Download Study Guide 2 - Mr. Gonzalez

4-1
NAME ______________________________________DATE __________PERIOD______
Study Guide
Student Edition
Pages 142–147
Parallel Lines and Planes
When planes do not intersect, they are said to be parallel. Also,
when lines in the same plane do not intersect, they are parallel.
But when lines are not in the same plane and do not intersect,
they are skew.
Example:
Name the parts of the triangular prism shown at
the right. Sample answers are given.
parallel planes:
planes PQR and NOM
苶 and R
苶Q
苶
parallel segments: 苶
MO
skew segments:
M
苶N
苶 and R
苶Q
苶
Refer to the figure in the example.
1. Name two more pairs of parallel segments.
Sample answers: M
苶N
苶 and R
苶P
苶, 苶
NO
苶 and P
苶Q
苶, 苶
NP
苶 and M
苶R
苶,
N
苶P
苶 and O
苶Q
苶, 苶
MR
苶 and O
苶Q
苶
2. Name two more segments skew to 苶
NM
苶.
苶Q
P
苶, O
苶Q
苶
3. Name a segment that is parallel to plane MRQ. N
苶P
苶
Name the parts of the hexagonal prism shown at the right.
苶C
苶 F
苶E
苶, Q
苶P
苶, K
苶L
苶
4. three segments that are parallel to B
5. three segments that are parallel to J
苶K
苶 A
苶B
苶, M
苶P
苶, D
苶E
苶
6. a segment that is skew to 苶
QP
苶
Sample answers: A
苶J
苶, A
苶F
苶, and D
苶E
苶
7. the plane that is parallel to plane AJQ
©
Glencoe/McGraw-Hill
plane CDM
T19
Geometry: Concepts and Applications
4-2
NAME ______________________________________DATE __________PERIOD______
Study Guide
Student Edition
Pages 148–153
Parallel Lines and Transversals
A line that intersects two or more lines in a plane at
different points is called a transversal. Eight angles are
formed by a transversal and two lines.
Types of Angles
Angle
Definition
Examples
interior
lie between the two lines
⬔1, ⬔2, ⬔5, ⬔6
alternate interior
on opposite sides of the transversal
⬔1 and ⬔6, ⬔2 and ⬔5
consecutive interior
on the same side of the transversal
⬔1 and ⬔5, ⬔2 and ⬔6
exterior
lie outside the two lines
⬔3, ⬔4, ⬔7, ⬔8
alternate exterior
on opposite sides of the transversal
⬔3 and ⬔7, ⬔4 and ⬔8
Identify each pair of angles as alternate interior, alternate
exterior, consecutive interior, or vertical.
1. ⬔6 and ⬔10
2. ⬔14 and ⬔13
3. ⬔14 and ⬔6
4. ⬔1 and ⬔5
5. ⬔12 and ⬔15
6. ⬔2 and ⬔16
alternate exterior
alternate interior
consecutive interior
consecutive interior
alternate exterior
vertical
In the figure, 苶
AB
苶 苶
苶 and 苶
BC
苶 苶
苶.
DC
AD
7. For which pair of parallel lines are ⬔1 and ⬔4
alternate interior angles? B
苶C
苶 and A
苶D
苶
8. For which pair of parallel lines are ⬔2 and ⬔3
alternate interior angles? A
苶B
苶 and D
苶C
苶
©
Glencoe/McGraw-Hill
T20
Geometry: Concepts and Applications
NAME ______________________________________DATE __________PERIOD______
4-3
Study Guide
Student Edition
Pages 156–161
Transversals and Corresponding Angles
If two parallel lines are cut by a transversal, then the following
pairs of angles are congruent.
corresponding angles
alternate interior angles
alternate exterior angles
If two parallel lines are cut by a transversal, then
consecutive interior angles are supplementary.
In the figure m 储 n and p is a transversal. If m⬔2 ⫽ 35,
find the measures of the remaining angles.
Example:
Since m⬔2 ⫽ 35, m⬔8 ⫽ 35 (corresponding angles).
Since m⬔2 ⫽ 35, m⬔6 ⫽ 35 (alternate interior angles).
Since m⬔8 ⫽ 35, m⬔4 ⫽ 35 (alternate exterior angles).
m⬔2 ⫹ m⬔5 ⫽ 180. Since consecutive interior angles
are supplementary, m⬔5 ⫽ 145, which implies that
m⬔3, m⬔7, and m⬔1 equal 145.
In the figure at the right p q, m⬔1 ⴝ 78, and m⬔2 ⴝ 47.
Find the measure of each angle.
1. ⬔3
2. ⬔4
3. ⬔5
4. ⬔6
5. ⬔7
6. ⬔8
102
102
133
78
47
7. ⬔9
47
133
Find the values of x and y, in each figure.
8.
9.
(3x 5) °
(y 8)°
(6x 14)°
10.
6x °
5x ° 9y °
21, 60
(8x 40)°
18, 10
7y °
(3y 10)°
10, 19
Find the values of x, y, and z in each figure.
11.
12.
