Section 2­8 Proving Angle Relationships
Postulate 2.10 Protractor Postulate
Given AB and a number r between 0 and 180, there is exactly one ray with endpoint A, extending on either side of AB, such that the measure of the angle formed is r.
­ In the last section, we learned about the Segment
Addition Postulate.
­ A similar relationship exists between the measures
of angles.
Postulate 2.11 Angle Addition Postulate
If R is in the interior of , then If , then R is in the interior of
.
1
Ex 1 In the figure below, and
, find A
B
D
C
­ The Angle Addition Postulate can be used with
other angle relationships to provide additional
theorems relating to angles.
Theorem 2.3 Supplement Theorem
If two angles form a linear pair, then they are supplementary angles.
Theorem 2.4 Complement Theorem
If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles.
2
Ex 2 If and form a linear pair and , find Ex 3 Find the measure of each numbered angle.
a.
b.
c.
d.
3
­ The properties of algebra that applied
to the congruence of segments and the
equality of their measures also hold true
for the congruence of angles and the
equality of their measures.
Theorem 2.5 Congruence of angles is reflexive, symmetric, and transitive.
REFLEXIVE PROPERTY
SYMMETRIC PROPERTY If ,
then
TRANSITIVE PROPERTY If and , then ­ Algebraic properties can be applied to
prove theorems for congruence
relationships involving supplementary
and complementary angles.
4
Theorem 2.6 Angles supplementary to the same angle or to congruent angles are congruent.
Abbreviation: suppl. to same or
are
Example: If and ,
then
Theorem 2.7 Angles complementary to the same angle or to congruent angles are congruent.
Abbreviation: compl. to same or
are
Example: If and ,
then
5
Proof of Theorem 2.7
Given: and are complementary
and are complementary
Prove:
Statements Reasons 1. and are 1._______________
complementary
and are
complementary
2.
2._______________
3.
3._______________
4.
4._______________
5.
5._______________
6
Ex 3 Complete the following proof of Theorem 2.6.
Given: and are supplementary
and are supplementary
Prove: Statements Reasons 1. and are
1.______________
supplementary
and are
supplementary
2.
2.______________
3.
3.______________
4.
4.______________
5.
5.______________
6.
6.______________
7.
7.______________
7
Theorem 2.8 Vertical Angles Theorem
If two angles are vertical angles, then they are congruent.
Ex 4 If and are vertical angles and and , find and .
Ex 5 Find the measure of each numbered angle.
a.
b.
8
Theorem 2.9 Perpendicular lines intersect to form four right angles.
Theorem 2.10 All right angles are congruent.
A
B
9
Theorem 2.11 Perpendicular lines form congruent adjacent angles.
2
1
and are adjacent
Theorem 2.12 If two angles are congruent and supplementary, then each angle is a right angle.
1 2
If and , then and are right angles.
Theorem 2.13 If two congruent angles form a linear pair, then they are right angles.
1
2
If and and form a linear pair, then and are right angles.
10
Ex 6 Write a two­column proof.
Given: VX bisects
VY bisects
Prove:
Statements Reasons 11
Ex 7 Complete the following proof.
Given:
Prove:
Statements Reasons 1. _______________ 1. ______________
2.
2. ______________
3.
3. ______________
4. _______________
4. Substitution
5. _______________
5. ______________
6. _______________
6. ______________
Assign Pgs. 111 ­ 114 # 1, 3 ­ 5, 7 ­ 9, 12 ­ 24, 27 ­ 32,
38 ­ 41, 44, 47 ­ 54
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