the first of many fun lessons…. we will utilize class time and discussions to determine if a...

2.6: Proving Angles Congruent

The first of many fun lessons…

For many a lesson moving forward…

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


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.


One property we’ll use today is called the

TransitiveProperty

One property we’ll use today is called the

The transitive property states:


If a = b and b = c, then…


If a = b and b = c, then…a = c


If a = b and b = c, then…a = c

If 2 things are equal to the same thing, they are equal to each other.

Example:

1 hour = 60 minutes

Example:

1 hour = 60 minutes60 minutes = 3600 seconds

Example:

1 hour = 60 minutes60 minutes = 3600 seconds 1 hour = 3600 seconds.

Example:

We will first use it for the Vertical Angles Theorem

We will first use it for the Vertical Angles TheoremI’m confused: What is a theorem?

We will first use it for the Vertical Angles TheoremI’m confused: What is a theorem?

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

We will first use it for the Vertical Angles TheoremTheorem 2.1

We will first use it for the Vertical Angles TheoremTheorem 2.1

◦Vertical angles are congruent.

And this is where we justify

Theorem 2.1◦Vertical angles are congruent.

And this is where we justify

Theorem 2.1◦Vertical angles are congruent.

12 3

4

Sketch

12 3

4

m1 + m 2 =

Sketch

12 3

4

m1 + m 2 = 180

Sketch Linear Pair Property

12 3

4

m1 + m 2 = 180

For that same reason


12 3

4

m1 + m 2 = 180

For that same reason


12 3

4

m1 + m 2 = 180m1 + m 3 = 180

Which pair listed is a vertical angle pair?


12 3

4

m1 + m 2 = 180m1 + m 3 = 180

We solve for these variables, since we’re trying to create a relationship


12 3

4

m1 + m 2 = 180m1 + m 3 = 180



12 3

4

m1 + m 2 = 180m1 + m 3 = 180

m 2 = 180 – m1



12 3

4

m1 + m 2 = 180m1 + m 3 = 180

m 2 = 180 – m1 m 3 = 180 – m1

Are angles 2 and 3 equal to the same thing?


12 3

4

m1 + m 2 = 180m1 + m 3 = 180

m 2 = 180 – m1 m 3 = 180 – m1

Are angles 2 and 3 equal to the same thing?

Sketch Transitive Property

12 3

4

m1 + m 2 = 180m1 + m 3 = 180

m 2 = 180 – m1 m 3 = 180 – m1

2 3

Note the difference between justifying a theorem and providing a counterexample.

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


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.


Problem Solving…

m + 92

3m - 14

m + 923m - 14

How does the approach differ?

m + 92

3m - 14

m + 923m - 14

In this diagram the angles are indeed vertical angles.


m + 92

3m - 14

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


m + 92

3m - 14

3m – 14 = m + 922m = 106 m = 53

3(53) – 14 = 14553 + 92 = 145

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


m + 92

3m - 14

This one returns us to linear pairs

m + 923m - 14

(3m – 14) + (m + 92) = 180


m + 923m - 14

(3m – 14) + (m + 92) = 1804m + 78 = 1804m = 102m = 25.5

3(25.5) – 14 = 62.5(25.5) + 92 = 117.5


m + 923m - 14

Other conjectures…

2.2 Congruent Supplements Conjecture


2.2 Congruent Supplements Conjecture

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


Proof (Long Version)

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:


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



Once again, we have two objects that are equal to the same thing, but we formalize it by saying◦mA = 180 – mB◦mC = 180 – mB



Once again, we have two objects that are equal to the same thing, but we formalize it by saying◦Since the mA is now the same expression as the mC, we can say A C



Other conjectures:

Theorem 2.3: Congruent Complements Theorem

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


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.


Theorems 2.4 and 2.5

2.4

All right angles are congruent


2.4


2.5

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


2.4


2.5

If 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.


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.


IF 2 angles are congruent and supplementary, then they are both right angles.


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:


IF 2 angles are congruent and supplementary, then they are both right angles.


mA + mB = 180x + x = 1802x = 180x = 90

which is a right angle measure.

Pages 124 – 125 6 – 12, 14, 18, 19

Homework:

the first of many fun lessons…. we will utilize class time and discussions to determine if a...

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