Download Study Guide

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

page
1
page
2
page
3
Document related concepts
no text concepts found
Transcript 
Name———————————————————————— Lesson
2.6
Date —————————————
Study Guide
For use with the lesson “Prove Statements about Segments and Angles”
goal
Write proofs using geometric theorems.
Vocabulary
A proof is a logical argument that shows a statement is true.
A two-column proof has numbered statements and corresponding
reasons that show an argument in a logical order.
A theorem is a statement that can be proven.
Theorem 1 Congruence of Segments: Segment congruence is
reflexive, symmetric, and transitive.
Theorem 2 Congruence of Angles: Angle congruence is reflexive,
symmetric, and transitive.
Write a two-column proof
D
Write a two-column proof for the following situation.
} }
GIVEN: AD 5 8, BC 5 8, BC​
​  > CD​
​  
} }
PROVE: ​AD​ > CD​
​  
Lesson 2.6
Statements
A
1. AD 5 8
C
B
Reasons                                             
1. Given
BC 5 8
2. AD 5 BC
2. Transitive Property of Equality
} }
3. AD​
​  > BC​
​  
} }
4. BC​
​  > CD​
​  
} }
5. AD​
​  > CD​
​  
3. Definition of congruent segments
4. Given
5. Transitive Property of Equality
Exercise for Example 1
1. Write a two-column proof for B
the following situation.
}
} }
C
}
GIVEN: AD 5 12, AB 5 12, BC​
​  > CD​
​  , AD​
​  > CD​
​  
}
}
PROVE: ​BC​ > BA​
​  
A
D
2-82
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
example 1
Geometry
Chapter Resource Book
CS10_CC_G_MECR710761_C2L06SG.indd 82
4/27/11 5:35:09 PM
Name———————————————————————— Lesson
2.6
Date —————————————
Study Guide continued
For use with the lesson “Prove Statements about Segments and Angles”
example 2
Name the property shown
Name the property illustrated by the statement.
} }
} }
} }
a. If XY​
​  > WZ​
​  and WZ​
​  > PQ​
​  , then XY​
​  > PQ​
​  .
b. If ∠ M > ∠ N, then ∠ N > ∠ M.
Solution
a. Transitive Property of Segment Congruence
b. Symmetric Property of Angle Congruence
Exercises for Example 2
Name the property illustrated by the statement.
} }
} }
2. ∠ R > ∠ R
3. If ​XY​ > PQ​
​  , then PQ​
​  > XY​
​  .
} }
4.​XY​ > XY​
​  
example 3
5.If ∠ X > ∠ Y and ∠ Y > ∠ Z,
then ∠ X > ∠ Z.
Transitive Property of Congruence
C
B
GIVEN: ∠ A > ∠ B, ∠ B > ∠ C
A
PROVE: ∠ A > ∠ C
Statements
Reasons                                        
1. ∠ A > ∠ B
1. Given
2. m∠ A 5 m∠ B
2. Definition of congruent angles
3. m∠ B 5 m∠ C
3. Definition of congruent angles
4. m∠ A 5 m∠ C
4. Transitive Property of Equality
5. ∠ A > ∠ C
5. Definition of congruent angles
∠B > ∠C
Lesson 2.6
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Prove the Transitive Property of Angle Congruence.
Exercise for Example 3
6. Prove the Transitive Property of Segment Congruence.
}
} }
}
}
}
GIVEN: AB​
​  > BC​
​  , BC​
​  > CD​
​  
A
B
C
D
PROVE: ​AB​ > CD​
​  
Geometry
Chapter Resource Book
CS10_CC_G_MECR710761_C2L06SG.indd 83
2-83
4/27/11 5:35:09 PM
Equality, 10 5 2x by the Subtraction Property of
Equality, and 5 5 x by the Division Property of
Equality.
9. Because #EG​
​ ##$ bisects ∠ DEF,
∠ DEG > ∠ FEG. It is given that ∠ D > ∠ DEG,
so ∠ D > ∠ FEG by the Transitive Property of
Congruence. Then m∠ D 5 m∠ FEG,
4x 5 2x 1 30 by the Substitution Property of
Equality, 2x 5 30 by the Subtraction Property of
Equality and x 5 15 by the Division Property of
Equality.
10.
}
} }
}
​  , AB​
​  and CD​
​  bisect each other (Given)
1. AE​
​  > CE​
}
}
​  .
2. E is the midpoint of ​AB​ and of CD​
(Definition of segment bisector)
} } } }
​  , CE​
​  > ED​
​  (Definition of midpoint)
3. ​EB​ > AE​
} }
​  (Transitive Property of Equality)
4. ​AE​ > ED​
} }
​  (Transitive Property of Equality)
5. ​EB​ > ED​
11. Sample answers: a. Marge Jade Leon Ariel Clay
A
b.
A
E
A
B
C
B
C
D
D
E
E
}
Real-Life Application
1. 16 rods 2. Transitive Property of Segment
Congruence 3. 6 cuts 4. 150 in.
5. 14 rods; including the 5 that were already cut
6. 14 in. 7. 3 8. Yes; two pieces; one is 1 inch
in length and the other is 2 inches in length.
Challenge Practice
1. YZ 5 11, VZ 5 27.5 2. VW 5 1, VZ 5 5
3. The coordinate of X is 4, the coordinate of
Y is 6, and the coordinate of Z is 10.
4. The coordinate of V is 12, the coordinate of
X is 0, and the coordinate of Y is 26.
a1b
5. The coordinate of M is ​ }
   
