Download 2.6 Practice A

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Date ————————————
Practice A
LESSON
2.6
For use with pages 113–120
In Exercises 1–3, complete the proof.
1. GIVEN: m∠ A 5 m∠ B, m∠ B 5 m∠ C
B
PROVE: ∠ A > ∠ C
A
C
Statements
Reasons
1. m∠ A 5 m∠ B, m∠ B 5 m∠ C
1. Given
2. m∠ A 5 m∠ C
2.
3. ?
3. Definition of congruent angles
?
2. GIVEN: DE 5 EF, EF 5 DF
}
E
}
PROVE: DF > DE
F
Statements
Reasons
1. DE 5 EF, EF 5 DF
1.
2. ?
2. Transitive Property of Equality
3. DF 5 DE
3.
4. ?
4. Definition of congruent segments
?
?
3. GIVEN: ∠ 1 and ∠ 2 are a linear pair.
1
PROVE: m∠ 1 5 1808 2 m∠ 2
128
2
Statements
Reasons
1.
?
1. Given
2.
?
2. The angles in a linear pair are
supplementary angles.
3. m∠ 1 1 m∠ 2 5 1808
3.
4.
4. Subtraction Property of Equality
?
Geometry
Chapter 2 Resource Book
?
Copyright © Holt McDougal. All rights reserved.
LESSON 2.6
D
Name ———————————————————————
LESSON
2.6
Practice A
Date ————————————
continued
For use with pages 113–120
Use the property to complete the statement.
4. Reflexive Property of Congruence:
?
5. Symmetric Property of Congruence: If
> ∠4
?
>
} }
? , then CD > DX.
In Exercises 6–9, name the property illustrated by the statement.
} }
6. If ∠ 1 > ∠ 2 and ∠ 2 > ∠ 4, then ∠ 1 > ∠ 4.
7. XY > XY
8. If ∠ CDE > ∠ RST, then ∠ RST > ∠ CDE.
9. If AB > BC, then BC > AB.
}
}
}
}
10. Sketch a diagram that represents the following information.
∠ ABC and ∠ CBD are adjacent angles.
∠ ABD and ∠ DBE are a linear pair.
11. Use the given information and the diagram to prove the statement.
C
GIVEN: 2 • m∠ ABC 5 m∠ ABD
D
PROVE: ∠ ABC > ∠ CBD
A
B
Reasons
LESSON 2.6
Copyright © Holt McDougal. All rights reserved.
Statements
12. Bicycle Tour You take part in a three day bicycle tour. On the first day, you ride
95 miles. On the third (final) day, you also ride 95 miles. Use the following steps to
prove that the distance you ride in the first two days is equal to the distance that you
ride in the last two days.
a. Draw a diagram for the situation by using a line segment to represent the total
distance of the three days and dividing the line segment into three parts that
represent the daily distances.
b. State what is given and what is to be proved.
c. Write a two-column proof.
Geometry
Chapter 2 Resource Book
129
6. You are given that AB 5 CD. By the Addition
Property of Equality, you can write
AB 1 BC 5 BC 1 CD. You know that
AC 5 AB 1 BC and BD 5 BC 1 CD by the
Segment Addition Postulate. By the Substitution
Property of Equality, you have AC 5 BC 1 CD.
You are given AC 5 6x 2 12, BC 5 4, and
CD 5 3x 2 2. Substitute these expressions into
the equation AC 5 BC 1 CD to obtain
6x 2 12 5 4 1 3x 2 2. Simplify the right side of
the equation to obtain 6x 2 12 5 3x 1 2. By the
Subtraction Property of Equality you have
3x 2 12 5 2. Next, by the Addition Property of
Equality you have 3x 5 14. Finally, by the
14
. Substitute
Division Property of Equality, x 5 }
3
this value of x into the expression for CD to obtain
CD 5 12. Because you are given that AB 5 CD,
you know that AB 5 12 also.
7. m∠ RPQ 5 m∠ RPS
Given
m∠ SPQ 5 m∠ RPQ 1 m∠ RPS
Segment Addition Postulate
m∠ SPQ 5 m∠ RPQ 1 m∠ RPQ
Substitution Prop. of Equality
m∠ SPQ 5 2(m∠ RPQ)
Simplify.
8. a 5 b
Given
ac 5 bc
Multiplication Prop. of Equality
c5d
Given
bc 5 bd
Multiplication Prop. of Equality
ac 5 bd
Substitution Prop. of Equality
9. You are given that a is a positive integer.
Assume a is even. Then a 5 2k, where k is a
positive integer. Substitute 2k for a in a 1 1 to
obtain 2k 1 1. Because 2k is even, adding 1 to
this expression produces an odd number.
Therefore, a 1 1 is odd.
Lesson 2.6
Practice Level A
1. Transitive Property of Equality; ∠ A > ∠ C
2. Given; DE 5 DF; Symmetric Property of
} }
Equality; DF > DE 3. ∠ 1 and ∠ 2 are a linear
pair; ∠ 1 and ∠ 2 are supplementary; Definition of
Supplementary Angles; m∠ 1 5 1808 2 m∠ 2
} }
4. ∠ 4 5. DX; CD 6. Transitive Property of
Congruence 7. Reflexive Property of Congruence
8. Symmetric Property of Congruence
9. Symmetric Property of Congruence
A24
Geometry
Chapter 2 Resource Book
10. Sample sketch:
D
C
A
B
E
11. 1. 2m∠ ABC 5 m∠ ABD (Given)
2. m∠ ABC 1 m∠ CBD 5 m∠ ABD
(Angle Addition Postulate)
3. 2m∠ ABC 5 m∠ ABC 1 m∠ CBD
(Transitive Property of Equality)
4. m∠ ABC 5 m∠ CBD
(Subtraction Property of Equality)
5. ∠ ABC > ∠ CBD
(Definition of congruent angles)
B
12. Sample answer: a. A
95 mi
C
D
95 mi
b. Given: AB 5 95, CD 5 95 Prove: AC 5 BD
c. 1. AB 5 95, CD 5 95 (Given)
2. AB 1 BC 5 AC, CD 1 BC 5 BD (Segment
Addition Postulate) 3. 95 1 BC 5 AC,
95 1 BC 5 BD (Substitution Property of
Equality) 4. AC 5 95 1 BC (Symmetric Property
of Equality) 5. AC 5 BD (Transitive Property of
Equality)
Practice Level B
1. 1. Given 2. Given 3. Substitution Property
} }
of Equality 4. HI > IJ 5. Given 6. Transitive
Property of Congruence
2. 1. Given 2. Given 3. Definition of
complementary angles 4. Transitive Property of
Equality 5. Subtraction Property of Equality
6. Definition of congruent angles
3. 1. Given 2. Reflexive Property of Equality
3. Addition Property of Equality 4. Segment
Addition Postulate 5. Segment Addition Postulate
6. Substitution Property of Equality
4. 1. Given 2. Transitive Property of Angle
Congruence 3. m∠ 2 5 m∠ 4 4. Substitution
Property of Equality
5. x 5 6; Because the angles are congruent,
the measures of the angles are congruent by the
definition of congruent angles. Set the measure of
the angles equal to each other to find x.
} }
6. x 5 3; By the transitive property, FG > JH.
Set the lengths of the segments equal to each
other to find x.
7. x 5 5; By the transitive property,
∠ ABD > ∠ EBC. Because the angles are congruent,
the measures of the angles are congruent by the
definition of congruent angles. Set the measures
of the angles equal to each other to find x.
Copyright © Holt McDougal. All rights reserved.
ANSWERS
Lesson 2.5, continued