1. Chapter 4
Congruent Triangles

2. Chapter Objectives Classification of Triangles by Sides
Classification of Triangles by Angles
Exterior Angle Theorem
Triangle Sum Theorem
Adjacent Sides and Angles
Parts of Specific Triangles
5 Congruence Theorems for Triangles

3. Lesson 4.1
Triangles and Angles

4. Lesson 4.1 Objectives Identify the parts of a triangle
Classify triangles according to their sides
Classify triangles according to their angles
Calculate angle measures in triangles

5. Classification of Triangles by Sides
Name
Equilateral
Isosceles
Scalene
Looks Like
Characteristics
3 congruent sides
At least 2 congruent sides
No Congruent Sides

6. Classification of Triangles by Angles
Name
Acute
Equiangular
Right
Obtuse
Looks Like
Characteristics
3 acute angles
3 congruent angles
1 right angles
1 obtuse angle

7. Example 1
You must classify the triangle as specific as you possibly can.
That means you must name
Classification according to angles
Classification according to sides
In that order!
Example
Obtuse isosceles

8. Vertex
The vertex of a triangle is any point at which two sides are joined.
It is a corner of a triangle.
There are 3 in every triangle

9. Adjacent Sides and Adjacent Angles
Adjacent sides are those sides that intersect at a common vertex of a polygon.
These are said to be adjacent to an angle.
Adjacent angles are those angles that are right next to each other as you move inside a polygon.
These are said to be adjacent to a specific side.

10. Special Parts in a Right Triangle
Right triangles have special names that go with it parts.
For instance:
The two sides that form the right angle are called the legs of the right triangle.
The side opposite the right angle is called the hypotenuse.
The hypotenuse is always the longest side of a right triangle.
hypotenuse
legs

11. Special Parts of an Isosceles Triangle
An isosceles triangle has only two congruent sides
Those two congruent sides are called legs.
The third side is called the base.
legs
base

12. More Parts of Triangles
If you were to extend the sides you will see that more angles would be formed.
So we need to keep them separate
The three original angles are called interior angles because they are inside the triangle.
The three new angles are called exterior angles because they lie outside the triangle.

13. Example 2
Classify the following triangles by their sides and their angles.
Scalene
Scalene
Obtuse
Right
Isosceles
Acute

14. Theorem 4.1: Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180o. B
mA + mB + mC = 180o C A

15. Example 3
Solve for x and then classify the triangle based on its angles. 75
Acute 50
3x + 2x + 55 = 180
Triangle Sum Theorem
5x + 55 = 180
Simplify
5x = 125
SPOE
x = 25
DPOE

16. Theorem 4.2: Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. A B C
m A +m B = m C

17. Example 4 Solve for x x = 50 + 70 Exterior Angle Theorem x = 120
Simplify

18. Corollary to the Triangle Sum Theorem
A corollary to a theorem is a statement that can be proved easily using the original theorem itself.
This is treated just like a theorem or a postulate in proofs.
The acute angles in a right triangle are complementary. C A B
mA + mB = 90o

19. Homework 4.1 In Class In HW Due Tomorrow 1-9p
In HW
10-26, 31-39, 41-47, 49, 50, 52-68
Due Tomorrow

20. Congruence and Triangles
Lesson 4.2
Congruence and Triangles

21. Lesson 4.2 Objectives
Identify congruent figures and their corresponding parts.
Prove two triangles are congruent.
Apply the properties of congruence to triangles.

22. Congruent Triangles When two triangles are congruent, then
Corresponding angles are congruent.
Corresponding sides are congruent.
Corresponding, remember, means that objects are in the same location.
So you must verify that when the triangles are drawn in the same way, what pieces match up?

23. Naming Congruent Parts
Be sure to pay attention to the proper notation when naming parts.
 ABC   DEF
This is called a congruence statement. A B C F D
A   D
B   E
C   F
and
AB  DE
BC  EF
AC  DF E

24. Theorem 4.3: Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

25. Prove Triangles are Congruent
In order to prove that two triangles are congruent, we must
Show that ALL corresponding angles are congruent, and
Show that ALL corresponding sides are congruent.
We must show all 6 are congruent!

26. Example 5 Complete the following statements. Segment EF  ___________
segment OP
P  ________ F
G  ________ Q
mO = ________
110o
QO = ________
7 km
GFE  __________
QPO
Yes, the order is important!

27. Theorem 4.4: Properties of Congruent Triangles
Reflexive Property of Congruent Triangles
 ABC   ABC
Reflexive Property of 
Symmetric Property of Congruent Triangles
If  ABC   DEF, then  DEF   ABC.
Symmetric Property of 
Transitive Property of Congruent Triangles
If  ABC   DEF and  DEF   JKL, then  ABC   JKL.
Transitive Property of 

28. Homework 4.2
None!

29. Proving Triangles are Congruent: SSS & SAS
Lesson 4.3
Proving Triangles are Congruent:
SSS
&
SAS

30. Lesson 4.3 Objectives
Prove triangles are congruent using the SSS Congruence Postulate
Prove triangles are congruent using the SAS Congruence Postulate

31. Postulate 19: Side-Side-Side Congruence Postulate
If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

32. Postulate 20: Side-Angle-Side 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.

33. Which One Do I Use? Remember there are 6 parts to every triangle.
Identify which parts of the triangle do you know (100% sure) are congruent.
Rotate around the triangle keeping one thing in mind.
Cannot rotate so that 2 parts in a row are missed!
That means as you rotate by counting angle, then side, then angle, then side, then angle, and then side you cannot miss two pieces in a row!
You can skip 1, but not 2!!
Be sure the pattern that you find fits the same pattern in the same way from the other triangle.
If it fits, they are congruent.

