1. Geometry
6.3 Keep It in Proportion

2. 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

3. 10 4 = 5 2 𝑜𝑟 = 2 5
7 2.8 = 5 2 𝑜𝑟 = 2 5

4. 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.

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

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

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

8. 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

9. 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

10. 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

11. 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

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

13. Problem 5 Proportional Segments Theorem

14. Problem 5 Proportional Segments Theorem

15. Problem 5 Proportional Segments Theorem

16. 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

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

18. Problem 6 Triangle Midsegment Theorem

19. 3𝑥
18 12 = 3 2
3𝑥+2𝑥+12+18=55
5𝑥+30=55
5𝑥=25
𝑥=5
𝑄𝑇=2𝑥=2∗5=10 2𝑥
?????

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

21. 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