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Presentation transcript:

Warm Up Complete each sentence. 1. If the measures of two angles are ? , then the two angles are congruent. 2. If two angles form a ? , then they are supplementary. 3. If two angles are complementary to the same angle, then the two angles are ? . equal linear pair congruent

Learning Targets I will prove geometric theorems by using deductive reasoning.

Example Given: 2 and 3 are complementary 1  3 Use the given information to write a two-column proof. Given: 2 and 3 are complementary 1  3 Prove: 2 and 1 are complementary

Statements Reasons Example 1 Continued Two-column proof: 1  3 1. 2 and 3 are complementary 1  3 1. Given 2. m2 + m3 = 90° 2. Def. complementary angles 3. m1 = m3 3. Def. congruent angles 4. m2 + m1 = 90° 4. Substitution property 5. 2 and 1 are complementary 5. Def. complementary angles

Example Use the given information to write a two-column proof. Prove: m1  m3

Example Continued

Example Use the given paragraph proof to write a two-column proof. Given: m1 + m2 = m4 Prove: m3 + m1 + m2 = 180° Paragraph Proof: It is given that m1 + m2 = m4. 3 and 4 are supplementary by the Linear Pair Theorem. So m3 + m4 = 180° by definition. By Substitution, m3 + m1 + m2 = 180°.

Statements Reasons Example Continued Two-column proof: 1. m1 + m2 = m4 1. Given 2. 3 and 4 are supplementary 2. Linear Pair Theorem 3. m3 + m4 = 180° 3. Def. supplementary angles 4. m3 + m1 + m2 = 180° 4. Substitution Property

Example Use the given information to write a two-column proof. Given: WXY is a right angle. 1  3 Prove: 1 and 2 are complementary.

1. WXY is a right angle. 1  3 1. Given 2. mWXY = 90° Example Continued Statements Reasons 1. WXY is a right angle. 1  3 1. Given 2. mWXY = 90° 2. Def. right angle 3. m2 + m3 = mWXY 3. Angle addition postulate 4. m2 + m3 = 90° 4. Substitution property 5. m1 = m3 5. Def. congruent angles 6. m2 + m1 = 90° 6. Substitution property 7. 1 and 2 are comp. 7. Def. complementary angles

Example Provide the reasons for the two-column proof shown below. Given: 1 and 2 are complementary Prove: 3 and 4 are complementary m3 + m4 = 90° 3 and 4 are comp.

Example Use the information to write a two-column proof. Given: 1  4 Prove: 2  3

Example Continued

Homework Page 124, #9 – 16.