Download Reteach Geometric Proof

Name
LESSON
2-6
Date
Class
Reteach
Geometric Proof
To write a geometric proof, start with the hypothesis
of a conditional.
Hypothesis
Apply deductive reasoning.
Deductive Reasoning
• Definitions • Properties
• Postulates • Theorems
Prove that the conclusion of the conditional is true.
Conclusion
__›
Conditional: If BD is the angle bisector of ABC, and
ABD 1, then DBC 1.
__›
Given: BD is the angle bisector of ABC, and ABD 1.
!
$
Prove: DBC 1
Proof:
__›
1. BD is the angle bisector of ABC.
2. ABD DBC
3. ABD 1
4. DBC 1
"
1.
2.
3.
4.
Given
Def. of bisector
Given
Transitive Prop. of 1
#
_
1. Given:
_
Prove:
-
N is the midpoint
the
_, Q is
_ of MP
_
midpoint of RP , and PQ NM .
.
_
PN QR
2
0
Write a justification for each step.
Proof:
_
1. N is the midpoint of MP .
1.
Given
2.
Given
3.
Def. of midpoint
4.
Given
5.
Transitive Prop. of 6.
Def. of midpoint
7.
Transitive Prop. of _
2. Q is the midpoint of RP .
_
_
3. PN NM
_
_
4. PQ NM
_
_
5. PN PQ
_
_
6. PQ QR
_
_
7. PN QR
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46
1
Holt Geometry
Name
LESSON
2-6
Date
Class
Reteach
Geometric Proof
continued
A theorem is any statement that you can prove. You can use two-column proofs
and deductive reasoning to prove theorems.
Congruent Supplements
Theorem
If two angles are supplementary to the same angle (or to two
congruent angles), then the two angles are congruent.
Right Angle Congruence
Theorem
All right angles are congruent.
Here is a two-column proof of one case of the Congruent Supplements Theorem.
4 and 5 are supplementary and
5 and 6 are supplementary.
Given:
7
4
6
5
4 6
Prove:
Proof:
Statements
Reasons
1. 4 and 5 are supplementary.
1. Given
2. 5 and 6 are supplementary.
2. Given
3. m4 m5 180
3. Definition of supplementary angles
4. m5 m6 180
4. Definition of supplementary angles
5. m4 m5 m5 m6
5. Substitution Property of Equality
6. m4 m6
6. Subtraction Property of Equality
7. 4 6
7. Definition of congruent angles
Fill in the blanks to complete the two-column proof
of the Right Angle Congruence Theorem.
1
2. Given: 1 and 2 are right angles.
2
Prove: 1 2
Proof:
Statements
1. a.
Reasons
1 and 2 are right angles.
1. Given
2. m1 90
3. c.
2. b.
m2 90
3. Definition of right angle
4. m1 m2
5. e.
4. d.
1 2
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Definition of right angle
Transitive Property of Equality
5. Definition of congruent angles
47
Holt Geometry
Name
Date
LESSON
2-6
Class
Practice A
__›
A.
B.
C.
�
�
�
�
�
4. c.
�1 � �2
Given
2. �1 and �2 are supplementary, and
�3 and �4 are supplementary.
2. b.
Linear Pair Thm.
3. c.
m�1 � m�2 � 180°, and
m�3 � m�4 � 180°
Name
LESSON
2-6
4. d.
43
Seg. Add. Post.
5. 2AB � AC, and 2EF � DF.
Subst.
6. AB � EF
Given
7. 2AB � 2EF
Mult. Prop. of �
8. AC � DF
Subst. Prop. of �
�
�
�
�
Reasons
�HKJ is a straight angle.
1. Given
2. b.
2. m�HKJ � 180�
__›
3. c. KI
bisects �HKJ
4. �IKJ � �IKH
4. Def. of � bisector
5. Def. of � �
5. m�IKJ � m�IKH
6. d.
Def. of straight �
3. Given
m�IKJ � m�IKH � m�HKJ
6. � Add. Post.
