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Geometry

Geometric Proof

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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:

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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.

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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

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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

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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.

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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

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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.

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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

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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

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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:

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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.

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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

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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

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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!

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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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Now some problems for you to practice !

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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:

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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

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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

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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

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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

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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

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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

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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.

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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

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NEXT PAGE

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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

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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

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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

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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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You did a great job You did a great job today!today!