Download Sec 2.8: Proving Angle Relationships Description of the lesson: This

Sec 2.8: Proving Angle Relationships
Description of the lesson: This section is about proving angle relationships using different
theorems and postulates through two column proofs.
Subject Area: Geometry
Approximate amount of time to be spent on this lesson: 2 periods
National or District Standards to be addressed: Use visualization, spatial reasoning, and
geometric modeling to solve problems.
Resources used in this lesson: Geometry Book
Learning objectives:
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Students will create one proof involving supplementary and complementary angles
Students will create one proof involving congruent and right angles
Motivational Strategies: Start off by looking at a pair of scissors. Have the students notice that
when you open the pair of scissors the blades make an angle, and one of the blades and one of
the handles make a linear pair. Explain angles are everywhere.
Lesson Procedures
1) Opening lecture
a. Start discussing the angles in the pair of scissors.
b. Tell students that today’s lesson will be about angles
c. Postulate 2.11
i. If R is in the interior of ∠𝑃𝑄𝑆, then 𝑚∠𝑃𝑄𝑅 + 𝑚∠𝑅𝑄𝑆 = 𝑚∠𝑃𝑄𝑆.
ii. If 𝑚∠𝑃𝑄𝑅 + 𝑚∠𝑅𝑄𝑆 = 𝑚∠𝑃𝑄𝑆, then R is in the interior of ∠𝑃𝑄𝑆.
d. Where have we seen this postulate before?
i. Segment addition postulate
e. Ex: At four o’clock, the angle between the hour and the minute hands of the clock
is 120°. If the second hand stops where it 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 hour hands?
i. They are both are 60° by the definition of angle bisector and angle
addition postulate.
f. Ex: The Grand Union Flag at the left contains several angles. If 𝑚∠𝐴𝐵𝐷 = 44
and 𝑚∠𝐴𝐵𝐶 = 88, find 𝑚∠𝐷𝐵𝐶.
i. 𝑚∠𝐴𝐵𝐷 + 𝑚∠𝐷𝐵𝐶 = 𝑚∠𝐴𝐵𝐶
ii. 44 + 𝑚∠𝐷𝐵𝐶 = 88
iii. 𝑚∠𝐴𝐵𝐶 = 44
g. Theorems
i. Supplement Theorem
1. If two angles form a linear pair, then they are supplementary
angles.
ii. Complement Theorem
1. If the noncommon sides of two adjacent angles form a right angle,
then the angles are complementary angles.
h. Ex. If 𝑚∠1 and 𝑚∠2 form a linear pair and 𝑚∠2 = 67, find 𝑚∠1.
i. 𝑚∠1 + 𝑚∠2 = 180
ii. 𝑚∠1 + 67 = 180
iii. 𝑚∠1 = 113
i. Student Exercise
i. Have students work on “Check Your Progress” problems 2A and 2B
1. Each row works on 2A or 2B
a. Every other row
2. Have students discuss their solutions with another person from
another row who worked on the other problem.
j. Theorem 2.5
i. Congruence of angles in reflexive, symmetric, and transitive.
1. Reflexive
a. ∠1 ≅ ∠1
2. Symmetric
a. If ∠1 ≅ ∠2, then ∠2 ≅ ∠1.
3. Transitive
a. If ∠1 ≅ ∠2, and ∠2 ≅ ∠3, then ∠1 ≅ ∠3.
k. Ex. Prove the symmetric property of congruence
i. Given: ∠𝐴 ≅ ∠𝐵
Prove: ∠𝐵 ≅ ∠𝐴
Statements
Reasons
Given
∠𝐴 ≅ ∠𝐵
Definition of Congruence
𝑚∠𝐴 = 𝑚∠𝐵
Symmetric Property
𝑚∠𝐵 = 𝑚∠𝐴
Definition of Congruence
∠𝐴 ≅ ∠𝐵
l. Theorems
i. 2.6 - Angles supplementary to the same angle or to congruent angles are
congruent.
1. If 𝑚∠1 + 𝑚∠2 = 180 and 𝑚∠2 + 𝑚∠3 = 180, then ∠1 ≅ ∠3.
ii. 2.7- Angles complementary to the same angle or to congruent angles are
congruent.
1. If 𝑚∠1 + 𝑚∠2 = 90 and 𝑚∠2 + 𝑚∠3 = 90, then ∠1 ≅ ∠3.
m. Ex. Prove Theorem 2.7
i. Given: ∠1 and ∠3 are complementary.
∠2 and ∠3 are complementary.
Prove: ∠1 ≅ ∠2
Statements
Reasons
Given
∠1 and ∠3 are complementary.
∠2 and ∠3 are complementary.
Definition of complementary angles
𝑚∠1 + 𝑚∠3 = 90
𝑚∠2 + 𝑚∠3 = 90
Substitution
𝑚∠1 + 𝑚∠2 = 𝑚∠2 + 𝑚∠3
Reflexive Property
𝑚∠3 = 𝑚∠3
Subtraction
Property
𝑚∠1 = 𝑚∠2
Definition of Congruent Angles
∠1 ≅ ∠2
n. Ex. Using supplementary angles
i. Given: ∠1 and ∠2 form a linear pair.
∠2 and ∠3 form a linear pair.
Prove: ∠1 ≅ ∠3
Statements
∠1 and ∠2 form a linear pair.
∠2 and ∠3 form a linear pair.
∠1 and ∠2 form a supplementary.
∠2 and ∠3 form a supplementary.
∠1 ≅ ∠3
Reasons
Given
Supplement Theorem
Theorem 2.3
Theorem 2.6
o. Theorems
i. 2.8 If two angles are vertical angles, then they are congruent.
p. Student Exercise
i. If ∠1 and ∠2 are vertical angles and 𝑚∠ = 𝑥 and 𝑚∠2 = 228 − 3𝑥, find
𝑚∠1 and 𝑚∠2.
ii. Use a two column proof to solve this problem
1. Use what you know.
q. Theorems
i. 2.9 Perpendicular lines intersect to form four right angles
ii. 2.10 All right angles are congruent
iii. 2.11 Perpendicular lines form congruent adjacent angles.
iv. 2.12If two angles are congruent and supplementary, then each angle is a
right angle
v. 2.13 If two congruent angles form a linear pair, then they are right angles.
2) Closure
a. Discuss what we did for this lesson
i. What were some of our main ideas of this lesson
1. Congruence Properties for angles
2. Supplementary and Complementary angles
ii. What’s next?
1. Review day for Chapter 2 test
2. Chapter 2 Test
Adaptations needed to meet student’s needs: N/A
Assessment of Objectives:
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Completion of the proof for the symmetric property
Completion of the vertical angles proof
Participation in lecture
Reflection on lesson
Homework after lesson: Day 1: 5-13, 35, 37 Day 2: 14-16, 30, 32-36