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Notes 2-6: Geometric Proof
Objectives:

Geometry - Ch. 2: Geometric Reasoning
Write two-column proofs.
Prove geometric theorems by using deductive reasoning.
When writing a geometric proof, you use deductive
reasoning to create a chain of logical steps that move
from the ______________________________ to
the ______________________________.
Complete the following __________-____________________ proofs using some definitions from chapter 1.
EX 1: Given: A and B are supplementary & mA  45o .
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1.
2.
3.
4
5.
Statements
A and B are supplementary
mA  mB  180o
mA  45o
45o  mB  180o
mB  135o
Prove: mB  135o
Reasons
Prove: BC  EF
EX 2: Given: B is the midpoint of AC & AB  EF .
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1.
2.
3.
4.
Statements
B is the midpoint of AC
Reasons
AB  BC
AB  EF
BC  EF
1) Theorem: any statement that you can ____________________. Once proven, you can use it as a reason in later proofs.
2) Linear Pair Theorem: If two angles form a linear pair, then they are ___________________________________.
3) Congruent Complements Theorem: If two angles are complementary to the same angle, then the two angles are
_________________________.
4) Right Angle Congruence Theorem: all right angles are ________________________.
Complete the following two-column proof to prove a new theorem.
EX 3: Given: 1 and 2 are supplementary, and 2 and 3 are supplementary.
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Statements
Prove: 1  3
Reasons
1.
Given
2.
Given
3.
Def. of supp. s
4.
Def. of supp. s
5.
Substitution Property
6.
Subtraction Property
7.
Def. of  s
5) Congruent Supplements Theorem: If 2 s are supplementary to the same , then the two angles are _____.
Ex 4 : Given: N is the midpoint of MP , Q is the midpoint of RP , and PQ  NM .
Prove: PN  QR
Write a justification for each step.
Proof:
1. N is the midpoint of MP .
1. _________________________________
2. Q is the midpoint of RP .
2. _________________________________
3. PN  NM
3. _________________________________
4. PQ  NM
4. _________________________________
5. PN  PQ
5. _________________________________
6. PQ  QR
6. _________________________________
7. PN  QR
7. _________________________________
Ex 5: Here is a two-column proof of one case of the Congruent Supplements Theorem.
Given:
4 and 5 are supplementary and
5 and 6 are supplementary.
Prove:
4  6
Statements
Reasons
1. 4 and 5 are supplementary.
1.
2. 5 and 6 are supplementary.
2.
3. m4  m5  ________
3. Definition of ____________________________
4. m5  m6  ________
4. Definition of ____________________________
5. m4  m5  m5  m6
5. ______________________________ Property
6. m4  m6
6. ______________________________ Property
7. 4  6
7. Definition of _________________________
Ex 6: Fill in the blanks in either column to complete the proof.
Given: PR bisects QPS. PS bisects RPT .
Prove: m∠QPR ≅∠SPT
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1.
2.
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5.
6.
Statements
PR bisects QPS
 ____   ____
Reasons
Definition of _________________________
PS bisects RPT
 ____   ____
 ____   ____
Definition of _________________________
_________________________ Property
Definition of _________________________