Geometry 6.3 Keep It in Proportion

6.3 Theorems About Proportionality Objectives Prove the Angle Bisector/Proportional Side Theorem Prove the Triangle Proportionality Theorem Prove the Converse of the Triangle Proportionality Theorem Prove the Proportional Segments Theorem associated with parallel lines Prove the Triangle Midsegment Theorem

10 4 = 5 2 𝑜𝑟 4 10 = 2 5 7 2.8 = 5 2 𝑜𝑟 2.8 7 = 2 5

Problem 1: Proving the Angle Bisector/Proportional Side Theorem A bisector of an angle in a triangle divides the opposite side into two segments whose lengths are in the same ratio as the lengths of the sides adjacent to the angle.

Problem 2: Applying the Angle Bisector/Proportional Side Theorem Together 1-5 𝑥 300 = 1200 500 Cross-Multiply Then Divide 𝑥= 1200𝑥300 ÷500 𝑥=720 𝑓𝑒𝑒𝑡 1200 x 300 500

Problem 2: Applying the Angle Bisector/Proportional Side Theorem 𝑥 30 = 8 24 Cross-Multiply Then Divide 𝑥= 8𝑥30 ÷24 𝑥=10=𝐷𝐵 𝑥

Problem 2: Applying the Angle Bisector/Proportional Side Theorem 9 𝑥 = 11 22 Cross-Multiply Then Divide 𝑥= 9𝑥22 ÷11 𝑥=18=𝐴𝐶 𝑥

Problem 2: Applying the Angle Bisector/Proportional Side Theorem The Ratio of Sides 14 7 = 2 1 = 𝐴𝐵 𝐴𝐶 AB is twice AC AB=2AC AC + AB = 36 AC + 2AC = 36 3AC=36 AC=12 AB = 2AC AB = 24

Problem 2: Applying the Angle Bisector/Proportional Side Theorem 14 6 = 16 𝑥 Cross-Multiply Then Divide 𝑥= 16𝑥6 ÷14 𝑥=6.9=𝐷𝐶 𝐴𝐶=6+𝑥 𝐴𝐶=6+6.9=12.9 𝐴𝐶=6+𝑥 𝑥

Problem 3 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

Problem 4: Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side

Problem 5 Proportional Segments Theorem If three parallel lines intersect two transversals, then they divide the transversals proportionally Collaborate 1-4 (3 Minutes)

Problem 5 Proportional Segments Theorem

Problem 5 Proportional Segments Theorem

Problem 5 Proportional Segments Theorem

Problem 6 Triangle Midsegment Theorem The midsegment of a triangle is parallel to the third side of the triangle and is half the measure of the third side of the triangle If 𝐽𝐺 is given as a midsegment, then what can we conclude? 𝐽𝐺 𝑖𝑠 𝑎 𝑚𝑖𝑑𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝑜𝑓 ∆𝐷𝑀𝑆 𝐽𝐺 ∥ 𝐷𝑆 𝐽𝐺= 1 2 𝐷𝑆 𝐷𝑆=2𝐽𝐺

Problem 6 Triangle Midsegment Theorem Collaborate 3-4 (2 Minutes)

Problem 6 Triangle Midsegment Theorem

3𝑥 18 12 = 3 2 3𝑥+2𝑥+12+18=55 5𝑥+30=55 5𝑥=25 𝑥=5 𝑄𝑇=2𝑥=2∗5=10 2𝑥 ?????

Formative Assessment Skills Practice 6.3 Vocabulary Pg. 527 (1-5) Matching Problem Set Pg. 528-531 (1-12) Problem Set Pg. 532-535 (21-28)

Formative Assessment: Day 2 Assignment 6.2 and 6.3 Handout Collaborate as a group Turn in an individual assignment before we leave for the assembly