7. 2m�IKJ � 180�
7. e. Subst. (Steps
8. m�IKJ � 90�
8. Div. Prop. of �
9. �IKJ is a right angle.
9. f.
2, 5, 6
)
Def. of right �
Add. Prop. of �
Date
Class
Holt Geometry
2-6
Write a two-column proof.
4
Holt Geometry
Hypothesis
Deductive Reasoning
• Definitions • Properties
• Postulates • Theorems
Apply deductive reasoning.
Reasons
1. Given
2. Linear Pair Thm.
3. Def. of supp. �
4. Subst. Prop. of �
5. Subtr. Prop. of �
Class
Geometric Proof
To write a geometric proof, start with the hypothesis
of a conditional.
3
Statements
1. m�2 � m�3 � m�4 � 180�
2. �1 and �2 are supplementary.
3. m�1 � m�2 � 180�
4. m�1 � m�2 � m�2 � m�3 � m�4
5. m�1 � m�3 � m�4
Date
Reteach
LESSON
2
44
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Name
Geometric Proof
1
Def. of � segments
Statements
1. a.
Practice C
1. Given: The sum of the angle measures
in a triangle is 180°.
Prove: m�1 � m�3 � m�4
_
Proof:
3. Def. of supp. �
4. m�1 � m�2 � m�3 � m�4 � 360�
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All rights reserved.
Def. of � segments
4. AB � BC � AC, and DE � EF � DF.
10. Given: �HKJ
is a straight angle.
__›
KI bisects �HKJ.
Prove: �IKJ is a right angle.
Reasons
1. a.
Def. of mdpt.
Fill in the blanks to complete the two-column proof.
Follow the plan to fill in the blanks in the two-column proof.
7. Given: �1 and �2 form a linear pair, and
� �
�
�3 and �4 form a linear pair.
�
Prove: m�1 � m�2 � m�3 � m�4 � 360�
Plan: The Linear Pair Theorem shows that �1 and �2 are supplementary and
�3 and �4 are supplementary. The definition of supplementary says that
m�1 � m�2 � 180� and m�3 � m�4 � 180�. Use the Addition Property
of Equality to make the conclusion.
1. �1 and �2 form a linear pair, and
�3 and �4 form a linear pair.
Given
_
3. AB � BC, and DE � EF.
_
4. Def. of � �
Statements
_
�
9. AC � DF
Def. of straight �
3. Subst. Prop. of �
�
2. AB � BC, and DE � EF.
1. Given
3. m�1 � m�2
_
_
Reasons
2. b.
�
_
Fill in the blanks with the justifications and steps listed to complete the
two-column proof. Use this list to complete the proof.
�1 � �2
Def. of straight �
1
�1 and �2 are straight angles.
6. Given: �1 and �2 are straight angles.
Prove: �1 � �2
2
Proof:
2. m�1 � 180�, m�2 � 180�
�
�
�
1. B is the midpoint of AC, _
and E is the midpoint of DF.
two-column
5. In a
proof, each step in the proof is on the left and
the reason for the step is on the right.
�1 and �2 are straight angles.
_
Given: AB = EF, B is the midpoint
_of AC,
and E is the midpoint of DF.
�
Definition of � bisector
Given
Transitive Prop. of �
Statements
Geometric Proof
Write a justification for each step.
�
Class
Practice B
2-6
B
HJ is the bisector of �IHK.
A
�2 � �1
B
�1 � �3
C
�2 � �3
1. a.
Date
LESSON
Geometric Proof
Write the letter of the correct justification next to each step.
(Use one
__›justification twice.)
Given: HJ is the bisector of �IHK and �1 � �3.
1.
2.
3.
4.
Name
Prove that the conclusion of the conditional is true.
Conclusion
__›
Conditional: If BD is the angle bisector of �ABC, and
�ABD � �1, then �DBC � �1.
__›
Given: BD is the angle bisector of �ABC, and �ABD � �1.
2. Peter drives on a straight road and stops at an intersection. The intersecting road
is also straight. Peter notices that one of the angles formed by the intersection is
a right angle. He concludes that the other three angles must also be right angles.
