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Geometry Geometric Proof. Warm Up. Identify the property that justifies each statement:. Geometric Proof. - PowerPoint PPT Presentation
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CONFIDENTIAL 2
Warm Up
1) JK KL, so KL JK
2) I f m = n and n = p, then m = p.
Identify the property that justifies each statement:
CONFIDENTIAL 3
Geometric Proof
When writing a geometric proof, you use the deductive reasoning to create a chain of logical steps that move from
hypothesis to conclusion of the conjecture you are proving. By proving that the conclusion is true, you have proven that the
original conjecture is true.
Conclusion
Definition Postulate Properties Theorems
Hypothesis
When writing a geometric proof, it is important to justify each logical step with a reason. You can use symbols and
abbreviations, but they must be clear enough so that anyone who reads your proof will understand them.
CONFIDENTIAL 4
A B C
Writing justifications
Write a justification for each step, given that A and B are complementary and A C.
1. A and B are complementary 1. Given2. mA + mB = 90 2. Def. of comp.S3. A C 3. Given4. mA mB 4. Def. of S5. mC + mB = 90 5. Subtr. Prop. of = step 2,46. C and B are complementary 6. Def. of comp.S
CONFIDENTIAL 5
Now you try!
1) Write a justification for each step, giventhat B is the mid-point of AC and AB EF.
1. B is the mid-point of AC2. AB BC3. AB EF4. BC EF
E
B
A
C
F
CONFIDENTIAL 6
A theorem is any statement that you can prove. Once you have proven a theorem, you can use it as a reason in later proofs.
CONFIDENTIAL 7
Theorem Hypothesis Conclusion
Linear Pair Theorem:
If two angles form a linear pair, then they are supplementary.
/A and /B form a linear pair.
/A and /B are supplementary.
Congruent Supplements Theorem:
If two angles are supplementary to the same angle (or two congruent angles), then the two angles are congruent.
/1 and /2 are supplementary./2 and /3 are supplementary.
/1 /3
Theorems
CONFIDENTIAL 8
A geometric proof begins with a Given and prove statements, which restate the hypothesis
and conclusion of the conjecture. In a two-column proof, you list the steps of the proof in
the left column and you write the matching reason for each step in the right column.
CONFIDENTIAL 9
Completing a two-column proof
Fill in the blanks to complete a two-column proof of theLinear Pair Theorem Given: 1 and 2 form a linear pair.Proof: 1 and 2 are supplementary.
Proof:
STATEMENTS REASONS
1. 1 and 2 form a linear pair 1. Given2. BA and BC form a line 2. Def. of lin. pair3. mABC = 180 3. Def. of straight 4. a. __?___ 4. Add. Post.5. b. __?___ 5. Subst. steps 3,4
BA C
21
NEXT PAGE
CONFIDENTIAL 10
BA C
21
Use the existing statements and reasons in the proof to fill in the blanks:
a. m1 + m2 =mABC Add. Post. is given as the reasonb. m1 + m2 = 180 Subst. 180 for ABCc. Def. of supp. s The measure of supp.S add to 180 by def
CONFIDENTIAL 11
Now you try!2
1
32) Fill in the blanks to complete a two-column proofof the Linear Pair Theorem Given: 1 and 2 are supplementary.Prove: 2 and 3 are supplementary.
Proof:
STATEMENTS REASONS
1. a. __?___ 1. Given2. m1 + m2 = 180 2. Def. of supp. m2 + m3 = 1803. b. __?___ 3. Subst.4. m1 = m2 4. Reflex prop. of =5. m2 = m3 5. c. __?___6. d. __?___ 6. Def. of comp.S
/2
Prove: 1 3
Proof:
CONFIDENTIAL 12
Before you start writing a proof, you should plan out your logic. Sometimes,
you will be given a plan for a more challenging proof. This plan will detail the
major steps of the proof for you.
CONFIDENTIAL 13
Theorem Hypothesis Conclusion
Right Angle Congruence Theorem:
All right angles are congruent./A and /B are right angles.
Congruent Complements Theorem:
If two angles are complementary to the same angle (or two congruent angles), then the two angles are congruent.
/1 and /2 are complementary./2 and /3 are complementary.
/1 /3
Theorems
A B
CONFIDENTIAL 14
Writing a two-column proof from a plan
Use the given plan to write a two-column proof of theRight Angle Congruence Theorem.
Given: 1 and 2 are Right angles.Prove: 1 2 .
Proof:
STATEMENTS REASONS
1. 1 and 2 are Right angles 1. Given2. m1 = 90 2. Def. of rt. m2 = 903. m1 = m2 3. Trans. prop. of =4. m1 m2 4. Def. of S
12
CONFIDENTIAL 15
21
3
3) Use the given plan to write a two-column proof ofthe Right Angle Congruence Theorem. Given: 1 and 2 are Complementary. And 2 and 3 are Complementary.Prove: 1 3 .
Plan: The measures of complementary angles add to 90 by definition. Use substitution to show that the sums of both the pairs are equal. Use the Substraction Property and the definition of congruent angles to conclude that 1 3.
Now you try!
