1. 2.6 Proving Geometric Relationships
Geometry
2.6 Proving Geometric Relationships
How can you prove angles congruent?

2. Geometry 2.6 Proving Geometric Relationships
Topic/Objectives
Use angle congruence properties.
Prove theorems about special angle pairs.
June 1, 2018
Geometry 2.6 Proving Geometric Relationships

3. Theorem 2.2: Angle Congruence
Angle congruence is reflexive, symmetric and transitive.
Examples
Reflexive: ABC  ABC
Symmetric: If A  B, then B  A
Transitive: If A  B, and B  C, then A  C
The proofs are similar to those for segment congruence
June 1, 2018
Geometry 2.6 Proving Geometric Relationships

4. Geometry 2.6 Proving Geometric Relationships
Example
Given: 1  2, 3  4 and 2  3
Prove: 1  4
Statement
Reason
1. 1  2
1. Given
2. 2  3
2. Given
3. 3  4
3. Given 4 1
4. 1  4
4. Transitive 2 3
June 1, 2018
Geometry 2.6 Proving Geometric Relationships

5. Theorem 2.3 Right Angle Conguence
All Right Angles are congruent.
(This should be obvious: all right angles measure 90° and so they all have the same measure and hence are congruent. We won’t prove this.)
All Rt s 
June 1, 2018
Geometry 2.6 Proving Geometric Relationships

6. Theorem 2.4 Congruent Supplements Theorem
If two angles are supplementary to the same angle, or to congruent angles, then they are congruent.
 supp
Example
mA + mB = 180°
mA + mC = 180°
Thus, B  C.
They are both supplementary to the same angle, A.
June 1, 2018
Geometry 2.6 Proving Geometric Relationships

7. Theorem 2.5 Congruent Complements Theorem
If two angles are complementary to the same angle, or to congruent angles, then the angles are congruent.
 comp
Example
mR + mS = 90°
mT + mS = 90°
Thus, R  T
R and T are complementary to the same angle, S.
June 1, 2018
Geometry 2.6 Proving Geometric Relationships

8. Postulate 2.8 Linear Pair Postulate
The angles of a linear pair are supplementary. 1 2
m1 + m2 = 180°
June 1, 2018
Geometry 2.6 Proving Geometric Relationships

9. Geometry 2.6 Proving Geometric Relationships
Vertical Angles are congruent.
Prove: 1  2 1 2 3
June 1, 2018
Geometry 2.6 Proving Geometric Relationships

10. Geometry 2.6 Proving Geometric Relationships
Vertical Angles are congruent.
Prove: 1  2 1 2 3
1 and 3 form a linear pair and by Post 12, their sum is 180.
Similarly, the sum of 2 and 3 is 180.
Thus, 1  2 since they are supplementary to the same angle. (See Theorem 2.4.)
June 1, 2018
Geometry 2.6 Proving Geometric Relationships

11. Geometry 2.6 Proving Geometric Relationships
Vertical
Angles
are
Congruent
June 1, 2018
Geometry 2.6 Proving Geometric Relationships

12. Geometry 2.6 Proving Geometric Relationships
Practice Examples
Find m1, m2, m3.
135° (supp. s)
45° 1 2 3
45° (vert. s )
135° (vert. s )
June 1, 2018
Geometry 2.6 Proving Geometric Relationships

13. Geometry 2.6 Proving Geometric Relationships
Practice
Solve for x.
(3x + 20)°
(5x + 8)°
Vertical Angles are Congruent
5x + 8 = 3x + 20
2x = 12
x = 6
June 1, 2018
Geometry 2.6 Proving Geometric Relationships

14. Geometry 2.6 Proving Geometric Relationships
Practice
Solve for x, and find each angle.
Linear Pair: Sum is 180
4x x + 25 = 180
9x + 45 = 180
9x = 135
x = 15
4(15) + 20 = 80°
5(15) + 25 = 100°
(4x + 20)°
(5x + 25)°
June 1, 2018
Geometry 2.6 Proving Geometric Relationships