Prove geometric theorems by deductive reasoning.

1. Obj. 10 Geometric Proof
The student will be able to (I can):
• Prove geometric theorems by deductive reasoning

2. geometric
proof
A proof which uses geometric properties
and definitions
• A two-column geometric proof begins
with the GGGGiiiivvvveeeennnn statement and ends with
the PPPPrrrroooovvvveeee statement.
• List the steps of the proof in the left
column and the justifications (reasons)
in the right column.
• You may use definitions, postulates, and
previously proven theorems as reasons.
• Other types of proofs are
— Paragraph proofs
— Flowchart proofs

3. Line segments with equal lengths are
congruent, and angles with equal measures
are also congruent. Therefore, the reflexive,
symmetric, and transitive properties of
equality have corresponding pppprrrrooooppppeeeerrrrttttiiiieeeessss ooooffff
ccccoooonnnnggggrrrruuuueeeennnncccceeee.
Reflexive Property of Congruence
fig. A @ fig. A
Symmetric Property of Congruence
If fig. A @ fig. B, then fig. B @ fig. A.
Transitive Property of Congruence
If fig. A @ fig. B and fig. B @ fig. C,
then fig. A @ fig. C.

4. GGGGiiiivvvveeeennnn:::: ÐBAC is a right angle;
Ð2 @ Ð3
PPPPrrrroooovvvveeee:::: Ð1 and Ð3 are comp.
1
2
3
•
•
B
A C
SSSSttttaaaatttteeeemmmmeeeennnnttttssss RRRReeeeaaaassssoooonnnnssss
1. ÐBAC is a right angle 1. Given
2. mÐBAC = 90° 2. DDDDeeeeffff.... rrrriiiigggghhhhtttt Ð
_______________
3. mÐ1111 ++++ mmmmÐ2222 ==== mmmmÐBBBBAAAACCCC
_______________________ 3. Ð Add. post.
4. mÐ1 + mÐ2 = 90° 4. Subst. prop. =
5. Ð2 @ Ð3 5. Given
6. mÐ2222 ==== mmmmÐ3
_______________________ 6. Def. @ Ðs
7. mÐ1 + mÐ3 = 90° 7. SSSSuuuubbbbsssstttt.... pppprrrroooopppp.... ====
_______________
8. Ð1111 aaaannnndddd Ð3333 aaaarrrreeee ccccoooommmmpppp....
_______________________ 8. Def. comp. Ðs

5. Example: GGGGiiiivvvveeeennnn Ð1 and Ð2 are supplementary, and
Ð2 and Ð3 are supplementary
PPPPrrrroooovvvveeee Ð1 @ Ð3
1. Ð1 and Ð2 are supp. 1. Given
Ð2 and Ð3 are supp.
2. mÐ1 + mÐ2 = 180° 2. Def. supp. Ð
mÐ2 + mÐ3 = 180°
3. 180° = mÐ2 + mÐ3 3. Sym. prop =
4. mÐ1+mÐ2=mÐ2+mÐ3 4. Trans. prop =
5. mÐ1 = mÐ3 5. Subtr. prop.=
6. Ð1 @ Ð3 6. Def. @ Ðs

6. GGGGiiiivvvveeeennnn:::: Ð2 @ Ð3
PPPPrrrroooovvvveeee:::: Ð1 and Ð3 are supplementary
1 2 3
1. Ð2 @ Ð3 1. Given
2. Ð1 and Ð2 are supp. 2. Def. linear pair
3. mÐ1 + mÐ2 = 180° 3. Def. supp. Ðs
4. mÐ2 = mÐ3 4. Def. @ Ðs
5. mÐ1 + mÐ3 = 180° 5. Subst. prop. =
6. Ð1 and Ð3 are supp. 6. Def. supp. Ðs