1. Parallel Lines and Proportional Parts
Similar Triangles
Parallel Lines and Proportional Parts

2. Angle-Angle (AA) Similariry
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

3. Examples
Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

4. Examples
Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.
No, they are not similar; the angles are not congruent

5. Examples
Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

6. Examples
Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.
Yes; △LPQ ≅ △LJK

7. Side-Side-Side (SSS) Similarity
If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

8. Side-Angle-Side (SAS) Similarity
If the lengths of two sides of one triangle are proportional to the lengths of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.

9. Examples
Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.
Yes; △JKL ∼ △QPM; SSS, ratio 4:3

10. Examples
Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

11. Examples
Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.
Yes; △WTZ ∼ △WYX; SAS, ratio 1:2

12. Examples
Find each measure.
QP and MP

13. Examples Find each measure. QP and MP 5/(5+x) = 6/(9 3/5) 48 = 30+6x
QP = 3; MP = 8

14. Examples
Find each measure.
WR and RT

15. Examples Find each measure. WR and RT (x+6)/8 = (2x+6)/10
WR = 8, RT = 10

16. Triangle Proportionality Theorem
If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the sides into segments of proportional lengths.

17. Converse of Triangle Proportionality Theorem
If a line intersects two sides of a triangle and separates the sides into proportional corresponding segments, then the line is parallel to the third side of the triangle.

18. Examples
If PS = 12.5, SR = 5, and PT = 15, find TQ.

19. Examples If PS = 12.5, SR = 5, and PT = 15, find TQ. 12.5/5 = 15/x
x = 6; TQ = 6

20. Examples
In △DEF, DG is half the length of GF, EH = 6, and HF = 10. Is DE ∥ GH?

21. Examples
In △DEF, DG is half the length of GF, EH = 6, and HF = 10. Is DE ∥ GH?
No; GF:DG = 2:1; HF:EH = 5:3

22. Midsegment
A midsegment of a triangle is a segment with endpoints that are the midpoints of two sides of the triangle.
Every triangle has three midsegments.

23. Triangle Midsegment Theorem
A midsegment of a triangle is parallel to one side of the triangle, and its length is one half the length of that side.

24. Examples
Find the measure. DE

25. Examples
Find the measure. DE
15/2 = 7.5

26. Examples
Find the measure. DB

27. Examples
Find the measure. DB
FE = ½ AB
DB = ½ AB
FE = DB = 9.2

28. Examples
Find the measure.
m∠FED

29. Examples Find the measure. m∠FED FE ∥ AB ∠FED ≅ ∠EDB; Alt. Int. ∠

30. Proportional Parts of Parallel Lines
If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

31. Congruent Parts of Parallel Lines
If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

32. Examples
Find x.

33. Examples
Find x.
7x-2 = 4x+3
3x = 5
x = 5/3

34. Examples
Find x.

35. Examples
Find x.
2x+1 = 3x-5
6 = x