Download GEOMETRY 2.6 Proving Geometric Relationships

GEOMETRY
2.6 Proving Geometric
Relationships
October 1, 2015
GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS
ESSENTIAL QUESTION
How can we prove geometric relationships using
proofs?
October 1, 2015
GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS
HOW ARE THEOREMS CREATED?
A theorem is a statement that can be
proven.
This means proofs.
Once a theorem is proven using a proof, it
can be used as a reason in other proofs.
October 1, 2015
GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS
THEOREM 2.3
Right Angle Congruence Theorem
All Right Angles are congruent.
(This should be obvious: all right angles
measure 90° and so they all have the
same measure and hence are
congruent. Let’s formally prove this.)
All Rt s 
October 1, 2015
GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS
PROOF: RIGHT ANGLE CONGRUENCE THEOREM
Reasons
Statements
1. Given
2
2. Def. of right angle
3. Transitive Property
4. Def.  angles
QED
October 1, 2015
GEOMETRY 2.6 PROVING GEOMETRIC
RELATIONSHIPS
THEOREM 2.4 Congruent Supplements Theorem
If two angles are supplementary to the same angle,
or to congruent angles, then they are congruent.
Example
mA + mB = 180°
mA + mC = 180°
B
A
C
Thus, B  C.
They are both supplementary to the same angle, A.
 Supp Thm
October 1, 2015
GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS
PROOF: CONGRUENT SUPPLEMENTS THEOREM
Statements
Reasons
1. Given
2. Def. of supp. angles
3. Transitive Property
4. Subtraction Property
5. Def. congruent angles
October 1, 2015
QED
GEOMETRY 2.6 PROVING GEOMETRIC
RELATIONSHIPS
THEOREM 2.5
Congruent Complements Theorem
If two angles are complementary to the same
angle, or to congruent angles, then the angles are
congruent. The proof of this theorem is very similar to the proof we just wrote
Example
for Theorem 2.4. We will not work through this proof right now.
mR + mS = 90°
mT + mS = 90°
R
S
T
Thus, R  T
R and T are complementary to the
same angle, S.
 Comp Thm
October 1, 2015
GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS
POSTULATE 2.8
Linear Pair Postulate
The angles of a linear pair are supplementary.
1
2
m1 + m2 = 180°
October 1, 2015
GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS
THEOREM 2.6
Vertical Angles Congruence Theorem
Vertical Angles are congruent. Prove: 1  2
1
2
October 1, 2015
3
For this proof, we will use the
second type of proof called a
paragraph proof. Write it in a
natural style; like you’re explaining
it to someone. But be accurate!
GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS
PROOF: VERTICAL ANGLES CONGRUENCE THEOREM
Given: Vertical Angles are congruent. Prove: 1  2
1
2
October 1, 2015
1 and 3 form a linear pair and
by Linear Pair Post., their sum is
180.
3
Similarly, the sum of 2 and 3 is
180.
Thus, 1  2 by the ≅ Suppl. Th.
because they are supplementary
to the same angle.
GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS
October 1, 2015
GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS
EXAMPLE 1
Reasons
Statements
1. Given
October 1, 2015
QED
3. Substitution Property
4. Subtraction Property
5. Division Property
GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS
YOUR TURN.
Practice using the theorems shown today to solve the
following problems.
1. Find m1, m2, m3.
135° (supp. s)
45°
1
2 45° (vert. s )
3
135° (vert. s )
October 1, 2015
GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS
EXAMPLE 2
(5x + 8)°
Solve for x.
(3x + 20)°
Statement
Reason
1. 5x + 8 = 3x + 20
1. Vert. ∠𝑠 Thm
2. 2x = 12
2. Subtr. Prop.
3. x = 6
3. Div. Prop.
October 1, 2015
GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS
EXAMPLE 3
Solve for x, and find each angle.
4(15) + 20 = 80°
(4x + 20)°
5(15) + 25 = 100°
Subst. Prop.
(5x + 25)°
Statement
Reason
1. 4x + 20 + 5x + 25 = 180
1. Linear Pair Thm
2. 9x + 45 = 180
2. Simplify
3. 9x = 135
3. Subtr. Prop.
4. x = 15
4. Div. Prop.
October 1, 2015
GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS
ASSIGNMENT
October 1, 2015
GEOMETRY 2.6 PROVING GEOMETRIC RELATIONSHIPS