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 2x + 15 + 3x + 90 = 180 5x + 105 = 180

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. = y
158 = y

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

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

21. To define congruent triangles To write a congruent statement
4.1 Apply Congruence and Triangles 4.2 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. Will three congruent parts be sufficient? And if so….what three parts?
Congruent Shortcuts?
Will three congruent parts be sufficient?
And if so….what three parts?

33. Section 4.3 Proving 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. Before we start…let’s get a few things straightC X Z Y
INCLUDED SIDE

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

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

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

44. Alt Int Angles are congruent given parallel lines
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. Ex 2
What other pair of angles needs to be marked so that the two triangles are congruent by AAS? F D E M L N

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

48. ΔGIH  ΔJIK by AAS Ex 4 G I H J K
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 G I H J K
ΔGIH  ΔJIK by AAS

49. ΔABC  ΔEDC by ASA B A C E D Ex 5
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. B A C E D
Ex 5
ΔABC  ΔEDC by ASA

50. ΔACB  ΔECD by SAS Ex 6 E A C B D
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

51. ΔJMK  ΔLKM by SAS or ASA Ex 7 J K L M
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 7 J K L M
ΔJMK  ΔLKM by SAS or ASA

52. 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 8 J T L K V U
Not possible

53. Postulate 20:Side-Angle-Side (SAS) Congruence Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

54. Example 1:Use the SAS Congruence Postulate
Write a proof.

55. Example 2:Use SAS and properties of shapes

56. Checkpoint

57. Checkpoint

58. Theorem 4.5:Hypotenuse-Leg Congruence Theorem
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second triangle, then the two triangles are congruent.

59. Example 3:Use the Hypotenuse-Leg Theorem
Write a proof.

60. Example 3:Use the Hypotenuse-Leg Theorem

61. Example 3:Use the Hypotenuse-Leg Theorem

62. Example 4:Choose a postulate or theorem

63. Example 4:Choose a postulate or theorem

64. Using Congruent Triangles: CPCTC
Academic Geometry

65. Proving Parts of Triangles Congruent
You know how to use SSS, SAS, ASA, and AAS to show that the triangles are congruent.
Once you have triangles congruent, you can make conclusions about their other parts because, by definition, corresponding parts of congruent triangles are congruent. Abbreviated CPCTC

66. Proving Parts of Triangles Congruent
In an umbrella frame, the stretchers are congruent and they open to angles of equal measure.
Given SL congruent to SR
<1 congruent <2
Prove that the angles formed by the shaft
and the ribs are congruent
shaft
stretcher
rib l r 3 4 1 2 c s

67. Proving Parts of Triangles Congruent
Prove <3 congruent <4
Statement Reason
shaft
stretcher
rib l r 3 4 1 2 c s

68. Proving Parts of Triangles Congruent
Given <Q congruent <R
<QPS congruent <RSP
Prove SQ congruent PR
Statements Reasons r p q s

69. Proving Parts of Triangles Congruent
Given <DEG and < DEF are right angles.
<EDG congruent <EDF
Prove EF congruent EG
Statements Reasons d e f g

70. 4.7 Isosceles and Equilateral Triangles
Chapter 4
Congruent Triangles

71. 4.5 Isosceles and Equilateral Triangles
Isosceles Triangle:
Vertex Angle
Leg
Leg
Base Angles
Base
*The Base Angles are Congruent*

72. Isosceles Triangles Theorem 4-3 Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent B
<A = <C A C

73. Isosceles Triangles
Theorem 4-4 Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent B
Given: <A = <C
Conclude: AB = CB A C

74. Isosceles Triangles Theorem 4-5
The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base B
Given: <ABD = <CBD
Conclude: AD = DC and
BD is ┴ to AC A C D

75. Equilateral Triangles
Corollary: Statement that immediately follows a theorem
Corollary to Theorem 4-3:
If a triangle is equilateral, then the triangle
is equiangular
Corollary to Theorem 4-4:
If a triangle is equiangular, then the triangle
is equilateral

76. Using Isosceles Triangle Theorems
Explain why ΔRST is isosceles. T U
Given: <R = <WVS,
VW = SW
Prove: ΔRST is isosceles R W
Statement
Reason V
1. VW = SW
1. Given S
2. m<WVS = m<S
2. Isosceles Triangle Thm.
3. m<R = m<WVS
3. Given
4. m<S = m<R
4. Transitive Property
5. ΔRST is isosceles
5. Def Isosceles Triangle

77. Using Algebra Find the values of x and y: M ) ) ΔLMN is isosceles
27° y° y°
m<L = m< N = 63
m<LM0 = y = m<NMO
63° N
y + y = 180 x°
y = 180
63° O L
2y = 54
x = 180
y = 27
90 + x = 180
x = 90

78. Landscaping
A landscaper uses rectangles and equilateral triangles for the path around the hexagonal garden. Find the value of x. x°