CONFIDENTIAL 16
The Proof process
1) Write the conjecture to be proven.2) Draw a diagram to represent the hypothesis of the
conjecture.3) State the given information and mark it on the diagram.4) State the conclusion of the conjecture in terms of the
diagram.5) Plan your argument and prove the conjecture.
CONFIDENTIAL 18
Assessment
1) In a two-column proof, you list the __?__ in the left column and the __?__ in the right column.
2) A __?__ is a statement you can prove.
Fill in the correct word in the blanks given below:
CONFIDENTIAL 19
3) Write a justification for each step,given that mA = 60 and mB = 2mA.
1. mA = 60 , mB = 2mA 2. mB = 2(60 ) 3. mB = 120 4. mA + mB = 60 + 120 5. mA + mB = 180 6. A and B are supplementary
A B
CONFIDENTIAL 20
4) Fill in the blanks to complete the two-column proof. Given: 2 3.Prove: 1 and 2 are supplementary .
Proof:
STATEMENTS REASONS
1. 2 3 1. Given2. m2 = m3 2. a. __?___ 3. b. __?___ 3. Linear pair thm.4. m1 + m2 = 180 4. Def. of supp. S5. m1 + m3 = 180 5. b. __?___ step 2,46. d. __?___ 6. Def. of supp.S
1 23
c
CONFIDENTIAL 21
5) Use the given plan to write a two-column proof. Given: X is the mid point of AY, and Y is the midpoint of XB.Prove: AX YB.
Plan: By the definition of midpoint, AX XY, and XY YB. Use the Transitive property to conclude AX YB.
A X Y B
CONFIDENTIAL 22
Geometric Proof
When writing a geometric proof, you use the deductive reasoning to create a chain of logical steps that move from
hypothesis to conclusion of the conjecture you are proving. By proving that the conclusion is true, you have proven that the
original conjecture is true.
Conclusion
Definition Postulate Properties Theorems
Hypothesis
When writing a geometric proof, it is important to justify each logical step with a reason. You can use symbols and
abbreviations, but they must be clear enough so that anyone who reads your proof will understand them.
Let’s review
CONFIDENTIAL 23
A B C
Writing justifications
Write a justification for each step, given that A and B are complementary and A C.
1. A and B are complementary 1. Given2. mA + mB = 90 2. Def. of comp.S3. A C 3. Given4. mA mB 4. Def. of S5. mC + mB = 90 5. Subtr. Prop. of = step 2,46. C and B are complementary 6. Def. of comp.S
CONFIDENTIAL 24
Theorem Hypothesis Conclusion
Linear Pair Theorem:
If two angles form a linear pair, then they are supplementary.
/A and /B form a linear pair.
/A and /B are supplementary.
Congruent Supplements Theorem:
If two angles are supplementary to the same angle (or two congruent angles), then the two angles are congruent.
/1 and /2 are supplementary./2 and /3 are supplementary.
/1 /3
Theorems
CONFIDENTIAL 25
A geometric proof begins with a Given and prove statements, which restate the hypothesis
and conclusion of the conjecture. In a two-column proof, you list the steps of the proof in
the left column and you write the matching reason for each step in the right column.
CONFIDENTIAL 26
Completing a two-column proof
Fill in the blanks to complete a two-column proof of theLinear Pair Theorem Given: 1 and 2 form a linear pair.Proof: 1 and 2 are supplementary.
Proof:
STATEMENTS REASONS
1. 1 and 2 form a linear pair 1. Given2. BA and BC form a line 2. Def. of lin. pair3. mABC = 180 3. Def. of straight 4. a. __?___ 4. Add. Post.5. b. __?___ 5. Subst. steps 3,4
BA C
21
NEXT PAGE
CONFIDENTIAL 27
BA C
21
Use the existing statements and reasons in the proof to fill in the blanks:
a. m1 + m2 =mABC Add. Post. is given as the reasonb. m1 + m2 = 180 Subst. 180 for ABCc. Def. of supp. s The measure of supp.S add to 180 by def
CONFIDENTIAL 28
Theorem Hypothesis Conclusion
Right Angle Congruence Theorem:
All right angles are congruent./A and /B are right angles.
Congruent Complements Theorem:
If two angles are complementary to the same angle (or two congruent angles), then the two angles are congruent.
/1 and /2 are complementary./2 and /3 are complementary.
/1 /3
Theorems
A B
CONFIDENTIAL 29
Writing a two-column proof from a plan
Use the given plan to write a two-column proof of theRight Angle Congruence Theorem.
Given: 1 and 2 are Right angles.Prove: 1 2 .
Proof:
STATEMENTS REASONS
1. 1 and 2 are Right angles 1. Given2. m1 = 90 2. Def. of rt. m2 = 903. m1 = m2 3. Trans. prop. of =4. m1 m2 4. Def. of S
12
CONFIDENTIAL 30
The Proof process
1) Write the conjecture to be proven.2) Draw a diagram to represent the hypothesis of the
conjecture.3) State the given information and mark it on the diagram.4) State the conclusion of the conjecture in terms of the
diagram.5) Plan your argument and prove the conjecture.
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