Draw a diagram and write a two-column proof to show that Peter is correct.
Possible answer:
4
1
3
Prove: �DBC � �1
Proof:
__
›
Statements
1. �1 is a right angle.
2. �1 and �2, �1 and �4,
�2 and �3 are supplementary.
3. �1 � �3
4. �3 is a right angle.
5. m�1 � m�2 � 180�,
m�1 � m�4 � 180�
6. m�1 � 90�
7. 90� � m�2 � 180�,
90� � m�4 � 180�
8. m�2 � 90�, m�4 � 90�
9. �2 and �4 are right angles.
Reasons
1. Given
2. Linear Pair Thm.
1. Given
2. Def. of � bisector
3. �ABD � �1
3. Given
4. �DBC � �1
4. Transitive Prop. of �
1. Given:
Prove:
_
�
�
_
_
2. Q is the midpoint of RP .
_
_
_
_
_
_
_
_
3. PN � NM
4. PQ � NM
5. PN � PQ
6. PQ � QR
_
_
7. PN � QR
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
45
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
69
�
PN � QR
1. N is the midpoint of MP .
8. Subtr. Prop. of �
9. Def. of rt. �
�
_
Proof:
6. Def. of rt. �
7. Subst.
�
_
N is the midpoint
the
_, Q is
_ of MP
_
midpoint of RP , and PQ � NM.
Write a justification for each step.
3. Congruent Supps. Thm.
4. Rt. � � Thm.
5. Def. of supp. �
1
�
�
1. BD is the angle bisector of �ABC.
2. �ABD � �DBC
2
�
46
�
1.
Given
2.
Given
3.
Def. of midpoint
4.
Given
5.
Transitive Prop. of �
6.
Def. of midpoint
7.
Transitive Prop. of �
Holt Geometry
Holt Geometry
Name
LESSON
2-6
Date
Class
Name
Reteach
Date
Challenge
LESSON
Geometric Proof
2-6
continued
A theorem is any statement that you can prove. You can use two-column proofs
and deductive reasoning to prove theorems.
Prove It!
In a proof, you can often determine the Given information from the figure.
Congruent Supplements
Theorem
If two angles are supplementary to the same angle (or to two
congruent angles), then the two angles are congruent.
Right Angle Congruence
Theorem
All right angles are congruent.
Write the information that is given in each figure. Then make a conjecture
about what you could prove using the given information.
2.
1.
�4 and �5 are supplementary and
�5 and �6 are supplementary.
Prove:
�4 � �6
7
4
6
�
1
5
2 3
�
Proof:
Statements
Reasons
1. Given
2. Given
3. m�4 � m�5 � 180�
3. Definition of supplementary angles
4. m�5 � m�6 � 180�
4. Definition of supplementary angles
�
�
5. m�4 � m�5 � m�5 � m�6
5. Substitution Property of Equality
Given: EF � EJ and FG � JH.
right �. Possible answer: Prove
Possible answer: Prove EH � EG.
�KLM and �NML are right angles.
�2 � �3
�1 � �4
6. Subtraction Property of Equality
Prove:
Possible answer:
2
Prove: �1 � �2
Proof:
Statements
Reasons
1. Given
3. c.
2. b.
m�2 � 90�
5. e.
4. d.
�1 � �2
47
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All rights reserved.
2-6
Transitive Property of Equality
5. Definition of congruent angles
Name
LESSON
Definition of right angle
3. Definition of right angle
4. m�1 � m�2
Date
Holt Geometry
Class
Name
3
2-6
1. Refer to the diagram of the stained-glass window and use
the given plan to write a two-column proof.
4
�
Reasons
1. Given
2. Rt. � � Thm.
3. Def. of � �
4. � Add. Post.
5. Subst. Prop. of �
6. Given
7. Def. of � �
8. Subst. Prop. of �
9. Reflex. Prop. of �
10. Subtr. Prop. of �
11. Def. of � �
Date
Class
Follow a Procedure
The following five steps are used to give geometric proofs:
The Proof Process
Given: �1 and �3 are supplementary.
