Download 2.6 Prove Statements about Segments and Angles

2.6 Prove Statements about Segments and Angles
2.7 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!
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 Arguments
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 Equality
Maybe you remember these from Algebra.
Reflexive Property of
Equality
For any real number a,
a = a.
Symmetric Property of
Equality
For any real numbers a and
b, if a = b, then b = a.
Transitive Property of
Equality
For any real numbers a, b,
and c, if a = b and b = c,
then a = c.
Theorems of Congruence
Congruence of Segments
Segment congruence is reflexive, symmetric,
and transitive.
Theorems of Congruence
Congruence of Angles
Angle congruence is reflexive, symmetric, and
transitive.
Example 1
Prove the following:
1. Segment congruence is reflexive
2. Angle congruence is symmetric
To do these proofs, you have to turn “congruence”
into “equality” and then turn “equality” back into
“congruence.” In either case, just apply the
Definition of Congruent Segments or Angles.
Example 1a
Given: AB
Prove: AB  AB
Statements
Reasons
1. AB
1.Given
2. AB has length AB
2.Ruler Postulate
3. AB = AB
3.Reflexive Prop. of =
4. AB  AB
4.Definition of Congruent
Segments
Example 1b
Given: A  B
Prove: B  A
1. m<A=m<B
2. m<B=m<A
3. <B =῀ <A
Given
Reflexive property of =
Definition of congruency
Example 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
1.
2.
3.
4.
5.
6.
7.
M is midpt.
AM = MB
AM = MB
AM + MB = AB
AM + AM = AB
2AM = AB
AM = ½ AB
῀
Given
Def, of midpt.
Def. of Cong.
Seg. Add. Post.
Sub.
Addition
Division
Example 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?
They would both be 900
They would be congruent
Right Angle Congruence Theorem
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?
Linear Pair Postulate
If two angles form a linear pair, then they are
supplementary.
Example 4
Given: m1  68
Prove: m2  112
2
3
1
4
TRY before
you click!
1.
2.
3.
4.
5.
6.
m<1 = 680
<1 & <2 are linear pair
<1 & <2 are supp.
m<1 + m<2 = 180
68 + m<2=180
m<2 = 112
Given
Def. of Linear Pair
Linear Pair Post.
Def. of supp.
Subst.
Subtraction
Congruent 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?
2
3
1
4
They must be
congruent
Congruent Supplement Theorem
If two angles are supplementary to the same
angle (or to congruent angles), then they
are congruent.
Example 5
Prove the Congruent Supplement Theorem.
Given: < 1 and < 2 are supplementary
< 2 and < 3 are supplementary
Prove: 1  3
Just TRY IT! In your
notebook
What 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 Theorem
If two angles are complementary to the same
angle (or to congruent angles), then they
are congruent.
Vertical Angle Congruence Theorem
Vertical angles are congruent.
Example 6
Prove the Vertical Angles Congruence
Theorem.
Given: < 1 and < 3 are vertical angles
Prove: 1  3
Come on – You can DO THIS!! (In your notebook)