# Geometry Worksheet Angles by oza12622

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```									Honors Geometry

Lesson 2.6
Deductive Reasoning
What You Should Learn
Why You Should Learn It
Goal 1: How to identify a special
angle relationships
Goal 2: How to use deductive
reasoning to verify angle relationships
Deductive reasoning allows you to
build a mathematical system and to
solve real-life problems
Identifying Special Pairs of Angles

Vertical Angles
Two angles are vertical angles if
their sides form two pairs of
opposite rays

1

2
Identifying Special Pairs of Angles

Linear pair
Two adjacent angles are a linear
pair if their noncommon sides are
on opposite rays

1   2
Identifying Special Pairs of Angles

Complementary Angles
Two angles are complementary if
the sum of their measures is 90°.
Each angle is the complement of
the other.

60°
30°
Identifying Special Pairs of Angles

Supplementary Angles
Two angles are supplementary if the
sum of their measures is 180°. Each
angle is the supplement of the other.

80°   100°
Example 1

Use the terms vertical angles, linear
pair, complementary angles &
supplementary angles to describe the
relationships between the labeled
angles in the figure

3
6       4
5
Example 1 Solution
Vertical Angles: 3 and 5, 4 and 6
Linear Pairs: 3 and 4, 4 and 5,
5 and 6, 6 and 3

Complementary Angles: None
Supplementary Angles: 3 and 4,            4 and 5,
5 and 6, 6 and 3

3
6       4
5
Postulate 11: Linear Pair Postulate

If two angles form a linear
pair, then they are
supplementary, i.e., the sum
of their measures is 180°
Deductive Reasoning
To deduce means to reason from
known facts
When you prove a theorem, you are
using deductive reasoning
using the existing structure to deduce new
parts of the structure
In geometry, as in construction, new
parts of the structure are built upon
existing parts
Theorem 2.1:
Congruent Supplements Theorem
If two angles are
supplementary
to the same
angle or to
congruent
angles, then they
are congruent
Proof of Theorem 2.1:
Congruent Supplements Theorem
Given: 1 is the supplement of 2
3 is the supplement of 4
2  4
Prove: 1  3
m1 + m2 = 180 Def. of Supplementary Angles
m3 + m4 = 180 Def. of Supplementary Angles
m1 + m2 = m3 + m4 Transitive Prop. of =
2  4 Given
m2 = m4      Def. of Congruence
m1 + m2 = m3 + m2 Subsitution Prop. of =
m1 = m3     Subtraction Property of Equality
1  3   Definition of Congruence
Theorem 2.2
Congruent Complements Theorem
If two angles are
complementary
to the same
angle or to
congruent
angles, then they
are congruent
Proof of Theorem 2.2
Congruent Complements Theorem
Fill in the missing steps and missing reasons
Proof of Theorem 2.2 (Solution)
Congruent Complements Theorem
Theorem 2.3:
Vertical Angles Theorem
If two angles are
vertical angles, then
they are congruent.

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