Download Applied Geometry - South Harrison County R2

Geometry
Lesson 2 – 8
Proving Angle Relationships
Objective:
Write proofs involving supplementary angles.
Write proofs involving congruent and right angles.
Postulate 2.10
Protractor Postulate

Given any angle, the measure can be put into
one-to-one correspondence with real numbers
between 0 and 180.
Postulate 2.11
Angle Addition Postulate
Use Angle Addition
Postulate
Find m1if m2  56 and mJKL  145.
m1  m2  mJKL
m1  56  145
m1 89
Example
If m1  23 and mABC  131, find the measure of 3.
Justify each step.
m1  m2  m3  mABC
23  90  m3  131
113  m3  131
113  m3 113  131 113
m3  18
Angle Add. Post.
Sub
Sub
Subt. Prop.
Sub
Theorems
Supplement Theorem

If two angles form a linear pair, then they
are supplementary angles.
Complement Theorem

If the noncommon sides of two adjacent
angles form a right angle, then the angles
are complementary angles.
Example
Angles 6 & 7 form a linear pair.
If m6  3x  32 & m7  5 x  12, find x, m6, & m7.
Justify each step.
m6  m7  180
Supplement Thm.
3x + 32 + 5x + 12 = 180
Sub
8x + 44 = 180
8x + 44 - 44 = 180 - 44
8x = 136
m6  83
m7  97
8 x 136

8
8
x = 17
Sub
Subt. Prop.
Sub
Division Prop.
Sub
Properties of Angle
Congruence
Reflexive

1  1
Symmetric

If 1  2, then 2  1
Transitive

If 1  2 & 2  3, then 1  3.
Theorem
Congruent Supplement Theorem

Angles supplementary to the same angle
or to congruent angles are congruent.
Abbreviation:
Theorem
Congruent Complements Theorem

Angles complementary to the same angle
or congruent angles are congruent.
Abbreviation
Prove that the vertical angles 2
and 4 are congruent.
Given: 2 & 4 are vertical angles
Prove:  2   4
Theorem 2.8
Vertical Angle Theorem

If two angles are vertical angles, then they
are congruent.
Prove that if DB bisects ADC , then 2  3.
Right Angle Theorems
Theorem 2.9

Perpendicular lines intersect to form 4 right
angles
Theorem 2.10

All right angles are congruent.
Theorem 2.11

Perpendicular lines from congruent
adjacent angles.
Theorem 2.12

If two angles are congruent and
supplementary, then each angle is a right
angle.
Theorem 2.13

If two congruent angles form a linear pair,
then they are right angles.
Homework
Pg. 154 1 – 4 all, 6, 8 – 14,
45 - 48 all