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Transcript
Name:_________________________
2-8 Proving Angle Relationships Notes
Objectives:


Students will write proofs involving supplementary and complementary angles
Students will write proofs involving congruent and right angles.
Postulate 2.11: Angle Addition Postulate:

If S is in the interior of ∠, then ∠ + ∠ = ∠.
Example 1: Using the Angle Addition Postulate:
If ∠ = 103 find the value of x. Then find ∠ and ∠
x=
∠ =
∠ =
Supplementary and Complementary Angles

Theorem 2.3: If two angles form a ___________ ____________, then they are
___________________ angles.
1
2
∠1 + ∠2 = 180

Theorem 2.4: If the non-common sides of two _____________ angles form a
____________ angle, then the angles are complimentary angles.
1
1 2
∠1 + ∠2 = 90
Example 2: Using Supplementary Angles
If ∠1 and ∠2 form a linear pair, and ∠1 = 4 − 5 and the ∠2 = 14 + 5, find x and the measurements of ∠1 and
∠2.
x=
∠1 =
∠2 =
1
2
Theorem 2.5
Angle Congruence
Reflexive Property
∠ ≅ ∠
Symmetric Property
If ∠ ≅ ∠, ∠ ≅ ∠
Transitive Property
If ∠ ≅ ∠ and ∠ ≅ ∠, then ∠ ≅ ∠
Theorem 2.6: Angles supplementary to the same angle
or to congruent angles are congruent.
If ∠1 + ∠2 = 180, and ∠2 + ∠3 = 180, then
______ ≅ _________.
Example 3: Proof of Theorem 2.6
Given: ∠1 and ∠2 are supplementary
∠2 and ∠3 are supplementary
Prove: ∠1 ≅ ∠3
Statements
Reasons
1. ∠1 and ∠2 are supplementary
∠2 and ∠3 are supplementary
2. ∠1 + ∠2 = 180
∠2 + ∠3 = 180
1. Given
3.
3. Substitution.
4. ∠1 = ∠3
4.
5. ∠1 ≅ ∠3
5.
2.
Theorem 2.7: Angles complementary to the same angle or to congruent angles
are congruent.
1
If ∠1 + ∠2 = 90, and ∠2 + ∠3 = 90, then ______ ≅ _________.
2
3
Theorem 2.8: If two angles are vertical angles, then they are
congruent ( ≅ )
Example 4: Prove Vertical Angles are ≅
Given: ∠1 and ∠2 form a linear pair.
∠2 and ∠3 form a linear pair.
Prove: ∠1 ≅ ∠3
Statements
Reasons
1. ∠1 and ∠2 form a linear pair
∠2 and ∠3 form a linear pair
1. Given
2. ∠1 and ∠2 are supplementary
∠2 and ∠3 are supplementary
2. ______________________________
3. ∠1 ≅ ∠3
3. ______________________________
∠1 ≅ ∠3 and ∠2 ≅ ∠4
Example 5: Using Vertical Angles
Find the value of x using vertical angles.
Theorem 2.9:
Perpendicular lines intersect to four ______________________.
Theorem 2.10:
All right angles are congruent.
Theorem 2.11:
Perpendicular lines form congruent adjacent angles.
Theorem 2.12:
If two angles are congruent and supplementary, then each angle is a __________________.
Theorem 2.13:
If two congruent angles form a linear pair, then they are ___________________.