Applying Triangle Sum Properties

1. Applying Triangle Sum Properties Section 4.1

2. Triangles • Triangles are polygons with three sides. • There are several types of triangle: • Scalene • Isosceles • Equilateral • Equiangular • Obtuse • Acute • Right

3. Scalene Triangles • Scalene triangles do not have any congruent sides. • In other words, no side has the same length. 6cm 3cm 8cm

4. Isosceles Triangle • A triangle with 2 congruent sides. • 2 sides of the triangle will have the same length. • 2 of the angles will also have the same angle measure.

5. Equilateral Triangles • All sides have the same length

6. Equiangular Triangles • All angles have the same angle measure.

7. Acute Triangle • All angles are acute angles.

8. Right Triangle • Will have one right angle.

9. Obtuse Angle • Will have one obtuse angle.

10. Exterior Angles vs. Interior Angles • Exterior Angles are angles that are on the outside of a figure. • Interior Angles are angles on the inside of a figure.

11. Interior or Exterior?

12. Interior or Exterior?

13. Interior or Exterior?

14. Triangle Sum Theorem (Postulate Sheet) • States that the sum of the interior angles is 180. • We will do algebraic problems using this theorem. The sum of the angles is 180, so x + 3x + 56= 180 4x + 56= 180 4x = 124 x = 31

15. Find the Value for X 2x + 15 + 3x + 90 = 180 2x + 15 5x + 105 = 180 5x = 75 3x x = 15

16. Corollary to the Triangle Sum Theorem (Postulate Sheet) • Acute angles of a right triangle are complementary. 3x + 10 5x +16

17. Exterior Angle Sum Theorem • The measure of the exterior angle of a triangle is equal to the sum of the non-adjacent interior angles of the triangle

18. 88 + 70 = y 158 = y

19. 2x + 40 = x + 72 • 2x = x + 32 • x = 32

20. Find x and y 46o 8x - 1 2yo 3x + 13

21. 4.1 Apply Congruence and Triangles4.2 Prove Triangles Congruent by SSS, SAS Objectives: • To define congruent triangles • To write a congruent statement • To prove triangles congruent by SSS, SAS

22. Congruent Polygons

23. Congruent Triangles (CPCTC) Two triangles are congruent triangles if and only if the corresponding parts of those congruent triangles are congruent.

24. Congruence Statement When naming two congruent triangles, order is very important.

25. Example Which polygon is congruent to ABCDE? ABCDE  -?-

26. Properties of Congruent Triangles

27. Example What is the relationship between C and F?

28. Third Angle Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

29. Congruent Triangles Checking to see if 3 pairs of corresponding sides are congruent and then to see if 3 pairs of corresponding angles are congruent makes a total of SIX pairs of things, which is a lot! Surely there’s a shorter way!

30. Congruence Shortcuts? Will one pair of congruent sides be sufficient? One pair of angles?

31. Congruence Shortcuts? Will two congruent parts be sufficient?

32. Congruent Shortcuts? Will three congruent parts be sufficient? And if so….what three parts?

33. Section 4.3Proving Triangles are Congruents by SSS

34. Draw any triangle using any 3 size lines • For me I use lines of 5, 4, and 3 cm’s. • Now use the same lengths and see if you can make a different triangle. • Now measure both triangles angles and see what you get. 3cm 53 53 90 5cm 3cm 4cm 5cm 90 37 4cm 37

35. Are the following triangles congruent? Why? 10 6 6 6 6 YES, all sides are equal so SSS a. 10 9 10 8 10 No, all sides are not equal 8 ≠ 6, so fails SSS b. 6 9

36. Use the SSS Congruence Postulate Decide whether the congruence statement is true. Explain your reasoning. SOLUTION Given Given Reflexive Property So, by the SSS Congruence Postulate,

37. 4.4:Prove Triangles Congruent by SAS and HL Goal:Use sides and angles to prove congruence.

38. Vocabulary • Leg of a right triangle: In a right triangle, a side adjacent to the right angle is called a leg. • Hypotenuse:In a right triangle, the side opposite the right angle is called the hypotenuse. Hypotenuse Leg

39. 4.5 ASA and AAS

40. C Y A B X Z Before we start…let’s get a few things straight INCLUDED SIDE

41. Angle-Side-Angle (ASA) Congruence Postulate A A S S A A Two angles and the INCLUDED side

42. A A A A S S Angle-Angle-Side (AAS) Congruence Postulate Two Angles and One Side that is NOT included

43. SSS SAS ASA AAS NO BAD WORDS Your Only Ways To Prove Triangles Are Congruent

44. Things you can mark on a triangle when they aren’t marked. Overlapping sides are congruent in each triangle by the REFLEXIVE property Alt Int Angles are congruent given parallel lines Vertical Angles are congruent

45. Ex 1 DEF NLM

46. D L M F N E Ex 2 What other pair of angles needs to be marked so that the two triangles are congruent by AAS?

47. D L M F N E Ex 3 What other pair of angles needs to be marked so that the two triangles are congruent by ASA?

48. G K I H J Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 4 ΔGIH ΔJIK by AAS

49. B A C D E Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 5 ΔABC ΔEDC by ASA

50. Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 6 E A C B D ΔACB ΔECD by SAS