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Lesson 2-8
Proving Angle Relationships
Transparency 2-8
5-Minute Check on Lesson 2-7
Justify each statement with a property of equality or a
property of congruence.
1. If AB  CD and CD  EF, then AB  EF.
2. RS  RS
3. If H is between G and I, then GH + HI = GI.
State a conclusion that can be drawn from the statements
given using the property indicated.
4. W is between X and Z; Segment Addition Postulate
5. LM  NO and NO  PQ; Transitive Property of Congruence
6.
Standardized Test Practice:
Which statement is true, given that K is
between J and L?
A
C
JK + KL = JL
LJ + JK = LK
B
D
JL + LK = JK
JK  KL
5-Minute Check on Lesson 2-7
Transparency 2-8
Justify each statement with a property of equality or a
property of congruence.
1. If AB  CD and CD  EF, then AB  EF. Transitive Property
2. RS  RS Reflexive Property
3. If H is between G and I, then GH + HI = GI.
Segment Addition Postulate
State a conclusion that can be drawn from the statements
given using the property indicated.
4. W is between X and Z; Segment Addition Postulate
XW + WZ = XZ
5. LM  NO and NO  PQ; Transitive Property of Congruence
LM  PQ
6. Standardized Test Practice:
Which statement is true, given that K is
between J and L?
A
C
JK + KL = JL
LJ + JK = LK
B
D
JL + LK = JK
JK  KL
Objectives
• Write proofs involving supplementary and
complementary angles
• Write proofs involving congruent and right
angles
Vocabulary
• No new vocabulary
Postulates
Postulate 2.10, Protractor Postulate: Given ray AB and
a number r between 0 and 180, there is exactly one ray
with endpoint A, extending on either side of ray AB,
such that the angle formed measures r°.
Postulate 2.11, Angle Addition Postulate:
If R is in the interior of PQS, then mPQR + mRQS =
mPQS and if mPQR + mRQS = mPQS, then R is in
the interior of PQS
Theorems
Theorem 2.3, Supplement Theorem: if two angles form a
linear pair, then they are supplementary angles.
Theorem 2.4, Complement Theorem: if the non-common
sides of two adjacent angles form a right angle, then the
angles are complementary angles.
Theorem 2.5, Angles supplementary to the same angle or to
congruent angles are congruent.
Theorem 2.6, Angles complementary to the same angle or to
congruent angles are congruent.
Theorem 2.7, Vertical Angles Theorem: If two angles are
vertical angles, then they are congruent.
Theorems
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.
Angle Proof
Given: 1 and 3 are vertical angles
m1 = 3x + 5, m3 = 2x + 8
Prove: m1 = 14
2
1
Statement
Reason
1 and 3 are vertical angles
Given
1  3
Vertical Angles Theorem
m1 = m3
Congruence Definition
m1 = 3x + 5
Given
m3 = 2x + 8
Given
3x + 5 = 2x + 8
Substitution
3x – 2x + 5 = 2x – 2x + 8
Subtraction
x+5=8
Substitution
x+5–5=8–5
Subtraction
x=3
Substitution
m1 = 3 (3) + 5 = 14
Substitution (twice)
3
4
TIME At 4 o’clock, the angle between the hour and
minute hands of a clock is 120º. 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?
Solution: If the second hand stops where the angle is
bisected, then the angle between the minute
and second hands is one-half the measure of
the angle formed by the hour and minute
hands, or ½(120º) = 60º.
By the Angle Addition Postulate, the sum of the two
angles is 120º, so the angle between the
second and hour hands is also 60º.
Answer: They are both 60º by the definition of angle bisector
and the Angle Addition Postulate.
QUILTING The diagram below shows one square for
a particular quilt pattern. If mBAC = mDAE = 20,
and BAE is a right angle, find mCAD.
Answer: 50
If 1 and 2 form a linear pair, and m2 = 166, find m1.
Solution:
m1 + m2 = 180
Supplement Theorem
m2 = 166
Given
m1 + 166 = 180
Substitution
m1 + 166 – 166 = 180 – 166
Subtraction Property
m1 = 14
Substitution
Answer: 14
If 1 and 2 are complementary angles and
m1 = 62, find m2.
Answer: 28
Given: 1 and 4 form a linear pair
m3 + m1 = 180.
Prove: 3  4
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
Given: NYR and RYA form a linear pair,
AXY and AXZ form a linear pair,
RYA  AXZ.
Prove: RYN  AXY
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 = d - 32 and
m2 = 175 – 2d, find m1 and m2.
1  2
Vertical Angles Theorem
m1 = m2
Definition of congruent angles
d – 32 = 175 – 2d
Substitution
3d – 32 = 175
Add 2d to each side.
3d = 207
Add 32 to each side.
d = 69
Divide each side by 3.
Answer: m1 = 37 and m2 = 37
If A and Z are vertical angles and mA = 3b -23
and mZ = 152 – 4b, find mA and mZ.
Answer: mA = 52; mZ = 52
Summary & Homework
• Summary:
– Properties of equality and congruence can be
applied to angle relationships
• Homework:
– pg 112-3: 16-23, 27-32, 41