2.7 Prove Angle Pair Relationships
Goal  Use properties of special pairs of angles.
Your Notes
THEOREM 2.3 RIGHT ANGLES CONGRUENCE THEOREM
All right angles are _congruent_.
Example 1
Use right angle congruence
Write a proof.
Given JK  KL, ML  KL
Prove K  L
The given information in
Example 1 is about
perpendicular lines. You must
then use deductive reasoning to
show that the angles are right
angles.
Statements
1. JK  KL, ML  KL
2. _ K and  L are right angles_.
3. K  L
Reasons
1. _Given_
2. Definition of perpendicular lines
3. _Right Angles Congruence Theorem_
THEOREM 2.4 CONGRUENT SUPPLEMENTS THEOREM
If two angles are supplementary to the same angle (or to congruent angles), then they are
_congruent_.
If l and 2 are supplementary and 3 and 2 are supplementary, then _1  3_.
THEOREM 2.5 CONGRUENT COMPLEMENTS THEOREM
If two angles are complementary to the same angle (or to congruent angles), then they are
_congruent_.
If 4 and 5 are complementary and 6 and 5 are complementary, then _4  6_.
Your Notes
Example 2
Use the Congruent Supplements Theorem
Write a proof.
Given l and 2 are supplements. l and 4 are supplements. m2 = 45°
Prove m4 = 45°
Statements
Reasons
1. l and 2 are supplements. 1. _Given_
l and 4 are supplements.
2. _2  4_
2. Congruent Supplements
Theorem
3. m2 = m4
3. _Definition of congruent
angles_
4. m2 = 45°
4. _Given_
5. _m4 = 45°_
5. Substitution Property of
Equality
Checkpoint Complete the following exercises.
1. In Example 1, suppose you are given that K  L. Can you use the Right Angles
Congruence Theorem to prove that K and L are right angles? Explain.
No, you cannot prove that K and L are right angles, because the converse of the
Right Angles Congruence Theorem is not always true.
2. Suppose A and B are complements, and A and C are complements. Can B
and C be supplements? Explain.
No, B and C are complements by the Congruent Complements Theorem, so they
cannot be supplements.
Your Notes
POSTULATE 12 LINEAR PAIR POSTULATE
If two angles form a linear pair, then they are _supplementary_.
l and 2 form a linear pair, so l and 2 are supplementary and
m1 + m2 = _180°_.
THEOREM 2.6 VERTICAL ANGLES CONGRUENCE THEOREM
Vertical angles are _congruent_.
Example 3
Use the Vertical Angles Congruence Theorem
Write a proof.
Given 4 is a right angle.
Prove 2 and 4 are supplementary.
You can use
information labeled
in a diagram in your
proof.
Statements
1. 4 is a right angle.
Reasons
1. _Given_
2. _m4 = 90°_
2.
Definition of a right angle
3. 2  4
3.
_Vertical Angles_
Congruence Theorem_
4. _m2 = m4_
4. Definition of congruent
angles
5. m2 = 90°
5. _Substitution Property_
_of Equality_
6. _2 and 4 are
supplementary._
6. m2 + m4 = 180°
Checkpoint In Exercises 3 and 4, use the diagram.
3. If m4 = 63°, find m1 and m2.
m1 = 117°, m2 = 63°
4. If m3 = 121°, find m1, m2, and m4
m1 = 121°, m2 = 59°, m4 = 59°
Your Notes
Example 4
Find angle measures
Write and solve an equation to find x. Use x to find mFKG.
Solution
Because mFKG and mGKH form a linear pair, the sum of their measures is _180°_.
(4x  1)° + 113° = _180°_ Write equation.
4x + _112_ = _180_ Simplify.
Subtract _112___ from each
4x = _68_
side.
x = _17_ Divide each side by 4.
Use x = _17_ to find mFKG.
mFKG = (4x  1)°
= [4(_17_)  1]°
= [_68_  1]°
= _67°_
Write equation.
Substitute __17__ for x.
Multiply.
Simplify.
The measure of FKG is _67°_.
Checkpoint Complete the following exercise.
5. Find mAEB.
mAEB = 70°
Homework
________________________________________________________________________
________________________________________________________________________