The first of many fun lessons…

 We will utilize class time and discussions to determine if a statement is true.

 From there, we will use the drawn conclusions to problem solve.

Transitive Property

The transitive property states:

If a = b and b = c, then…

The transitive property states: If a = b and b = c, then…a = c

The transitive property states: If a = b and b = c, then…a = c If 2 things are equal to the same thing, they are equal to each other.

 1 hour = 60 minutes

 60 minutes = 3600 seconds

 1 hour = 60 minutes  60 minutes = 3600 seconds   1 hour = 3600 seconds.

I’m confused: What is a theorem?

Relax, a theorem is a conjecture or statement that you prove true

 Theorem 2.1

◦ Vertical angles are congruent.

 Theorem 2.1 ◦ Vertical angles are congruent.

 Theorem 2.1 ◦ Vertical angles are congruent

Sketch m  1 + m  2 =

Sketch m  1 + m  2 = 180

Sketch Linear Pair Property m  1 + m  2 = 180

Sketch Linear Pair Property m  1 + m  2 = 180

Sketch Linear Pair Property m  1 + m  2 = 180 m  1 + m  3 = 180

Sketch Linear Pair Property m  1 + m  2 = 180 m  1 + m  3 = 180

Sketch Linear Pair Property m  1 + m  2 = 180 m  1 + m  3 = 180

Sketch Linear Pair Property m  1 + m  2 = 180 m  1 + m  3 = 180

Sketch Linear Pair Property m  1 + m  2 = 180 m  1 + m  3 = 180 m  2 = 180 – m  1

Sketch Linear Pair Property m  1 + m  2 = 180 m  1 + m  3 = 180 m  2 = 180 – m  1 m  3 = 180 – m  1

Sketch Linear Pair Property m  1 + m  2 = 180 m  1 + m  3 = 180 m  2 = 180 – m  1 m  3 = 180 – m  1

Sketch Transitive Property m  1 + m  2 = 180 m  1 + m  3 = 180 m  2 = 180 – m  1 m  3 = 180 – m  1  2   3

 Justifying a theorem often involves algebra because it must be true for all cases.

 With counterexamples, we only have to prove it doesn’t work for one case.

In this diagram the angles are indeed vertical angles.

In this diagram the angles are indeed vertical angles. Therefore, we solve for x by setting them equal to each other.

3m – 14 = m m = 106  m = 53 3(53) – 14 = = 145 In this diagram the angles are indeed vertical angles. Therefore, we solve for x by setting them equal to each other.

(3m – 14) + (m + 92) = 180

4m + 78 = 180 4m = 102 m = (25.5) – 14 = 62.5 (25.5) + 92 = 117.5

 2.2 Congruent Supplements Conjecture

If 2 angles are supplements of the same angle (or of congruent angles), then the 2 angles are congruent.

 Let A, B, and C be 3 angles such that A is supplementary to B and C is supplementary to B.

 Let A, B, and C be 3 angles such that A is supplementary to B and C is supplementary to B. From previous definitions:

◦ m  A + m  B = 180

 Let A, B, and C be 3 angles such that A is supplementary to B and C is supplementary to B. From previous definitions: ◦ m  A + m  B = 180 ◦ m  C + m  B = 180

 Let A, B, and C be 3 angles such that A is supplementary to B and C is supplementary to B. From previous definitions: ◦ m  A + m  B = 180 ◦ m  C + m  B = 180 ◦ Solving for A and C:

 Let A, B, and C be 3 angles such that A is supplementary to B and C is supplementary to B. From previous definitions: ◦ m  A = 180 – m  B ◦ m  C = 180 – m  B ◦ Solving for A and C:

 Once again, we have two objects that are equal to the same thing, but we formalize it by saying ◦ m  A = 180 – m  B ◦ m  C = 180 – m  B ◦ Solving for A and C:

 Once again, we have two objects that are equal to the same thing, but we formalize it by saying ◦ Since the m  A is now the same expression as the m  C, we can say  A   C ◦ Solving for A and C:

 If 2 angles are complements of the same angle (or of congruent angles), then the two angles are congruent

 Proof is almost identical to the previous one, replacing 180 with 90.

2.4All right angles are congruent

2.4All right angles are congruent 2.5If 2 angles are congruent and supplementary, then each is a right angle.

2.4All right angles are congruent 2.5If 2 angles are congruent and supplementary, then each is a right angle. With your fellow classmates, justify each of these statements with a proof, either in paragraph form or listing the steps.

All right angles are congruent Let A and B be right angles. The m  A = 90  and the m  B = 90 . If 2 angles have the same measure, they’re congruent. Therefore, all right angles are congruent.

All right angles are congruent IF 2 angles are congruent and supplementary, then they are both right angles. Let A and B be right angles. The m  A = 90  and the m  B = 90 . If 2 angles have the same measure, they’re congruent. Therefore, all right angles are congruent. Let  A   B. Then they have the same measure x. If they’re also supplementary, then they’re sum is 180. Setting up an equation:

All right angles are congruent IF 2 angles are congruent and supplementary, then they are both right angles. Let A and B be right angles. The m  A = 90  and the m  B = 90 . If 2 angles have the same measure, they’re congruent. Therefore, all right angles are congruent. m  A + m  B = 180 x + x = 180 2x = 180 x = 90 which is a right angle measure.

 Pages 124 – – 12, 14, 18, 19