warm up simplify each expression. 1. 90 – (x + 20) 2. 180 – (3x – 10) write an algebraic...

Warm UpSimplify 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 UpDetermine 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.

ObjectivesIdentify 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.

Hypothesis Conclusion

• Definitions• Postulates• Properties• Theorems

When writing a proof, it is important to justify each logical step with a reason.

Adjacent angles are two coplanar angles with a common vertex and a common side, but no common interior points.

Linear pair of angles are two adjacent angles whose noncommon sides are opposite rays.

3 and 4 are a linear pair.

1 and 2 are adjacent angles.

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 5

1 and 2

1 and 4

12 3

45

1 and 3

3 and 5 4 and 53 and 4

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.

Two angles are complementary angles if the sum of their measures is 90°.Each angle is the complement of the other.

complementary adjacent

complementary nonadjacent

supplementary adjacent

supplementary nonadjacent

1

4 2

3

Example 2: Use the figure to complete the statements.

If , then ______

If , then ______

If , then ______

If , then ______

If , then ______Finished w/ 1st and 3rd hour

Statements Reasons and form a linear pair. Given form a line. Def. of a lin. pair

Def of a straight + = Add. Post.

Subst. Prop. and are supplementary Def. of Suppl.

Example 3:Proof of the Linear Pair TheoremGiven: and form a linear pair.Prove: and are supplementary. 1 2

A B C

6x°

(3x + 45)°

Example 4:Find x.

(5y – 50)°

(4y – 10)°

Example 5:Find y.

are complementary. Find .

𝑚∠𝐴=8 𝑥−7

𝑚∠𝐵=𝑥−11

Example 6:

are supplementary. Find .𝑚∠𝐴=12 𝑥+1

𝑚∠𝐵=𝑥+10

Example 7:

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:

1 and 2 are supp., & 2 and 3 are supp.

m1 + m2 = m2 + m3

Subtr. Prop. of =

1 3

Statements Reasons

1. 1. Given

2. m1 + m2 = 180° m2 + m3 = 180°

2.

3. 3. Subst.

4. m1 = m3 4.5. 5. Def. of

Def. of supp.