Applied Geometry - South Harrison County R2
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Transcript 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 m1if m2 56 and mJKL 145.
m1 m2 mJKL
m1 56 145
m1 89
Example
If m1 23and mABC 131, find the measureof 3.
Justify each step.
m1 m2 m3 mABC Angle Add. Post.
23 90 m3 131
113 m3 131
113 m3 113 131 113
m3 18
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 m6 3x 32& m7 5x 12, find x, m6, & m7.
Justify each step.
m6 m7 180
Supplement Thm.
3x + 32 + 5x + 12 = 180
Sub
8x + 44 = 180
8x + 44 - 44 = 180 - 44
8x = 136
m6 83
m7 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, then2 1
Transitive
If 1 2 & 2 3, then1 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 verticalangles
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, then2 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