2.6 Prove Statements about Segments and Angles2.6 Prove Statements about Segments and Angles2.7 Prove Angle Pair Relationships2.7 Prove Angle Pair Relationships

Objectives:

1.To write proofs using geometric theorems

2.To use and prove properties of special pairs of angles to find angle measurements

Thanks a lot, Euclid!Thanks a lot, Euclid!

Recall that it was the development of civilization in general and specifically a series of clever ancient Greeks who are to be thanked (or blamed) for the insistence on reason and proof in mathematics.

Premises in Geometric Premises in Geometric ArgumentsArguments

The following is a list of premises that can be used in geometric proofs:

1.Definitions and undefined terms

2.Properties of algebra, equality, and congruence

3.Postulates of geometry

4.Previously accepted or proven geometric conjectures (theorems)

Properties of EqualityProperties of Equality

Maybe you remember these from Algebra.

Reflexive Property of Reflexive Property of EqualityEquality

For any real number a, a = a.

Symmetric Property of Symmetric Property of EqualityEquality

For any real numbers a and b, if a = b, then b = a.

Transitive Property of Transitive Property of EqualityEquality

For any real numbers a, b, and c, if a = b and b = c, then a = c.

Theorems of CongruenceTheorems of Congruence

Congruence of SegmentsCongruence of SegmentsSegment congruence is reflexive, symmetric,

and transitive.

Congruence of AnglesCongruence of AnglesAngle congruence is reflexive, symmetric, and

transitive.

Theorems of CongruenceTheorems of Congruence

Example 1aExample 1a

Given:

Prove:

Statements Reasons

1. 1.Given

2. has length AB 2.Ruler Postulate

3. AB = AB 3.Reflexive Prop. of =

4. 4.Definition of Congruent Segments

AB

AB AB

AB

AB

AB AB

Example 1bExample 1b

Given:

Prove:

A B B A

Example 2Example 2

Prove the following:If M is the midpoint of AB, then AB is twice AM

and AM is one half of AB.

Given: M is the midpoint of AB

Prove: AB = 2AM and AM = (1/2)AB

Example 3aExample 3a

If there was a right angle in Denton, TX, and other right angle in that place in Greece with all the ruins (Athens), what would be true about their measures?

Right Angle Congruence Right Angle Congruence TheoremTheorem

All right angles are congruent.

Yes, it seems obvious, but can you prove it? What would be your Given information? What would you have to prove?

Example 3bExample 3b

Given: < A and < B are right angles

Prove: A B

Linear Pair PostulateLinear Pair Postulate

If two angles form a linear pair, then they are supplementary.

Do we have to prove this?

Example 4Example 4

Given:

Prove:

1 68m 2 112m

4

3

2

1

Congruent SupplementsCongruent Supplements

Suppose your angles were numbered as shown. Notice angles 1 and 2 are supplementary. Notice also that 2 and 3 are supplementary. What must be true about angles 1 and 3?

4

3

2

1

Congruent Supplement Congruent Supplement TheoremTheoremIf two angles are supplementary to the same

angle (or to congruent angles), then they are congruent.

Example 5 Example 5

Prove the Congruent Supplement Theorem.

Given: < 1 and < 2 are supplementary< 2 and < 3 are supplementary

Prove: 1 3

What to ProveWhat to Prove

Notice that you can essentially have two kinds of proofs:

1.Proof of the Theorem– Someone has already proven this. You are

just showing your peerless deductive skills to prove it, too.

– YOU CANNOT USE THE THEOREM TO PROVE THE THEOREM!

2.Proof Using the Theorem (or Postulate)

Congruent Complement Congruent Complement TheoremTheoremIf two angles are complementary to the same

angle (or to congruent angles), then they are congruent.

You’ll have to prove this in your homework.

Vertical Angle Congruence Vertical Angle Congruence TheoremTheorem

Vertical angles are congruent.

Example 6Example 6

Prove the Vertical Angles Congruence Theorem.

Given: < 1 and < 3 are vertical angles

Prove: 1 3

Example 7Example 7

Given:

Prove:

1 53m 3 53m

4

3

2

1

Example 8Example 8

Given:

Prove: < 3 and < 4 are supplements

1 2

Example 9Example 9