1. Triangles and Lines - Proportional Relationships A proportion is just a comparison of two ratios or fractions. They are equivalent fractions. As one dimension increases or decreases by a factor, the other dimension increases or decreases by the same factor. It is widely used in blue prints when scaling drawings. You can put a ten story building on a 8.5 x 11 sheet of paper. Where one inch on the paper represents ten feet in actual size.

2. Triangles and Lines - Proportional Relationships A proportion is just a comparison of two ratios or fractions. They are equivalent fractions. As one dimension increases or decreases by a factor, the other dimension increases or decreases by the same factor. It is widely used in blue prints when scaling drawings. You can put a ten story building on a 8.5 x 11 sheet of paper. Where one inch on the paper represents ten feet in actual size. Here is a typical proportion…

3. Triangles and Lines - Proportional Relationships A proportion is just a comparison of two ratios or fractions. They are equivalent fractions. As one dimension increases or decreases by a factor, the other dimension increases or decreases by the same factor. It is widely used in blue prints when scaling drawings. You can put a ten story building on a 8.5 x 11 sheet of paper. Where one inch on the paper represents ten feet in actual size. Here is a typical proportion… You solve a proportion by cross multiplying…

4. Triangles and Lines - Proportional Relationships A proportion is just a comparison of two ratios or fractions. They are equivalent fractions. As one dimension increases or decreases by a factor, the other dimension increases or decreases by the same factor. It is widely used in blue prints when scaling drawings. You can put a ten story building on a 8.5 x 11 sheet of paper. Where one inch on the paper represents ten feet in actual size. Here is a typical proportion… You solve a proportion by cross multiplying…

5. Triangles and Lines - Proportional Relationships A proportion is just a comparison of two ratios or fractions. They are equivalent fractions. As one dimension increases or decreases by a factor, the other dimension increases or decreases by the same factor. It is widely used in blue prints when scaling drawings. You can put a ten story building on a 8.5 x 11 sheet of paper. Where one inch on the paper represents ten feet in actual size. Here is a typical proportion… You solve a proportion by cross multiplying… You try one…

6. Triangles and Lines - Proportional Relationships A proportion is just a comparison of two ratios or fractions. They are equivalent fractions. As one dimension increases or decreases by a factor, the other dimension increases or decreases by the same factor. It is widely used in blue prints when scaling drawings. You can put a ten story building on a 8.5 x 11 sheet of paper. Where one inch on the paper represents ten feet in actual size. Here is a typical proportion… You solve a proportion by cross multiplying… You try one…

7. Triangles and Lines - Proportional Relationships The proportionality theorem describes relationship of parallel lines that go thru triangles. The line is parallel with one of the sides of the triangle. A E D CB

8. Triangles and Lines - Proportional Relationships The proportionality theorem describes relationship of parallel lines that go thru triangles. The line is parallel with one of the sides of the triangle. A E D CB Theorem : If a line is parallel with one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

9. Triangles and Lines - Proportional Relationships The proportionality theorem describes relationship of parallel lines that go thru triangles. The line is parallel with one of the sides of the triangle. A E D CB Theorem : If a line is parallel with one side of a triangle and intersects the other two sides, then it divides those sides proportionally. So the measure of AD and DC is proportional to the measure of AE and EB.

10. Triangles and Lines - Proportional Relationships The proportionality theorem describes relationship of parallel lines that go thru triangles. The line is parallel with one of the sides of the triangle. A ED CB This works because DE creates two similar triangles; ∆ADE and ∆ABC. A

11. Triangles and Lines - Proportional Relationships The proportionality theorem describes relationship of parallel lines that go thru triangles. The line is parallel with one of the sides of the triangle. A ED CB A This works because DE creates two similar triangles; ∆ADE and ∆ABC. Similar Triangles have equal angles and proportional sides.

