Name
Date
Class
Practice B
LESSON
3-4
Perpendicular Lines
For Exercises 1– 4, name the shortest segment from the point to the line and write an inequality for x. (Hint: One answer is a double inequality.) 1.
2.
0 3.5
' (
X
)
11
X4
2
1
*
_
_
PR ; x 3.5 3.
HJ ; x 7
! 6
4. X3
"
3
21
#
$
5
6
_
4 X X4
7
_
AB ; x 9
UT ; x 17
Complete the two-column proof.
M
5. Given: m n Prove: 1 and 2 are a linear pair of congruent angles. Proof: Statements
m⬜n
1. a. 2. b.
m⬔1 90, m⬔2 90
1 2
N
Reasons 1. Given 2. Def. of
Def. of ⬵ ⭄
3. 1 2
3. c.
4. m1 m2 180
4. Add. Prop. of
5. d.
⬔1 and ⬔2 are a linear pair.
5. Def. of linear pair
6. The Four Corners National Monument is at the intersection of the borders of Arizona, Colorado, New Mexico, and Utah. It is called the four corners because the intersecting borders are perpendicular. If you were to lie down on the intersection, you could be in four states at the same time—the only place in the United States where this is possible. The figure shows the Colorado-Utah border extending north in a straight line until it intersects the Wyoming border at a right angle. Explain why the Colorado-Wyoming border must be parallel to the Colorado–New Mexico border. Possible answer:
Wyoming Utah Arizona
Colorado New Mexico
All the borders are straight lines, and the Colorado–Utah border is a transversal to the Colorado–Wyoming and the Colorado–New Mexico borders. Because the transversal is perpendicular to both borders, the borders must be parallel. Copyright © by Holt, Rinehart and Winston. All rights reserved.
28
Holt Geometry
Name
Date
Class
Name
Practice A
LESSON
3-4
3-4
perpendicular
midpoint
to a segment at the segment’s
.
1.
�
�
�
�
8
_
�
�
�
_
� � ��4
_
�
UT; x � 17 �
5. Given: m � n Prove: �1 and �2 are a linear pair of congruent angles.
congruent
1 2
Statements
angles,
�
2. b.
7. In a plane, if a transversal is perpendicular to one of two parallel lines, then it is to the other line. hoop
Use the drawing of a basketball goal for Exercises 8–10. In each exercise, justify Esperanza’s conclusion with one of the completed theorems from Exercises 5–7. Write the number 5, 6, or 7 in each blank to tell which theorem you used.
6
9. Esperanza knows that the hoop and the court are both perpendicular to the pole. She concludes that the hoop and the court are parallel to each other.
5
Date
Def. of � �
3. c.
4. m�1 � m�2 � 180�
4. Add. Prop. of �
�1 and �2 are a linear pair.
5. Def. of linear pair
Wyoming Utah Arizona
Colorado New Mexico
to the Colorado–Wyoming and the Colorado–New Mexico borders. Because the 7 Class
transversal is perpendicular to both borders, the borders must be parallel.
Holt Geometry
28
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Name
Practice C
LESSON
3-4
Perpendicular Lines
1. Draw a segment a little more than half the width of this page. Label this segment with length x, then use a compass and straightedge to construct a segment that has length _5_ x. 4
Date
Class
Holt Geometry
Reteach Perpendicular Lines
The perpendicular bisector of a segment is a line perpendicular to the segment at the segment’s midpoint.
Line b is the_ perpendicular bisector of RS.
�
�
1 –� 2
2. Def. of �
All the borders are straight lines, and the Colorado–Utah border is a transversal
10. Esperanza knows that the hoop and the court are parallel to each other. She also knows that the hoop is perpendicular to the pole. Esperanza concludes that the pole and the court are perpendicular.