�2 and �4 are supplementary.
�3 � �4
1. Write the conjecture to be proven.
3
2. Draw a diagram if one is not provided.
1
Prove: �1 � �2
3. State the given information and mark it on the diagram.
4
2
Use the definition of supplementary angles to write
the given information in terms of angle measures.
Then use the Substitution Property of Equality and
the Subtraction Property of Equality to conclude
that �1 � �2.
4. State the conclusion of the conjecture in terms of the diagram.
5. Plan your argument and prove your conjecture.
Mark the diagram and answer the questions about the following proof.
__›
Statements
Reasons
1. �1 and �3 are supplementary.
1. Given
�2 and �4 are supplementary.
2. m�1 � m�3 � 180�
2. Def. of supp. �
m�2 � m�4 � 180�
3. m�1 � m�3 � m�2 � m�4 3. Subst. Prop. of �
4. �3 � �4
4. Given
5. m�3 � m�4
Given: FD bisects �EFC.
�
__›
�
FC bisects �DFB.
Prove: �EFD � �CFB
�
�
�
Proof:
__›
__›
1. FD bisects �EFC, FC bisects �DFB.
1. Given
2. �EFD � �DFC, �DFC � �CFB
2. Definition of � bisector
5. Def. of � �
3. m�EFD � m�DFC, m�DFC � m�CFB
3. Definition of � �
6. m�1 � m�4 � m�2 � m�4
6. Subst. Prop. of �
4. m�EFD � m�CFB
4. Transitive Property of Equality
7. m�4 � m�4
7. Reflex. Prop. of �
5. �EFD � �CFB
5. Definition of � �
8. m�1 � m�2
8. Subtr. Prop. of �
9. �1 � �2
9. Def. of � �
1. What
__
__ was the given information?
›
›
FD bisects �EFC, FC bisects �DFB
2. What should be marked in the diagram?
The position of a sprinter at the starting blocks is shown in the diagram.
Which statement can be proved using the given information? Choose the
best answer.
All three angles should be marked congruent to each other.
3. What was the conjecture to be proved?
�
2. Given: �1 and �4 are right angles.
A �3 � �5
C m�1 � m�4 � 90�
B �1 � �4
D m�3 � m�5 � 180�
�EFD � �CFB
4. What titles should be put above the two columns?
�
Statements and Reasons
3. Given: �2 and �3 are supplementary.
�2 and �5 are supplementary.
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Holt Geometry
Reading Strategies
LESSON
Geometric Proof
F �3 � �5
G �2 � �5
_
�
2
48
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Problem Solving
Plan:
1
�
Statements
1. �KLM and �NML are right angles.
2. �KLM � �NML
3. m�KLM � m�NML
4. m�KLM � m�1 � m�2,
m�NML � m�3 � m�4
5. m�1 � m�2 � m�3 � m�4
6. �2 � �3
7. m�2 � m�3
8. m�1 � m�2 � m�2 � m�4
9. m�2 � m�2
10. m�1 � m�4
11. �1 � �4
1
2. Given: �1 and �2 are right angles.
�
�
Given:
7. Definition of congruent angles
�1 and �2 are right angles.
_
3. Write a two-column proof.
7. �4 � �6
2. m�1 � 90�
�
Given: �1 � �4 and �ABC is a
6. m�4 � m�6
1. a.
�
4
�2 � �3.
1. �4 and �5 are supplementary.
2. �5 and �6 are supplementary.
Fill in the blanks to complete the two-column proof
of the Right Angle Congruence Theorem.
�
�
Here is a two-column proof of one case of the Congruent Supplements Theorem.
Given:
Class
�
H �3 and �5 are complementary.
J �1 and �2 are supplementary.
49
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All rights reserved.
�
5. What property of equality was used to prove the angles congruent?
�
Transitive Property of Equality
Holt Geometry
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All rights reserved.
70
50
Holt Geometry
Holt Geometry