Download Practice B 3-3

Name
LESSON
3-3
Date
Class
Practice B
Proving Lines Parallel
Use the figure for Exercises 1–8. Tell whether lines m and n
must be parallel from the given information. If they are, state
your reasoning. (Hint: The angle measures may change for
each exercise, and the figure is for reference only.)
1. 7 3
M
N
1 8
2 7
3 6
4 5
2. m3 (15x 22)°, m1 (19x 10),
x8
m 储 n; Conv. of Alt. Int. ⭄ Thm.
m 储 n; Conv. of Corr. ⭄ Post.
3. 7 6
4. m2 (5x 3)°, m3 (8x 5),
x 14
m and n are parallel if and only if
m 储 n; Conv. of Same-Side
m⬔7 90°.
Int. ⭄ Thm.
5. m8 (6x 1)°, m4 (5x 3)°, x 9
6. 5 7
m 储 n; Conv. of Corr. ⭄ Post.
m and n are not parallel.
7. 1 5
8. m6 (x 10)°, m2 (x 15)
m 储 n; Conv. of Alt. Ext. ⭄ Thm.
m and n are not parallel.
9. Look at some of the printed letters in a textbook. The small horizontal and
vertical segments attached to the ends of the letters are called serifs. Most of the
letters in a textbook are in a serif typeface. The letters on this page do not have
serifs, so these letters are in a sans-serif typeface. (Sans means “without” in French.)
The figure shows a capital letter A with serifs.
Use the given information
to write a
_
_
paragraph proof that the serif, segment HI , is parallel to segment JK .
Given: 1 and 3 are supplementary.
_
_
Prove: HI JK
*
(
1 2
3
+
)
Sample answer: The given information states that ⬔1 and ⬔3 are
supplementary. ⬔1 and ⬔2 are also supplementary by the Linear Pair
Theorem. Therefore ⬔3 and ⬔2 must be congruent by the_
Congruent
_
Supplements Theorem. Since ⬔3 and ⬔2 are congruent, HI and JK
are parallel by the Converse of the Corresponding Angles Postulate.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
20
Holt Geometry
Name
Date
LESSON
3-3
Class
Name
Practice A
LESSON
3-3
Proving Lines Parallel
1. The Converse of the Corresponding Angles Postulate states that if two coplanar
lines are cut by a transversal so that a pair of corresponding angles is congruent,
parallel
.
then the two lines are
Use the figure for Exercises 2 and 3. Given
the information in each exercise, state the
reason why lines b and c are parallel.
2. �4 � �8
1 2
3 4
5 6
7 8
�
�
Practice B
Proving Lines Parallel
1. �7 � �3
4. m�2 � (5x � 3)°, m�3 � (8x � 5)�,
x � 14
Corr. � Post.
so that a pair of
congruent
alternate interior angles are
supplementary
, then the two lines are parallel.
parallel
angles are congruent, then the two lines are
�
3
2
Given: �1 and �3 are supplementary.
Prove: m � n
�
1
Proof:
.
�2 and �3 are supplementary.
_
�
�
1 2
3
�
�
Sample answer: The given information states that �1 and �3 are
supplementary. �1 and �2 are also supplementary by the Linear Pair
Congruent
Theorem. Therefore �3 and �2 must be congruent by the_
_
Supplements Theorem. Since �3 and �2 are congruent, HI and JK
are parallel by the Converse of the Corresponding Angles Postulate.
2. Linear Pair Thm.
m�n
_
Prove: HI � JK
Given
� Supps. Thm.
3. c.
m and n are not parallel.
Given: �1 and �3 are supplementary.
Reasons
1. a.
3. �1 � �2
4. Conv. of Corr. � Post.
19
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Name
3-3
m � n; Conv. of Corr. � Post.
8. m�6 � (x � 10)°, m�2 � (x � 15)�
m � n; Conv. of Alt. Ext. � Thm.
m�n
�2 and �3 are supplementary.
Given
� Supps. Thm.
Statements
1. �1 and �3 are supplementary.
LESSON
6. �5 � �7
9. Look at some of the printed letters in a textbook. The small horizontal and
vertical segments attached to the ends of the letters are called serifs. Most of the
letters in a textbook are in a serif typeface. The letters on this page do not have
serifs, so these letters are in a sans-serif typeface. (Sans means “without” in French.)
The figure shows a capital letter A with serifs.
Use the given information
to write a
_
_
paragraph proof that the serif, segment HI, is parallel to segment JK.
7. Shu believes that a theorem is missing from the lesson. His conjecture is that if
two coplanar lines are cut by a transversal so that a pair of same-side exterior
angles are supplementary, then the two lines are parallel. Complete the
two-column proof with the statements and reasons provided.
4. d.
Int. � Thm.
