3 ≠ 3 1
MN is not parallel to GH.

6. Corollary to Theorem 7-4
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 UW VX = WY XZ

7. Triangle-Angle-Bisector Thm
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 AD CA = DB CB

8. Ex. 3: Using Triangle-Angle-Bisector Theorem
In the diagram 1  2  3, and PQ = 9, QR = 15, and ST = 11. What is the length of TU?

9. 9 ● TU = 15 ● 11 Cross Multiply TUPQ ST
Parallel lines divide transversals proportionally. = QR TU 9 11 = 15 TU
9 ● TU = 15 ● 11 Cross Multiply
15(11) 55 TU = = 9 3

10. Ex. 4: Using the Proportionality Theorem
In the diagram, CAD  DAB. Use the given side lengths to find the length of DC.

11. Since AD is an angle bisector of CAB, you can apply Theorem 7. 5
Since AD is an angle bisector of CAB, you can apply Theorem Let x = DC. Then BD = 14 – x.
Solution: AB BD = AC DC 9
14-X = 15 X

12. Ex. 4 Continued . . . 9 ● x = 15 (14 – x) 9x = 210 – 15x 24x= 210

13. Ex. 6: Finding Segment Lengths
In the diagram KL ║ MN. Find the values of the variables.

14. Solution To find the value of x, you can set up a proportion.9 x =
13.5 x
13.5(37.5 – x) = 9x
– 13.5x = 9x
= 22.5 x
22.5 = x

15. Solution To find the value of y, you can set up a proportion.9
7.5 = y
9y = 7.5(22.5)
y = 18.75

16. Homework
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