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. Definitions Postulates Properties Theorems Hypothesis 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 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

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

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

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. ∠

6x° (3x + 45)° Example 4: Find x.

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

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