Beginning Proofs Lesson 1 4 TwoColumn Proofs Theorems

Beginning Proofs Lesson 1. 4 Two-Column Proofs & Theorems

Theorem: a mathematical statement that can be proven.

Theorem Procedures to follow: 1. We present a theorem or theorems. 2. We prove theorem(s) We shall omit the proofs of certain theorems, even though all have proofs.

3. We use theorems to help prove sample problems. 4. You are then given the challenge of using theorems to prove homework problems. Theorems will save you much time if you learn them and then use them!

Theorem 1 If two angles are right angles, then they are congruent. Given: <A is a right angle <B is a right angle Prove: <A is congruent to <B A B

Set Up for Proof Statements Reasons 1. <A is right < 1. Given 2. m<A = 90 2. If right < then 90 3. <B is right < 3. Given 4. m<B = 90 4. If right < then 90 5. <A congruent <B 5. If 2 angles have the same measure, then they are congruent.

Theorem 2 If two angles are straight angles, then they are congruent. Given: <ABC is a straight angle <DEF is a straight angle Prove: <ABC is congruent to <DEF A B C D E F

Proof Statements Reasons 1. <ABC is straight < 1. Given 2. m<ABC = 180˚ 2. If < is straight then m=180˚ 3. <DEF is straight < 3. Given 4. m<DEF = 180˚ 4. Same as #2 5. <ABC congruent <DEF 5. If 2 angles have the same measure, then they are congruent. Steps 2&4

Solve with a partner The Archimedean Club of Higher Mathematics is planning its annual banquet. The club gives awards to members of 5, 10, and 15 years. The club’s officers, Mrs. Martinez, Mr. Washington, and Ms. Chung, are due to receive awards. From the following information, which award will each receive and which post does each hold?

More information… 1. Ms. Chung has been a member for 10 years. 2. The president has been a member for fewer years than Ms. Chung. 3. Mrs. Martinez has been a member for more years than the secretary.

Answer… Mrs. Martinez is the vice-president and has been a member for 15 years. Mr. Washington is the president and has been a member for 5 years. Ms. Chung is the secretary and has been a member for 10 years.