Parallel Lines and Proportional Parts Notes 7.4 Parallel Lines and Proportional Parts

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

∆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

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

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

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

Triangle Midsegment Theorem 𝐶 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

Example #2 𝐹𝑖𝑛𝑑 𝐴𝐸 10 8 10 3 8 𝐵𝐷 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝐵𝐷= 1 2 𝐴𝐸 𝐶 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

Example #3 8 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥 𝑥 2 𝑥 6 = 8 10 6 𝑥= 48 10 = 24 5 𝑜𝑟 4.8 𝐶 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= 48 10 = 24 5 𝑜𝑟 4.8 10𝑥=48