1. Parallel Lines and Proportional Parts
Notes 7.4
Parallel Lines and Proportional Parts

2. Review 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝐴𝐵 𝐵𝐶 𝐴𝐶
Corresponding sides of similar triangles are _____________
Corresponding angles of similar triangles are __________
According to the segment addition postulate, ___ + ___ = ___
𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙
𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝐴𝐵 𝐵𝐶 𝐴𝐶 𝐴 𝐵 𝐶

3. ∆𝐴𝐶𝐸~∆𝐵𝐶𝐷 by AA Similarity
𝐴𝐶 𝐵𝐶 =
𝐸𝐶 𝐷𝐶
𝐴𝐵+𝐵𝐶 𝐵𝐶 = 𝐸𝐷+𝐷𝐶 𝐷𝐶 𝐵 𝐷
𝐴𝐵 𝐵𝐶 +
𝐵𝐶 𝐵𝐶 =
𝐸𝐷 𝐷𝐶 +
𝐷𝐶 𝐷𝐶 𝐴 𝐸
𝐴𝐶=
𝐴𝐵+𝐵𝐶
𝐴𝐵 𝐵𝐶 +
𝐸𝐷 𝐷𝐶 + 1= 1
𝐸𝐶=
𝐸𝐷+𝐷𝐶

4. 𝐶
𝐴𝐵 𝐵𝐶 +
𝐸𝐷 𝐷𝐶 + 1= 1
𝐴𝐵 𝐵𝐶 = 𝐸𝐷 𝐷𝐶 𝐵 𝐷 𝐴 𝐸

5. 𝐴𝐵 𝐵𝐶 = 𝐸𝐷 𝐷𝐶 Triangle Proportionality Theorem
If a line is parallel to one side of a triangle and intersects the other two sides, then it separates these sides into proportional segments 𝐵 𝐷 𝐴 𝐸
𝐴𝐵 𝐵𝐶 = 𝐸𝐷 𝐷𝐶

6. Example #1 9 21
𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥
21 9 = 𝑥 6 𝑥 6
9𝑥=126
𝑥=14

7. Triangle Midsegment Theorem𝐶
𝐼𝑓 𝐵 𝑎𝑛𝑑 𝐷 𝑎𝑟𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡𝑠
𝑜𝑓 𝐴𝐶 𝑎𝑛𝑑 𝐸𝐶 ,
𝑡ℎ𝑒𝑛 𝐵𝐷 ‖ 𝐴𝐸 𝐷 𝐵
𝑎𝑛𝑑 𝐵𝐷= 1 2 𝐴𝐸
*Only applies if B and D are midpoints 𝐸 𝐴

8. Example #2 𝐹𝑖𝑛𝑑 𝐴𝐸 10 8 10 3 8 𝐵𝐷 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝐵𝐷= 1 2 𝐴𝐸𝐶
Example #2 10
𝐹𝑖𝑛𝑑 𝐴𝐸 8 𝐵 10 3
𝐵𝐷 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝐷
𝐵𝐷= 1 2 𝐴𝐸 𝐴 8
3= 1 2 𝐴𝐸
6=𝐴𝐸 𝐸

9. Example #3 8 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥 𝑥 2 𝑥 6 = 8 10 6 𝑥= 48 10 = 24 5 𝑜𝑟 4.8𝐶
Example #3 8
𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥
𝐵 𝑎𝑛𝑑 𝐷 𝑎𝑟𝑒 𝑁𝑂𝑇 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡𝑠 𝑥
𝑠𝑜 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑀𝑖𝑑𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝐵 𝐷
𝑇ℎ𝑒𝑜𝑟𝑒𝑚 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑎𝑝𝑝𝑙𝑦 2
𝑥 6 = 8 10 𝐸 𝐴 6
𝑥= = 𝑜𝑟 4.8
10𝑥=48