27
m�1 � 90�, m�2 � 90�
6. The Four Corners National Monument is at the intersection of the borders of Arizona, Colorado, New Mexico, and Utah. It is called the four corners because the intersecting borders are perpendicular. If you were to lie down on the intersection, you could be in four states at the same time—the only place in the United States where this is possible. The figure shows the Colorado-Utah border extending north in a straight line until it intersects the Wyoming border at a right angle. Explain why the Colorado-Wyoming border must be parallel to the Colorado–New Mexico border. Possible answer:
court
8. Esperanza knows that the basketball pole intersects the court to form a linear pair of angles that are congruent. She concludes that the pole and the court are perpendicular.
1. Given
3. �1 � �2
5. d. pole
Reasons
m�n
1. a.
Name
�
Complete the two-column proof.
then the lines are perpendicular.
3-4
21
AB; x � 9
to each other.
Copyright © by Holt, Rinehart and Winston. All rights reserved.
�
��3
Proof:
6. If two intersecting lines form a linear pair of
LESSON
4.
�
FE; x � 8
5. If two coplanar lines are perpendicular to the same line, then the two lines are
perpendicular
_
HJ ; x � 7
� 6
Fill in the blanks to complete these theorems about parallel and perpendicular lines.
parallel
�
_
3.
_
��4
PR; x � 3.5
�
AB; x � 23
�
11
�
�
�
�
� �
�
3.5
4. 23
2.
�
to the line.
For Exercises 3 and 4, name the shortest segment from the point to the line and write an inequality for x. �
Perpendicular Lines
For Exercises 1– 4, name the shortest segment from the point to the line and write an inequality for x. (Hint: One answer is a double inequality.)
perpendicular
2. The shortest segment from a point to a line is
Class
Practice B
LESSON
Perpendicular Lines
1. The perpendicular bisector of a segment is a line
3.
Date
�
1 –� 4
�
The distance from a point to a line is the length of the shortest segment from the point to the line. It is the length of the perpendicular segment that joins them. �
_ _ _
_
5 –� 4
�
� ��9
2. Among segments BA, BC, BD, and BE, which is the shortest segment in the figure? Name the second shortest segment. Explain your answers.
_
_
�
_
The shortest segment from _ ‹__› W to SU is WT.
�
11 �
���
�
20
�
�
Because BD must be shorter than BE, x � 11. Therefore BC is the shortest
�
�
You can write and solve an inequality for x.
_
WU � WT
segment. If x � 1, then BD would be the second shortest segment, but if _
x�1� 8
x � 3, then AB would be the second shortest segment. So there is not enough
� 1 �1
information given in the figure to say which is the second shortest segment.
_
WT is the shortest segment. Substitute x � 1 for WU and 8 for WT. Subtract 1 from both sides of the equality.
x� 7
3. Use a straightedge to draw a triangle. Construct the perpendicular ���������������� bisector of each side of the triangle, and extend the bisectors into the interior of the triangle. Mark the point of intersection of the three bisectors. This is the circumcenter of the triangle. Use your compass to compare the distance from the circumcenter to each vertex of the triangle. What is remarkable about the distances?
Use the figure for Exercises 1 and 2.
‹__›
�
1. Name the shortest segment from point K to LN.
_
KM
The distances are equal.
��
2. Write and solve an inequality for x.
x � 5 � 14; x � 9
Now construct a circle completely around the triangle through all three vertices with your compass. You have circumscribed a circle around a triangle. 4. An architect designs a triangular jogging track around a circular pond. Each side of the track just touches the pond. The circle is inscribed in the triangle. The center of the circle is called the incenter of the triangle. The diameter of the circle has length 41. DA � 8x � 2z � 1 _1_, DB � 6x � y � 1, DC � 11y � 2z � 2. 2 Find x, y, and z. 3
Use the figure for Exercises 3 and 4.
�
�
‹___›
3. Name the shortest segment from point Q to GH.
_
�
QH
� �
�
�����
����� �
4. Write and solve an inequality for x.
�
�
�
�
x � 2 � 9; x � 11
x � 3, y � __ , z � �1 2
Copyright © by Holt, Rinehart and Winston. All rights reserved.
29
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Holt Geometry
Copyright © by Holt, Rinehart and Winston. All rights reserved.
57
30
Holt Geometry
Holt Geometry
