Geometry:
Section 2.6: Proving Angles Congruent
Goal: Students will understand how to defend claims about angle relationships.
A theorem is a conjecture or statement that you PROVE true using deductive reasoning.
You prove each step using any of the following information:
1. Definitions
2. Properties
3. Postulates
4. Previously Proven Theorems
Vertical Angle Theorem: Vertical angles are congruent
Proof of the vertical angle theorem. Claim: Vertical Angles are Congruent.
Given ∠1 and ∠3 are vertical angles
Prove: ∠1 ≅ ∠3
Statements
Reasons
Problem 1: Use the vertical angle theorem:
Find the value of x.
Problem 2: Proof Using the vertical angle theorem:
Given: ∠1 ≅ ∠4
Prove: ∠2 ≅ ∠3
Congruent Supplements Theorem: If two angles are supplements of the
same angle (or of congruent angles), then the two angles are congruent.
Problem 3: Writing a paragraph proof to prove
The Congruent Supplement Theorem
Given ∠1 and ∠3 are supplementary
Prove: ∠2 and ∠3 are supplementary
Prove: ∠1 ≅ ∠2
Congruent Complements Theorem: If two angles are complements of the
same angle (or of congruent angles), then the two angles are congruent.
Problem 4: Name a pair of congruent angles in each figure. Justify your
answer.
5. Given: 2 is complementary to 3.
6. Given: AYZ  BYW
1.Complete the proofs by filling in the blanks
Given: A  BDA
Prove: x = 5
Statements
Reasons
1)
1) Given
2)
2) Vertical Angles are .
3) A  CDE
3)
4)
4) Definition of Congruence
5) 11x + 20 = 12x + 15
5)
6)
6) Subtraction Property of Equality
7)
7)
Given: 5 2
Prove: 8  4
Statements
Reasons
1)
1) Given
2) 2  4
2)
3)
3)
4)
4)
5) 8  4
5)
5. Use a two column proof or a paragraph proof to
develop the proof below:
Given: 1 and 2 are complementary
2 and 3 are complementary
BD bisects ABC
Prove: m1 = 45