1. Warm Up
Simplify each expression.
1. 90 – (x + 20)
– (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

2. 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.

3. 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.

4. 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.

5. When writing a proof, it is important to justify each logical step with a reason.
Definitions
Postulates
Properties
Theorems
Hypothesis
Conclusion

6. 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.

7. 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.

8. Example 1:
Tell whether the angles are only adjacent, adjacent and form a linear pair, vertical angles or none. 1 2 3 4 5
1 and 3
1 and 2
1 and 4
1 and 5
3 and 5
3 and 4
4 and 5

9. Each angle is the complement of the other.
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

11. Use the figure to complete the statements.
Example 2:
Use the figure to complete the statements.
If 𝑚∠1=105°, then 𝑚∠3=______ 1 3
If 𝑚∠1=105°, then 𝑚∠4=______
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

12. Proof of the Linear Pair Theorem
Example 3:
Proof of the Linear Pair Theorem
Given: ∠1 and ∠2 form a linear pair.
Prove: ∠1 and ∠2 are supplementary. 1 2 A B C
Statements
Reasons
∠1 and ∠2 form a linear pair.
Given
𝐵𝐴 𝑎𝑛𝑑 𝐵𝐶 form a line.
Def. of a lin. pair
𝑚∠𝐴𝐵𝐶=180°
Def of a straight ∠
𝑚∠1 + m∠2 = 𝑚∠𝐴𝐵𝐶
∠ Add. Post.
𝑚∠1+𝑚∠2=180°
Subst. Prop.
∠1 and ∠2 are supplementary
Def. of Suppl. ∠

13. 6x°
(3x + 45)°
Example 4:
Find x.

14. Example 5:
Find y.
(5y – 50)°
(4y – 10)°

15. Example 6:
∠𝐴 and ∠𝐵 are complementary. Find 𝑚∠𝐴 and 𝑚∠𝐵.
𝑚∠𝐵=𝑥−11
𝑚∠𝐴=8𝑥 −7

16. Example 7:
∠𝐴 and ∠𝐵 are supplementary. Find 𝑚∠𝐴 and 𝑚∠𝐵.
𝑚∠𝐴=12𝑥+1
𝑚∠𝐵=𝑥+10

17. 1. Given 2. m1 + m2 = 180° m2 + m3 = 180° 2. 3. 3. Subst.
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.
Given
2. m1 + m2 = 180°
m2 + m3 = 180° 2. 3.
3. Subst.
4. m1 = m3 4. 5.
5. Def. of ≅∠𝑠
1 and 2 are supp., & 2 and 3 are supp.
Def. of supp. ∠𝑠
m1 + m2 = m2 + m3
Subtr. Prop. of =
1  3