1. 4-2 Angles in a Triangle
Mr. Dorn
Chapter 4

2. 4-2 Angles in a Triangle Angle Sum Theorem:
The sum of the angles in a triangle is 180.
x + y + z = 180 y x z

3. 4-2 Angles in a Triangle Third Angle Theorem:
If two angles of a triangle are congruent to two angles of another triangle, then the third angles are congruent. A Y
Given: C B
Conclusion: Z X

4. 4-2 Angles in a Triangle Exterior Angle Theorem:
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
m = x + y y x z m

5. 4-3 Congruent Triangles

6. Corresponding Parts of Congruent Triangles are Congruent.
CPCTC:
Corresponding Parts of Congruent Triangles are Congruent. A X Z B C Y

7. Example 1
Complete the sentence. Q P N L M R

8. Example 2 Given: CA = 14, AT = 18, TC = 21, and DG = 2x + 7 Find x.O
14 = 2x
7 = x 14 18 G C T D 21
2x + 7

9. Example 3 Given: AC = 7, BC = 10, DF = 2x + 4, and DE = 4x.
Find x and AB.
AB = 4x
7 = 2x + 4
AB = 4(1.5)
3 = 2x
AB = 6 B
1.5 = x E 10 4x F A C D 7
2x + 4

10. 4-4 Proving Triangles Congruent

11. 4-4 Proving Congruent Triangles
SSS Postulate:
(Side-Side-Side)
If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. Q M P R

12. 4-4 Proving Congruent Triangles
SAS Postulate:
(Side-Angle-Side)
If two sides of one triangle and the included angle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent. B C X Z Y

13. 4-4 Proving Congruent Triangles
ASA Postulate:
(Angle-Side-Angle)
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 triangles are congruent. B C X Z Y

14. 4-4 Proving Congruent Triangles
AAS Postulate:
(Angle-Angle-Side)
If two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of a second triangle, then the triangles are congruent. B C X Z Y

15. 4-4 Proving Congruent Triangles
Four Postulates that will work for any type of triangle:
SSS, SAS, ASA, AAS
*Just remember: Any Combination works as longs as it does not spell a bad word forward or back.

16. Example 1 Find x. x = 30 Use the Angle Sum Theorem! 3x
3x + x +2x = 180 2x
6x = 180 x
x = 30

17. Example 2 Find x. x = 22 Use the Exterior Angle Theorem!
(103 – x) +2x = 6x - 7
(103-x) 2x
103 + x = 6x - 7
103 = 5x - 7
110 = 5x
(6x-7)
x = 22

18. Example 3 Find x. x = 65 Use the Angle Sum Theorem! y + 53 + 80 = 18050
n = 180 53
n = 180 x
n = 68
Vertical angles are congruent, so… 62 47 y n 80 68
x = 180
x = 180

19. Example 4
Are the two triangles congruent? If so, what postulate identifies the two triangles as congruent?
Yes, SAS

20. Example 5
Are the two triangles congruent? If so, what Postulate identifies the two triangles as congruent?
Not Congruent!

21. Example 6 Given: Prove: ; S is the midpoint of Statements Reasons 1.
QRS  TRS Q T S
Statements
Reasons 1.
1. Given
; S is the midpoint of 2.
2. Midpoint Theorem 
3. Reflexive Property 3.  4.
QRS  TRS
4. SSS

22. Example 7 Given: bisects BAC and BDC. Prove: BAD  CAD Proof:
Since
bisects BAC and BDC, BAD  CAD and BDA  CDA. 
by the Reflexive Property.
By ASA, BAD  CAD.