Download Geometry: Lesson 2.5 – Proving Angle Relationships

Geometry: Lesson 2.5 – Proving Angle Relationships
Geometry Oklahoma Academic Standards:
G.2D.1.2 Apply the properties of angles, including
corresponding, exterior, interior, vertical, complementary,
and supplementary angles to solve real world and
mathematical problems using algebraic reasoning and
proofs.
Lesson Objectives:
1. To prove angle relationships with definitions.
2. To use the proven theorems to solve
problems.
Introduction: Now that we have proven segment relationships, we can start to prove some
angle relationships.
Vocabulary: Definition – Adjacent
An adjacent angle is one that shares a side with another angle.
Which angle pairs represent adjacent angles in the above diagram?
____________________________________________________________________
The opposite of adjacent is vertical.
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Vocabulary: Definition – Vertical
A vertical angle is one is directly across from another angle.
Which angle pairs represent vertical angles in the above diagram?
____________________________________________________________________
Vocabulary: Definition – Complementary
A complementary angle is an angle that adds to be 90 degrees with another angle.
Visually:
Symbolically: _________________________
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Vocabulary: Definition – Supplementary
A supplementary angle is an angle that adds to be 180 degrees with another angle.
Visually:
Symbolically: _________________________
Vocabulary: Definition – Linear Pair
A Linear Pair are a couple of angles that are both adjacent and supplementary.
Visually:
Symbolically: _________________________
Using these definitions, we can now begin writing some theorems about how the various kinds
of angles relate to each other.
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Conjecture: If two angles are vertical, then they are congruent.
Conjecture: If two angles are supplementary to a third angle, then the two angles are
congruent.
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Conjecture: If an angle is complementary to itself, then the angle must measure 45 degrees.
Conjecture: If two angles are complementary and supplementary, then each angle is a right
angle.
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Conjecture: If two angles are right, then they are congruent.
Conjecture: If two angles are congruent and supplementary, then each angle is a right angle.
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Now that we have proven some theorems about angles, let’s use them to solve some problems.
Example 1: Find the measure of angle 6.
Example 2: Find the value of x.
Example 3: Find the measures of angles 2 and 3.
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Assignment 2.5
You have the choice of the following:
1. Prove or disprove the following conjectures:



“If two angles are straight angles, then they must be congruent.”
“If an angle is complementary to a third angle and another angle is
complementary to that third angle, then the two angles are congruent.”
“If an angle is vertical to a second angle and the second angle is adjacent to a
third angle, then the first and third angles must also be adjacent.”
2. Complete the handout, “Geometry: Handout 2.5” and turn it in.
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