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Warm Up
Simplify each expression.
1. 90 – (x + 20)
2. 180 – (3x – 10)
Write an algebraic expression for each of the
following.
3. 4 more than twice a number
4. 6 less than half a number
Warm Up
Determine whether each statement is true or false. If false, give a
counterexample.
1. It two angles are complementary, then they are not congruent.
2. If two angles are congruent to the same angle, then they are congruent to
each other.
3. Supplementary angles are congruent.
Objectives
Identify adjacent angles, linear pair of angles,
vertical angles, complementary, and supplementary
angles.
Find measures of pairs of angles.
Prove geometric theorems by using deductive
reasoning.
A postulate is a statement that you accept as
true without proof.
A theorem is any statement that you can
prove. Once you have proven a theorem, you
can use it as a reason in later proofs.
When writing a proof, it is important to justify
each logical step with a reason.
Hypothesis
•
•
•
•
Definitions
Postulates
Properties
Theorems
Conclusion
Adjacent angles are two coplanar angles
with a common vertex and a common
side, but no common interior points.
1 and 2 are adjacent angles.
Linear pair of angles are two adjacent
angles whose noncommon sides are
opposite rays.
3 and 4 are a linear pair.
Two angles are vertical angles if their
sides form two pairs of opposite rays.
1 and 3 are vertical angles, as are 2 and 4.
Example 1:
Tell whether the angles are only adjacent, adjacent and
form a linear pair, vertical angles or none.
1 and 2
1 and 3
1
2
5
1 and 4
1 and 5
3 and 5
3 and 4
3
4
4 and 5
Two angles are complementary angles if the sum of their measures is 90°.
Each angle is the complement of the other.
Two angles are supplementary angles if the sum of their measure is 180°.
Each angle is the supplement of the other.
Complementary angles and supplementary angles can be adjacent or
nonadjacent.
complementary
adjacent
complementary
nonadjacent
supplementary
adjacent
supplementary
nonadjacent
Example 2:
Use the figure to complete the statements.
If 𝑚∠1 = 105°, then 𝑚∠3 =______
1
If 𝑚∠1 = 105°, then 𝑚∠4 =______
4
2
3
If 𝑚∠2 = 67°, then 𝑚∠3 =______
If 𝑚∠3 = 112°, then 𝑚∠1 =______
If 𝑚∠2 = 50°, then 𝑚∠3 =______
Finished w/ 1st and
3rd hour
Example 3:
Proof of the Linear Pair Theorem
Given: ∠1 and ∠2 form a linear pair.
Prove: ∠1 and ∠2 are supplementary.
1
A
Statements
∠1 and ∠2 form a linear pair.
𝐵𝐴 𝑎𝑛𝑑 𝐵𝐶 form a line.
𝑚∠𝐴𝐵𝐶 = 180°
𝑚∠1 + m∠2 = 𝑚∠𝐴𝐵𝐶
𝑚∠1 + 𝑚∠2 = 180°
∠1 and ∠2 are supplementary
Reasons
Given
Def. of a lin. pair
Def of a straight ∠
∠ Add. Post.
Subst. Prop.
Def. of Suppl. ∠
2
B
C
Example 4:
Find x.
6x°
(3x + 45)°
Example 5:
Find y.
(5y – 50)°
(4y – 10)°
Example 6:
∠𝐴 and ∠𝐵 are complementary.
Find 𝑚∠𝐴 and 𝑚∠𝐵.
𝑚∠𝐵 = 𝑥 − 11
𝑚∠𝐴 = 8𝑥 − 7
Example 7:
∠𝐴 and ∠𝐵 are supplementary. Find 𝑚∠𝐴 and 𝑚∠𝐵.
𝑚∠𝐴 = 12𝑥 + 1
𝑚∠𝐵 = 𝑥 + 10
Example 8:
Fill in the blanks to complete a two-column proof of the
Congruent Supplements Theorem.
Given: 1 and 2 are supplementary, and
2 and 3 are supplementary.
Prove: 1  3
Proof:
Statements
Reasons
1. 1 and 2 are supp., &
1. Given
2 and 3 are supp.
2. Def. of supp. ∠𝑠
2. m1 + m2 = 180°
m2 + m3 = 180°
3. m1 + m2 = m2 + m3 3. Subst.
4. Subtr. Prop. of =
4. m1 = m3
5. 1  3
5. Def. of ≅ ∠𝑠