34. Example 6 Decide whether or not the congruence statement is true.
Explain your reasoning.
The statement is not true because the vertices are
out of order.
Reflexive Property of Congruence
Reflexive Property of Congruence
Because the segment is shared between two triangles, and yet it is the same segment
The statement is not true because the vertices are
out of order.
The statement is true because of
SSS Congruence

35. Example 7
Decide whether or not there is enough information to conclude SAS Congruence.
Yes! No
Reflexive Property of Congruence
Yes!

36. Homework 4.3
In Class
1-5 p HW
6-20
Due Tomorrow

37. Proving Triangles are Congruent: ASA & AAS
Lesson 4.4
Proving Triangles are Congruent:
ASA
&
AAS

38. Lesson 4.4 Objectives
Prove that triangles are congruent using the ASA Congruence Postulate
Prove that triangles are congruent using the AAS Congruence Theorem

39. Postulate 21: Angle-Side-Angle Congruence
If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.

40. Theorem 4.5: Angle-Angle-Side Congruence
If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of the second triangle, then the two triangles are congruent.

41. Example 8
Complete the proof
Given
Given
Reflexive POC
SSS Congruence

42. Homework 4.4
In Class
1-7 p HW
8-18 evens
Due Tomorrow

43. Using Congruent Triangles
Lesson 4.5
Using Congruent Triangles

44. Lesson 4.5 Objectives
Observe that corresponding parts of congruent triangles are congruent

45. Showing Triangles are Congruent
You only have 4 shortcuts right now to show that two triangles are congruent to each other.
SSS Congruence
SAS Congruence
ASA Congruence
AAS Congruence
Otherwise you need to show all 6 parts of a triangle have matching congruent parts to another triangle.
If you can use one of the above 4 shortcuts to show triangle congruency, then we can assume that all corresponding parts of the triangles are congruent as well.

46. Surveying MNP  MKL Segment NM  Segment KM Definition of a midpoint
Given
Segment NM  Segment KM
Definition of a midpoint
LMK  PMN
Vertical Angles Theorem
KLM  NPM
ASA Congruence
Segment LK  Segment PN
Corresponding Parts of Congruent Triangles

47. Example 9
Tell which triangles you show to be congruent in order to prove the statement is true.
What postulate or theorem would help you show the triangles are congruent.
Show:
STV  UTV
Show:
Segment XY  Segment ZW
Alternate Interior Angles Theorem (Parallel Lines)
Reflexive Property of Congruence
Reflexive Property of Congruence
STV  UTV
WXZ  YZX
SSS Congruence
ASA Congruence
Corresponding Parts of Congruent Triangles
Corresponding Parts of Congruent Triangles

48. Homework 4.5
In Class
1-3 p HW
4-18, 25-36
Due Tomorrow

49. Isosceles, Equilateral, and Right Triangles
Lesson 4.6
Isosceles,
Equilateral,
and
Right Triangles

50. Lesson 4.6 Objectives
Use properties of isosceles and equilateral triangles.
Identify more properties based on the definitions of isosceles and equilateral triangles.
Use properties of right triangles.

51. Isosceles Triangle Theorems
Theorem 4.6: Base Angles Theorem
If two sides of a triangle are congruent, then the angles opposite them are congruent.
Theorem 4.7: Converse of Base Angles Theorem
If two angles of a triangle are congruent, then the sides opposite them are congruent.

52. Example 10 Solve for x Theorem 4.7 Theorem 4.6 4x + 3 = 15
7x + 5 = x + 47
4x = 12
6x + 5 = 47
x = 3
6x = 42
x = 7

53. Equilateral Triangles
Corollary to Theorem 4.6
If a triangle is equilateral, then it is equiangular.
Corollary to Theorem 4.7
If a triangle is equiangular, then it is equilateral.

54. Example 11 Solve for x Corollary to Theorem 4.6
It does not matter which two sides you set equal to each other, just pick the pair that looks the easiest!
Corollary to Theorem 4.6
In order for a triangle to be equiangular, all angles must equal…
2x + 3 = 4x - 5
3 = 2x - 5
5x = 60
8 = 2x
x = 12
x = 4

55. Theorem 4.8: 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 right triangle, then the two triangles are congruent.
Abbreviate using HL

56. Example 12
Determine if enough information is given to conclude the triangles are congruent using HL Congruence
Neither triangle is a right triangle, so…
Reflexive Property of Congruence
Reflexive Property of Congruence
Not congruent
Yes they are congruent!

57. Homework 4.6
In Class
1-7 p HW
8-28 even, 33, 34
Due Tomorrow

58. Triangles And Coordinate Proof
Lesson 4.7
Triangles
And
Coordinate Proof

59. Lesson 4.7 Objectives Place geometric figures in a coordinate plane.
Use the Distance Formula to verify congruent triangles.

60. Coordinate Proof
A coordinate proof involves placing geometric figures in a coordinate plane.
Then you employ the following tools to prove concepts from your picture
Distance Formula
Midpoint Formula
(x2 – x1)2 + (y2 – y1)2
(x1 + x2)
(y1 + y2) ( ) , 2

61. Homework 4.7 WS