7. �1 � �5
6. If two coplanar lines are cut by a transversal so that a pair of alternate exterior
2. b.
m � n; Conv. of Same-Side
m�7 � 90°.
m and n are not parallel.
5. If two coplanar lines are cut by a transversal so that a pair of same-side interior
angles are
m and n are parallel if and only if
5. m�8 � (6x � 1)°, m�4 � (5x � 3)°, x � 9
, then the two lines are
parallel.
1 8
2 7
3 6
4 5
m � n; Conv. of Corr. � Post.
3. �7 � �6
Fill in the blanks to complete these theorems about parallel lines.
transversal
�
�
2. m�3 � (15x � 22)°, m�1 � (19x � 10)�,
x�8
m � n; Conv. of Alt. Int. � Thm.
m�7 = 68°, �3 � �7, Conv. of
4. If two coplanar lines are cut by a
Class
Use the figure for Exercises 1–8. Tell whether lines m and n
must be parallel from the given information. If they are, state
your reasoning. (Hint: The angle measures may change for
each exercise, and the figure is for reference only.)
3. m�3 � 68�, m�7 � (5x + 3)�, x � 13
Conv. of Corr. � Post.
Date
Date
Holt Geometry
Class
Name
Practice C
LESSON
3-3
Proving Lines Parallel
1. p � q, m�1 � (6x � y � 4)�, m�2 �
(x � 9y � 1)�, m�3 � (11x � 2)�
Find x, y, and the measures of
�1, �2, and �3.
1
�
20
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
2. Use the figure and the given information to write a paragraph proof
that the sum of the measures
of the three angles in a triangle is _
180�.
‹__›
(Hint: Begin by constructing FG through point C and parallel to AB.)
� 1
You can use the Converse of the
Corresponding Angles Postulate
to show that two lines are parallel.
�
Given: ABC is a triangle.
4
�
Prove: m�1 � m�2 � m�3 � 180�
‹__›
2
�1 � �3
�
q || r
Possible answer: Construct FG through point C and parallel to AB . �3 and
�4 are a linear pair, so m�3 � m�4 � 180� by the Linear Pair Theorem.
But the Angle Addition Postulate shows that m�4 � m�ACF � m�FCD,
so by substitution m�3 � m�ACF � m�FCD � 180�. m�1 � m�ACF
by the Alternate Interior Angles Theorem and m�2 � m�FCD by the
Corresponding Angles Postulate. Therefore m�1 � m�2 � m�3 � 180�
by substitution.
3. In an isosceles triangle, at least two of the‹__
angles
are
__› congruent. To construct
›
�FDE
and let
isosceles triangle DEH, begin by drawing DE and DF. If you copy __
›
the angle open in the same
be› parallel to DF . Instead,
___› direction, the ray would __
intersects
copy �FDE and draw EG so that the ray_
_DF. Label the intersection
point H. Use your compass to measure DH and EH. What is remarkable about
the measures of these segments?
The measures of the segments are equal.
1 2
�
34
�
�1 � �3 are corresponding angles.
Converse of the Corresponding Angles Postulate
Given: m�2 � 3x°, m�4 � (x � 50)°, x � 25
_
m�2 � 3(25)° � 75°
Substitute 25 for x.
m�4 � (25 � 50)° � 75° Substitute 25 for x.
m�2 � m�4
Transitive Property of Equality
�2 � �4
Definition of congruent angles
q || r
Converse of the Corresponding Angles Postulate
For Exercises 1 and 2, use the Converse of the Corresponding
Angles Postulate and the given information to show that c || d.
1. Given: �2 � �4
�2 � �4
c || d
�
�
If two coplanar lines are cut by a transversal
so that a pair of corresponding angles are
congruent, then the two lines are parallel.
Given: �1 � �3
�
3
�
�
Holt Geometry
Proving Lines Parallel
�
x � 11; y � �5; m�1 � 57°; m�2 � 57°; m�3 � 123�
Class
Reteach
Converse of the Corresponding
Angles Postulate
2 3
Date
�2 and �4 are corr. �.
Conv. of Corr. � Post.
1
3
2
4
�
�
�
2. Given: m�1 � 2x°, m�3 � (3x � 31)°, x � 31
�
m�1 � 2x°
� 2(31)° � 62°
m�3 � (3x � 31)°
� 3(31)° � 31° � 62°
m�1 � m�3
�1 � �3
c || d
�
4. Construct another isosceles triangle with angles and side lengths different from
the triangle you drew in Exercise 3. Again measure the lengths of the sides
opposite the congruent angles. Write a conjecture about the measures of the
side lengths in isosceles triangles.
Possible answer: If a triangle is
isosceles, then the sides opposite the
��������
�������
congruent angles are congruent.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
21
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
55
Substitute 31 for x.
Substitute 31 for x.
Trans. Prop. of �
Def. of � �
Conv. of Corr. � Post.
22
Holt Geometry
Holt Geometry