 
​, the
2
3a 1 b
coordinate of P is }
​  4   
​, and the coordinate
5a 1 3b
of Q is ​ }
   
​. 
8
6. x 5 10, y 5 2 7. x 5 18, y 5 8
8.
D
c. Given: C is the midpoint of ​AE​ , B is the
}
​  ,
Prove: AB 5 DE d. 1. C is the midpoint of AE​
}
}
B is the midpoint of ​AC​,  D is the midpoint of ​CE​ 
} } } } } }
​  > BC​
​  , CD​
​  > DE​
​  
(Given) 2. ​AC​ > ​CE​ , AB​
(Definition of midpoint) 3. AC 5 CE, AB 5 BC,
CD 5 DE (Definition of congruent segments)
4. AC 5 AB 1 BC, CE 5 CD 1 DE
(Segment Addition Postulate)
5. AC 5 AB 1 AB, CE 5 DE 1 DE
(Substitution Property of Equality)
6. AB 1 AB 5 DE 1 DE (Substitution Property of
Equality) 7. 2AB 5 2DE (Simplify.)
8. AB 5 DE (Division Property of Equality)
Study Guide
}
}
} }
​  ,
(Definition of congruent segments); BC​
​  > CD​
} }
} }
AD​
​  (Given); ​CD​ > BA​
​ ( Transitive Property
​  > CD​
} }
​  (Transitive
of Segment Congruence) BC​
​  > BA​
1. AD 5 12, AB 5 12 (Given); ​AD​ > AB​
​  
Property of Segment Congruence)
2. Reflexive Property of Angle Congruence
3. Symmetric Property of Segment Congruence
4. Reflexive Property of Segment Congruence
5. Transitive Property of Angle Congruence
A28
}
(Definition of congruent segments); BC 5 CD
(Definition of congruent segments); AB 5 CD
} }
​  
(Transitive Property of Equality); ​AB​ > CD​
(Definition of congruent segments)
}
}
}
midpoint of AC​
​ ,  D is the midpoint of ​CE​ 
} }
6. ​AB​ > BC​
​  , BC​
​  > CD​
​  , (Given); AB 5 BC
A
C
1 2
B
3 4
G
E
F
Statements
Reasons
​###$ bisects ∠ ABC. 1. Given
1. BD​
2. ∠ 1 > ∠ 2 2. Definition of angle
bisector
3. m∠ 1 5 m∠ 2 3. Definition of
congruent angles
4. m∠ 2 5 m∠ 3 4. Measures of vertical
angles are equal.
5. m∠ 1 5 m∠ 3 5. Transitive Property of
Equality
6. m∠ 1 5 m∠ 4 6. Measures of vertical
angles are equal.
7. m∠ 3 5 m∠ 4 7. Substitution Property
of Equality
##$ bisects ∠ EBG. 8. Definition of
8. #​BF ​
angle bisector
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
answers
Lesson 2.6 Prove Statements
about Segments and Angles,
continued
Geometry
Chapter Resource Book
CS10_CC_G_MECR710761_C2AK.indd 28
4/27/11 6:42:32 PM
Related documents