Geometry Section 2-8 1112

Section 2-8
Proving Angle Relationships

Wednesday, November 16, 2011

Essential Questions

How do you write proofs involving supplementary and
complementary angles?

How do you write proofs involving congruent and right
angles?


More Postulates and Theorems
Protractor Postulate:

Angle Addition Postulate:

Theorem 2.3 - Supplement Theorem:


Protractor Postulate: Given any angle, the measure
can be put into one-to-one correspondence with
real numbers between 0 and 180




This means we can measure angles in degrees



Angle Addition Postulate: D is in the interior of ∠ABC
IFF m∠ABD + m∠DBC = m∠ABC


Angle Addition Postulate: D is in the interior of ∠ABC
IFF m∠ABD + m∠DBC = m∠ABC
Theorem 2.3 - Supplement Theorem: If two angles form
a linear pair, then they are supplementary angles


Theorem 2.4 - Complement Theorem:

Theorem 2.5 - Properties of Angle Congruence
Reﬂexive Property of Congruence:
Symmetric Property of Congruence:

Transitive Property of Congruence:


Theorem 2.4 - Complement Theorem: If the noncommon
sides of two adjacent angles form a right angle,
then the angles are complementary angles
Reﬂexive Property of Congruence:



Reﬂexive Property of Congruence: ∠1 ≅ ∠1



If ∠1 ≅ ∠2, then ∠2 ≅ ∠1


If ∠1 ≅ ∠2, then ∠2 ≅ ∠1
If ∠1 ≅ ∠2 and ∠2 ≅ ∠3, then ∠1 ≅ ∠3


Theorem 2.6 - Congruent Supplements Theorem:

Theorem 2.7 - Congruent Complements Theorem:

Theorem 2.8 - Vertical Angles Theorem:


Theorem 2.6 - Congruent Supplements Theorem: Angles
supplementary to the same angle or to congruent
angles are congruent
Theorem 2.7 - Congruent Complements Theorem:



Theorem 2.7 - Congruent Complements Theorem: Angles
complementary to the same angle or to congruent


Theorem 2.7 - Congruent Complements Theorem: Angles
complementary to the same angle or to congruent
Theorem 2.8 - Vertical Angles Theorem: If two angles
are vertical angles, then they are congruent


EN
EV
Right Angle Theorems
Theorem 2.9:

Theorem 2.10:
Theorem 2.11:

Theorem 2.12:

Theorem 2.13:


EN
EV
Theorem 2.9: Perpendicular lines intersect to form
four right angles
Theorem 2.10:
Theorem 2.11:

Theorem 2.12:

Theorem 2.13:


EN
EV
four right angles
Theorem 2.10: All right angles are congruent
Theorem 2.11:

Theorem 2.12:

Theorem 2.13:


EN
EV
four right angles
Theorem 2.11: Perpendicular lines form congruent
adjacent angles
Theorem 2.12:

Theorem 2.13:


EN
EV
four right angles
adjacent angles
Theorem 2.12: If two angles are congruent and
supplementary, then each angle is a right angle
Theorem 2.13:


EN
EV
four right angles
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


Example 1
Using a protractor, a construction worker measures that
the angle a beam makes with the ceiling is 42°. What is
the measure of the angle that the beam makes with the
wa%?


Example 1
Ceiling
wa%?
42°
Be
am
Wa!


Example 1
Ceiling
wa%?
42°
Be
am
90°-42°
Wa!


Example 1
Ceiling
wa%?
42°
Be
am
90°-42°
Wa!

48°


Example 2
At 4:00 on an analog clock, the angle between the hour and
minute hands of a clock is 120°. When the second hand
bisects the angle between the hour and minute hands, what
are the measures of the angles between the minute and
second hands and between the second and the hour hands?


Example 2
At 4:00 on an analog clock, the angle between the hour and
minute hands of a clock is 120°. When the second hand
bisects the angle between the hour and minute hands, what
are the measures of the angles between the minute and
second hands and between the second and the hour hands?

Since the larger angle of 120° is bisected, two smaller
angles of 60° are formed, and since those two angles
add up to the larger one, both angles we are looking
for are 60°.


Example 3
In the ﬁgure, ∠1 and ∠4 form a linear pair, and m∠3 +
m∠1 = 180°. Prove that ∠3 and ∠4 are congruent.


Example 3
∠1 and ∠4 form a linear pair,
and m∠3 + m∠1 = 180°


Example 3
∠1 and ∠4 form a linear pair, Given
and m∠3 + m∠1 = 180°


Example 3
and m∠3 + m∠1 = 180°
∠1 and ∠4 are supplementary


Example 3
and m∠3 + m∠1 = 180°
Linear pairs are
∠1 and ∠4 are supplementary supplementary


Example 3
and m∠3 + m∠1 = 180°
Linear pairs are


Example 3
and m∠3 + m∠1 = 180°
Linear pairs are
Def. of
supplementary


Example 3
and m∠3 + m∠1 = 180°
Linear pairs are
Def. of
supplementary
∠3 ≅ ∠4


Example 3
and m∠3 + m∠1 = 180°
Linear pairs are
Def. of
supplementary
∠3 ≅ ∠4 Angles supplementary to same ∠ are ≅


Check Your Understanding

Check out problems #1-7 on page 154 to see what you
understand (or don’t) and formulate some questions on the
ideas.


Problem Set


Problem Set

p. 154 #8-20

“Compassion for others begins with kindness to ourselves.”
- Pema Chodron


Geometry Section 2-8 1112

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Geometry Section 2-8 1112