31
CONFIDENTIAL 1 Geometry Geometric Proof

CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

Embed Size (px)

Citation preview

Page 1: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

CONFIDENTIAL 1

Geometry

Geometric Proof

Page 2: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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:

Page 3: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up 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.

Page 4: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 5: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 6: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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.

Page 7: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 8: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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.

Page 9: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 10: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 11: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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:

Page 12: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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.

Page 13: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 14: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 15: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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!

Page 16: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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.

Page 17: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

CONFIDENTIAL 17

Now some problems for you to practice !

Page 18: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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:

Page 19: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 20: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 21: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 22: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 23: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 24: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 25: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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.

Page 26: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 27: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 28: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 29: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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

Page 30: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

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.

Page 31: CONFIDENTIAL 1 Geometry Geometric Proof. CONFIDENTIAL 2 Warm Up Identify the property that justifies each statement:

CONFIDENTIAL 31

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