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Lesson 2-1 Inductive Reasoning and Conjecture
Lesson 2-2 Logic
Lesson 2-3 Conditional Statements
Lesson 2-4 Deductive Reasoning
Lesson 2-5 Postulates and Paragraph Proofs
Lesson 2-6 Algebraic Proof
Lesson 2-7 Proving Segment Relationships
Lesson 2-8 Proving Angle Relationships
Example 3 Use Supplementary Angles
Example 4 Vertical Angles
OBJECTIVE: To write proofs involving congruent
and right angles (2.9.8J) (M8.D.1.1)
THEOREMS
Theorem 2.5: Congruence of angles is
reflexive, symmetric, and transitive.
• REFLEXIVE: 1  1
• SYMMETRIC: If 1  2, then 2  1
• TRANSITIVE: If 1  2 , and 2  3
then 1  3
,
THEOREMS
• 2.6 Angles supplementary to the same
angle or to congruent angles are
congruent.
• 2.7 Angles complementary to the same
angle or to congruent angles are
congruent.
• 2.8 Vertical Angle Theorem: If two angles
are vertical angles, then they are
congruent.
THEOREMS
• 2.9 Perpendicular lines intersect to form 4 right
angles.
• 2.10 All right angles are congruent.
• 2.11 Perpendicular lines form congruent
adjacent angles.
• 2.12 If 2 angles are congruent and
supplementary, then each angles is a right
angle.
• 2.13 If 2 congruent angles for a linear pair, then
they are right angles.
In the figure,
Given:
Prove:
and
form a linear pair, and
and
Prove that
are congruent.
form a linear pair.
Proof:
Statements
Reasons
1.
1. Given
2.
2. Linear pairs are
supplementary.
3. Definition of
supplementary angles
3.
4.
4. Subtraction Property
5.
5. Substitution
6.
6. Definition of congruent
angles
In the figure, NYR and RYA form a linear pair,
AXY and AXZ form a linear pair, and RYA and
AXZ are congruent. Prove that RYN and AXY
are congruent.
Proof:
Statements
Reasons
1.
1. Given
linear pairs.
2.
2. If two s form a
linear pair, then
they are suppl. s.
3.
3. Given
4.
4.
If 1 and 2 are vertical angles and m1
m2
find m1 and m2.
1
m1
2
Vertical Angles Theorem
m2
Definition of congruent angles
Substitution
Add 2d to each side.
Add 32 to each side.
Divide each side by 3.
and
Answer: m1 = 37 and m2 = 37
If
and
are vertical angles and
find
and
Answer: mA = 52; mZ = 52
and
REVIEW
1. Two angles that are nonadjacent are
(always, sometimes, or never) vertical?
SOMETIMES
2. Two angles that are congruent are
(always, sometimes, or never)
complementary to the same angle?
SOMETIMES
REVIEW
3. The measures of 2 complementary angles are
in the ratio 4:1. What is the measure of the
smaller angle?
A. 15 B. 18 C. 24 D. 36
B
4. T is the set of all positive numbers n such that
n < 50 and square root of n is an integer. What
is the median of the members of set T?
A. 4 B. 16 C. 20 D. 25
B
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