Download Geometry 2-8 - Proving Angle Relationships

Geometry 2-8 - Proving Angle Relationships
A. Supplementary and Complementary Angles
1. Postulate 2-11 - Angle Addition Postulate - If
R is in the interior of ∠PQS , then m∠PQR +
m∠ _____ = m∠PQS . The converse is also true:
If m∠PQR + m∠RQS = m∠PQS , then R is in
the interior of ∠PQS .
Example 1: If m∠PQR = 2 x + 12 ,
m∠RQS = 3x − 2 , and m∠PQS = 50,
find the value of x and m∠PQR .
P
Q
R
S
P
Q
R
S
2. Theorem 2-3 – Supplemental Theorem - If two angles form a linear pair, then
they are ________________.
3. Theorem 2-4 - Complement Theorem - If the noncommon sides of two
adjacent angles form a right angle, then the angles are complimentary angles.
Example 2: If ∠1 and ∠2 form a linear
pair, and m ∠1 = 4 x − 5 and
m ∠2 = 14 x + 5 , find x, and m ∠1 .
2
1
4. Theorem 2-5 - Congruence of angles is reflexive, symmetric, and transitive
a. Reflexive Property - ∠1 ≅ ∠1
b. Symmetric Property - If ∠1 ≅ ∠2 , then_______.
c. Transitive Property - If ∠1 ≅ ∠2 , and ∠2 ≅ ∠3 , then ∠1 ≅ ∠3 .
5. Theorem 2-6 - Angles supplementary to the
same angle or to congruent angles are congruent.
a. If m∠1 + m∠2 = 180 , and
m∠2 + m∠3 = 180 , then ___________
2
1
3
Example 3: Proof of Theorem 2-6
Given: ∠1 and ∠2 are supplementary
∠2 and ∠3 are supplementary
Prove: ∠1 ≅ ∠3
Proof:
Statements
1. ∠1 and ∠2 are supplementary
∠2 and ∠3 are supplementary
2. m∠1 + m∠2 = 180
m∠2 + m∠3 = 180
3.
4. m∠1 = m∠3
5. ∠1 ≅ ∠3
6. Theorem 2-7 - Angles complementary
To the same angle or to congruent
angles are congruent.
a. If m∠1 + m∠2 = 90 , and
m∠2 + m∠3 = 90 , then __________.
3
2
1
Reasons
1. Given
2.
3. Substitution
4.
5.
1
2
3
7. Theorem 2-8 - If two angles are vertical angles, then they are congruent.
Example 4: If ∠1 and ∠2 are vertical angles, and m∠1 = d − 32 , and
m∠2 = 175 − 2d , find m∠1 and m∠2 .
8. Theorem 2-9 - Perpendicular lines intersect to form four _______________.
9. Theorem 2-10 - All right angles are congruent.
10. Theorem 2-11 - Perpendicular lines form congruent adjacent angles.
11. Theorem 2-12 - If 2 angles are congruent and supplementary, then each
angles is a right angle.
12. If two congruent angles form a linear pair, then they are _____________.
HW: Geometry 2-8 p. 112-114
16-32 all, 39, 44-45, 47-55 odd