12. Triangles and Lines - Proportional Relationships The proportionality theorem describes relationship of parallel lines that go thru triangles. The line is parallel with one of the sides of the triangle. A ED CB A This works because DE creates two similar triangles; ∆ADE and ∆ABC. Similar Triangles have equal angles and proportional sides. So if AC = 8 and AD = 4, we have a factor of 2. ∆ABC’s sides are all two times larger than ∆ADE’s sides. 8 4

13. Triangles and Lines - Proportional Relationships The proportionality theorem describes relationship of parallel lines that go thru triangles. The line is parallel with one of the sides of the triangle. A ED CB A This works because DE creates two similar triangles; ∆ADE and ∆ABC. Similar Triangles have equal angles and proportional sides. So if AC = 8 and AD = 4, we have a factor of 2. ∆ABC’s sides are all two times larger than ∆ADE’s sides. What would AE equal if AB = 14 ? 8 4 14

14. Triangles and Lines - Proportional Relationships The proportionality theorem describes relationship of parallel lines that go thru triangles. The line is parallel with one of the sides of the triangle. A ED CB A This works because DE creates two similar triangles; ∆ADE and ∆ABC. Similar Triangles have equal angles and proportional sides. So if AC = 8 and AD = 4, we have a factor of 2. ∆ABC’s sides are all two times larger than ∆ADE’s sides. What would AE equal if AB = 14 ? AE = 7 8 4 14 7

15. Triangles and Lines - Proportional Relationships Let’s solve some problems… A E D CB Example # 1 : Find the measure of AD if AB ║ DE. x 6 4 9

16. Triangles and Lines - Proportional Relationships Let’s solve some problems… A E D CB Example # 1 : Find the measure of AD if AB ║ DE. x 6 4 9 You don’t really have to memorize the rule, just multiply ACROSS the parallel line…

17. Triangles and Lines - Proportional Relationships Let’s solve some problems… A E D CB Example # 1 : Find the measure of AD if AB ║ DE. x 6 4 9 You don’t really have to memorize the rule, just multiply ACROSS the parallel line…

18. Triangles and Lines - Proportional Relationships Let’s solve some problems… A E D CB Example # 2 : Find the measure of EB if CB ║ DE. x 16 9 11

19. Triangles and Lines - Proportional Relationships Let’s solve some problems… A E D CB Example # 2 : Find the measure of EB if CB ║ DE. x 16 9 11

20. Triangles and Lines - Proportional Relationships Let’s solve some problems… A E D CB Example # 3 : Is AB ║ DE ?. 8 3 6 2

21. Triangles and Lines - Proportional Relationships Let’s solve some problems… A E D CB Example # 3 : Is AB ║ DE ?. NO !!! 8 3 6 2

22. Triangles and Lines - Proportional Relationships Parallel Proportional Segments theorem - if three or more parallel line are cut by two transversals, intercepted segments on the transversals are proportional.

23. Triangles and Lines - Proportional Relationships Parallel Proportional Segments theorem - if three or more parallel line are cut by two transversals, intercepted segments on the transversals are proportional. a b c - Lines a, b, and c are parallel

24. Triangles and Lines - Proportional Relationships Parallel Proportional Segments theorem - if three or more parallel line are cut by two transversals, intercepted segments on the transversals are proportional. a b c -Lines a, b, and c are parallel -Transversals t 1 and t 2 cut lines a, b, and c t1t1 t2t2 A B C D E F

25. Triangles and Lines - Proportional Relationships Parallel Proportional Segments theorem - if three or more parallel line are cut by two transversals, intercepted segments on the transversals are proportional. a b c -Lines a, b, and c are parallel -Transversals t 1 and t 2 cut lines a, b, and c t1t1 t2t2 A B C D E F

26. Triangles and Lines - Proportional Relationships Parallel Proportional Segments theorem - if three or more parallel line are cut by two transversals, intercepted segments on the transversals are proportional. a b c Example # 1 : Find the measure of segment AB. t1t1 t2t2 A B C D E F 1012 9 x

27. Triangles and Lines - Proportional Relationships Parallel Proportional Segments theorem - if three or more parallel line are cut by two transversals, intercepted segments on the transversals are proportional. a b c Example # 1 : Find the measure of segment AB. t1t1 t2t2 A B C D E F 1012 9 Again, just multiply across the parallel line… x

28. Triangles and Lines - Proportional Relationships Parallel Proportional Segments theorem - if three or more parallel line are cut by two transversals, intercepted segments on the transversals are proportional. a b c Example # 1 : Are lines a, b, and c parallel ?. t1t1 t2t2 A B C D E F 10 20 126

29. Triangles and Lines - Proportional Relationships Parallel Proportional Segments theorem - if three or more parallel line are cut by two transversals, intercepted segments on the transversals are proportional. a b c Example # 1 : Are lines a, b, and c parallel ?. t1t1 t2t2 A B C D E F 10 20 126 YES !!!