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TOPIC 9
Geometry Proofs
Steps for proving a theorem, or accomplishing a proof.
a. State the theorem
b. Make a diagram
c. Write the “Given” and “Prove”
d. Write a two-column proof
When you solve an equation in algebra, you are doing a step-by-step process to convince
someone that an answer is valid. That is the process of writing a proof. So we will begin by
looking at the steps in the solution of an equation, adding the algebra reasons for each step.
These reasons will come from postulates and theorems in algebra
Example 1:
Given:
Solve: 2(x – 4) = 18
Proof:
1.
2(x – 4) = 18
2. 2x – 8 = 18
3.
2x = 26
4.
x = 13
In order to provide the reasons, we need to look at what these algebra properties are for equality:
PROPERTIES of EQUALITY
Addition
Subtraction
Multiplication
Division
Substitution
Distribution
Example:
TOPIC 8: Introduction to Proof
page 2
Some more algebra examples, and then some with geometry and algebra together.
Example 2:
Proof:
Given:
2x = 5 -
Prove:
x=
2
x
3
15
8
Statements
Reasons
1.
2x = 5 -
2
x
3
1.
2.
6x = 15 – 2x
2.
3.
8x = 15
3.
4.
x=
15
8
4.
Example 3:
________________________
A B
C
Given:
ABC , AB = x BC = 3x + 2 AC = 30
Prove:
x= 7
Proof:
Statements
1.
ABC
2.
AB + BC = AC
3.
AB = x
4.
BC = 3x + 2
5.
AC = 30
6.
x + 3x + 2 = 30
7.
4x + 2 = 30
8.
4x = 28
Reasons
TOPIC 8: Introduction to Proof
9.
page 3
x=7
When you are doing a proof, there are FOUR kinds of reasons that you can use:
A.
B.
C.
D.
Here is your first geometry proof:
4.
Given: mABG  mDEH
mGBC  mHEF
C
G
F
H
Prove: mABC  mDEF
A
B
E
Proof:
Statements
Reasons
1. mABG  mDEH
mGBC  mHEF
1.
2. mABG  mGBC  mDEH  mHEF
2.
3. mABG  mGBC  mABC
mDEH  mHEF  mDEF
3.
4. mABC  mDEF
4.
D
TOPIC 8: Introduction to Proof
5.
page 4
Given: FL = AT
A
F
Prove:
T
L
FA = LT
Proof:
Statements
6.
Reasons
1.
1. Given
2. LA = LA
2.
3. FL + LA = AT + LA
3.
4. FL + LA = FA
AT + LA = LT
4.
5.
5. Substitution Property
Given: GO = EM
O
Prove: GE = OM
G
M
E
Proof:
Statements
Reasons
1.
1.
2. GE + EO = G0
OM + EO = EM
2.
3. GE + EO = OM + EO
3.
4. EO = EO
4.
5.
5. Subtraction Property of Equality
TOPIC 8: Introduction to Proof
7.
page 5
Given: m < 1 + m < 3 = 180
1
2
3
Prove: m < 2 = m < 3
Proof:
1.
1. Given
2.
m < 1 + m < 3 = 180
2.
3.
m < 1 + m < 2 = m < 1 + m < 3 3.
4.
m<1=m<1
4.
5.
5.
Here is the first theorem we will state and then prove.
8.
Theorem: Congruent Supplements Theorem
If two angles are supplements of the same angle
(or of congruent angles), then the two angles
are congruent.
Given: A and B are supplementary
C and B are supplementary
A
C
B
Prove: A  C
Statements
Reasons
1. A and B are supplementary
C and B are supplementary
1.
2.
2. Definition of supplementary angles
.
TOPIC 8: Introduction to Proof
9.
Prove that linear pairs are supplementary.
10.
Theorem: Vertical Angle Theorem
page 6
TOPIC 8: Introduction to Proof
page 7
11. Theorem: Congruent Complements Theorem
If two angles are complements of the same angle (or of congruent angles), then the two
angles are congruent.
12.
Prove that right angles are congruent to each other
TOPIC 8: Introduction to Proof
13.
page 8
Prove that perpendicular pairs are complementary.
Calculation problems based on the new theorems
Find the value of the variables.
14.
3x+y
15.
y-16
y-15
3x-16
2x+20
2x-16
x
16.
An angle and its supplement are in the ratio 2 : 3. Find the measure of the angle and the
measure of the supplement.