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Transcript
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
2-8 Study Guide and Intervention
Proving Angle Relationships
Supplementary and Complementary Angles There are two basic postulates for working with angles. The Protractor
Postulate assigns numbers to angle measures, and the Angle Addition Postulate relates parts of an angle to the whole angle.
Angle Addition
Postulate
R is in the interior of ∠PQS if and only if m∠PQR + m∠RQS = m∠PQS.
The two postulates can be used to prove the following two theorems.
Supplement
Theorem
If two angles form a linear pair, then they are supplementary angles.
Example: If ∠1 and ∠2 form a linear pair, then m∠1 + m∠2 = 180.
Complement
Theorem
If the noncommon sides of two adjacent angles form a right angle,
then the angles are complementary angles.
⃡ , then m∠3 + m∠4 = 90.
Example: If ⃡𝐺𝐹 ⊥ 𝐺𝐻
Example 1: If ∠1 and ∠2 form a linear pair
Example 2: If ∠1 and ∠2 form a right angle
and m∠2 = 115, find m∠1.
m∠1 + m∠2 = 180
m∠1 + 115 = 180
- 115
-115
m∠1 = 65
and m∠2 = 20, find m∠1.
m∠1 + m∠2 = 90
m∠1 + 20 = 90
Supplement Theorem
Substitution
Subtraction Prop.
Substitution
- 20
-20
m∠1 = 70
Complement Theorem
Substitution
Subtraction Prop.
Substitution
Congruent and Right Angles The Reflexive Property of Congruence, Symmetric Property of Congruence, and
Transitive Property of Congruence all hold true for angles. The following theorems also hold true for angles.
Congruent Supplements Theorem
Angles supplement to the same angle or congruent angles are congruent.
Congruent Compliments Theorem
Angles compliment to the same angle or to congruent angles are congruent.
Vertical Angles Theorem
If two angles are vertical angles, then they are congruent.
Theorem 2.9
Perpendicular lines intersect to form four right angles.
Theorem 2.10
All right angles are congruent.
Theorem 2.11
Perpendicular lines form 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.
Chapter 2
49
Glencoe Geometry
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
2-8 Practice
Proving Angle Relationships
Find the measure of each numbered angle and name the theorems that justify your work.
1. m∠ 1 = x + 10
m∠ 2 = 3x + 18
2. m∠ 4 = 2x – 5
m∠ 5 = 4x – 13
3. m∠ 6 = 7x – 24
m∠ 7 = 5x + 14
4. Write a two-column proof.
Given: ∠ 1 and ∠ 2 form a linear pair.
∠ 2 and ∠ 3 are supplementary.
Prove: ∠ 1 ≅ ∠ 3
Chapter 2
49
Glencoe Geometry