1. Proving Angles Congruent
During this lesson, you will:
Determine and apply conjectures about angle relationships
Prove and apply theorems about angles
Mrs. McConaughy
Geometry

2. Part I: Discovering Angle Relationships
Mrs. McConaughy
Geometry

3. Definitions: Special Angle Pairs
complementary angles
Two angles are ___________________ if their measures add up to 90.
Two angles are ___________________ if their measures add up to 180.
supplementary angles
Mrs. McConaughy
Geometry

4. Vocabulary Review: Pairs of Angles Formed By Intersecting Lines
Opposite (non-adjacent) angles, formed by intersecting lines, which share a common vertex and whose sides are opposite rays are called ______________.
Adjacent angles formed by intersecting lines which share a common vertex, a common side, and with one side formed by opposite rays are called ____________.
vertical angles
linear pairs
Mrs. McConaughy
Geometry

5. Given the following diagram, identify all vertical angle pairs:1 2 4 3
∠ 1 & ∠ 3
∠ 2 & ∠ 4
Mrs. McConaughy
Geometry

6. Given the following diagram, identify all linear pairs of angles:2 4 8 6
∠ 6 & ∠ 8
∠ 8 & ∠ 2
∠ 4 & ∠ 6
∠ 2 & ∠ 4
Mrs. McConaughy
Geometry

7. Investigative Results:
If two angles are vertical angles, then
the angles are _________.
(VERTICAL ANGLES CONJECTURE)
If two angles are a linear pair of angles, then the angles are ______________ (____).
(LINEAR PAIR CONJECTURE)
congruent
supplementary
Mrs. McConaughy
Geometry

8. If two angles are equal and supplementary, what must be true of the two angles?
If two angles are both equal in measure and supplementary, then each angle measures ____.
(EQUAL SUPPLEMENTS CONJECTURE) 90
Mrs. McConaughy
Geometry

9. Examples: Use your conjectures to find the measure of each lettered angle.
Example A
a b c
Example B a
b c 70 30
Mrs. McConaughy
Geometry

10. Examples: Use your conjectures to a. find the value of the variable.
EXAMPLE C
(3y + 20)
(5y – 16)
EXAMPLE D
(2x – 6)
Vertical Angles
Are Congruent
Linear Pairs Are
Supplementary
(3x + 31)
5y – 16 = 3y + 20
3x x – 6 = 180
5y = 3y + 36
5y = 3y + 36
5x + 25 = 180
2y = 36
5x = 155
Mrs. McConaughy
Geometry
y = 18
x = 31

11. Homework Assignment:
Discovering Angle Relationships WS 1-5 all, plus select problems from text.
Mrs. McConaughy
Geometry

12. Part 2: Proving and Applying Theorems About Angles
Mrs. McConaughy
Geometry

13. Congruent Supplements Theorem
If two angles are supplements of congruent angles, then the two angles are congruent.
Mrs. McConaughy
Geometry

14. Given: ∠A supp ∠B; ∠C supp ∠D; ∠B  ∠C Prove: ∠A  ∠D
STATEMENT
REASON 1. 2. 3. 4. 5. 6. 7.
∠A supp ∠B; ∠C supp ∠D
Given.
m ∠A + m ∠B = 180; m ∠C + m ∠D = 180
Def. of supp. ∠’s
m ∠A + m ∠B = m ∠C + m ∠D
. Substitution Prop. of =
∠B  ∠C
Given.
Def. of 
m ∠B = m ∠C
m ∠A = m ∠ D
Subtraction Prop. of =
∠A  ∠C
Def. of 
Mrs. McConaughy
Geometry

15. Vocabulary: Corollary
A _________ of a theorem is a theorem whose proof contains only a few additional statements in addition to the original proof.
EXAMPLE:
If two angles are supplements of the same angle, then the two angles are congruent.
Mrs. McConaughy
Geometry

16. Congruent Complements Theorem
If two angles are complements of congruent angles, then the two angles are congruent.
COROLLARY: If two angles are complements of the same angle, then the two angles are congruent.
Mrs. McConaughy
Geometry

17. Given: ∠A comp . ∠B; ∠C comp. ∠D; ∠B  ∠C Prove: ∠A  ∠D
STATEMENT
REASON 1. 2. 3. 4. 5. 6. 6 7.
∠A comp ∠B; ∠C comp ∠D
Given.
m ∠A + m ∠B = 90; m ∠C + m ∠D = 90
Def. of supp. ∠’s
m ∠A + m ∠B = m ∠C + m ∠D
Substitution Prop. of =
∠B  ∠C
Given.
m ∠B = m ∠C
Def. of 
m ∠A = m ∠ D
(-) Prop. of =
∠A  ∠C
Def. of 
Mrs. McConaughy
Geometry

18. Vertical angles are congruent.
Theorem
Vertical angles are congruent.
Given: ∠ 1 and ∠ 3 are
vertical angles
Prove: ∠ 1  ∠ 3
Mrs. McConaughy
Geometry

19. Given: ∠ 1 and ∠ 3 are vertical angles Prove: ∠ 1  ∠ 3
STATEMENT
REASON 1. 2. 3. 4. 5.
∠ 1 and ∠ 3 are vertical angles
Given.
Def. of linear pair
∠ 1 and ∠2 are a linear pair
∠ 2 and ∠3 are a linear pair
Def. of linear pair
∠1 supp ∠2; ∠3 supp ∠2
Linear pairs are supp.
∠ 1  ∠ 3
Supp. of same ∠ 
Mrs. McConaughy
Geometry

20. All right angles are congruent.
Theorem
All right angles are congruent.
Mrs. McConaughy
Geometry

21. Final Checks for Understanding
In the following exercises, ∠ 1 and ∠ 3 are a linear pair, ∠ 1 and ∠ 4 are a linear pair, and ∠ 1 and ∠ 2 are vertical angles. Is the statement true?
∠ 1  ∠ b. ∠ 1  ∠ 2
c. ∠ 1  ∠ d. ∠ 3  ∠ 2
e. ∠ 3  ∠ 4
f. m∠ 2 + m ∠ 3 = 180
Mrs. McConaughy
Geometry

22. Homework Assignment
Pages : all all. Prove: 19 & 35 all.
Mrs. McConaughy
Geometry