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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