(3z 18)°
(y 18)°
72°
3y °
x°
108, 36, 30
©
Glencoe/McGraw-Hill
(y 12)°
x°
z°
90, 93, 15
T21
Geometry: Concepts and Applications
NAME ______________________________________DATE __________PERIOD______
4-4
Study Guide
Student Edition
Pages 162–167
Proving Lines Parallel
Suppose two lines in a plane are cut by a transversal. With
enough information about the angles that are formed, you
can decide whether the two lines are parallel.
IF
THEN
corresponding angles are congruent,
alternate interior angles are congruent,
alternate exterior angles are congruent,
consecutive interior angles are supplementary,
the lines are perpendicular to the same line,
the lines are parallel.
Example:
If ⬔1 ⫽ ⬔2, which lines must be
parallel? Explain.
៭៮៬
AC 储 ៭៮៬
BD because a pair of corresponding
angles are congruent.
Find x so that a b.
1.
2.
a
110 °
a
b
3. a
(3x 50)°
b
(2x 5)°
b
(4x 10)°
4.
15
(4x 22)°
a
b
17
45
5.
a
6.
57 °
2x °
21
a
(6x 7) °
(3x 38)°
b
b
(3x 9)°
(6x 12)°
44
15
Given the following information, determine which lines, if any,
are parallel. State the postulate or theorem that justifies your
answer.
7. ⬔1 ⬵ ⬔8
c d, alternate exterior
angles congruent
9. m⬔7 ⫹ m⬔13 ⫽ 180
e f, consecutive
interior angles
supplementary
©
Glencoe/McGraw-Hill
8. ⬔4 ⬵ ⬔9
e f, alternate
interior angles
congruent
10. ⬔9 ⬵ ⬔13
c d,
corresponding
angles congruent
T22
e
f
c
d
1 2
3 4
5 6
7 8
9 10
11 12
13 14
15 16
Geometry: Concepts and Applications
4-5
NAME ______________________________________DATE __________PERIOD______
Study Guide
Student Edition
Pages 168–173
Slope
To find the slope of a line containing two points with
coordinates (x1, y1) and ( x2, y2), use the following formula.
y ⫺y
x2 ⫺ x1
2
1 , where x1 ⫽ x2
m⫽
The slope of a vertical line, where x1 ⫽ x2, is undefined.
Two lines have the same slope if and only if they are parallel
and nonvertical.
Two nonvertical lines are perpendicular if and only if the
product of their slopes is ⫺1.
Example:
Find the slope of the line ᐉ passing through A(2, ⫺5)
and B(⫺1, 3). State the slope of a line parallel to ᐉ.
Then state the slope of a line perpendicular to ᐉ.
Let ( x1, y1) ⫽ (2, ⫺5) and ( x2, y2) ⫽ (⫺1, 3).
3 ⫺ (⫺5)
8
⫽ ⫺ᎏᎏ.
Then m ⫽ ⫺
1⫺2
3
Any line in the coordinate plane parallel to ᐉ has
slope ⫺ᎏ8ᎏ.
3
Since
8
3
⫺ᎏᎏ
⭈ ᎏ3ᎏ ⫽ ⫺1, the slope of a line perpendicular to
8
the line ᐉ is ᎏ3ᎏ.
8
Find the slope of the line passing through the given points.
1. C(⫺2, ⫺4), D(8, 12)
8
5
2. J(⫺4, 6), K(3, ⫺10)
3. P(0, 12), R(12, 0)
16
ⴚ
7
4. S(15, ⫺15), T(⫺15, 0)
1
2
ⴚ
ⴚ1
5. F (21, 12), G(⫺6, ⫺4)
6. L(7, 0), M(⫺17, 10)
16
27
5
12
ⴚ Find the slope of the line parallel to the line passing through
each pair of points. Then state the slope of the line
perpendicular to the line passing through each pair of points.
7. I(9, ⫺3), J(6, ⫺10)
7
3
, ⴚ 3
7
©
Glencoe/McGraw-Hill
8. G(⫺8, ⫺12), H(4, ⫺1)
11
12
, ⴚ
12
11
9. M(5, ⫺2), T(9, ⫺6)
ⴚ1,
T23
1
Geometry: Concepts and Applications
NAME ______________________________________DATE __________PERIOD______
4-6
Study Guide
Student Edition
Pages 174–179
Equations of Lines
You can write an equation of a line if you are given
• the slope and the coordinates of a point on the line, or
• the coordinates of two points on the line.
Example: Write the equation in slope-intercept form of the line
that has slope 5 and an x-intercept of 3.
Since the slope is 5, you can substitute 5 for m in
y ⫽ mx ⫹ b.
y ⫽ 5x ⫹ b
Since the x-intercept is 3, the point (3, 0) is on the line.
y ⫽ 5x ⫹ b
0 ⫽ 5(3) ⫹ b
y ⫽ 0 and x ⫽ 3
0 ⫽ 15 ⫹ b
Solve for b.
b ⫽ ⫺15
So the equation is y ⫽ 5x ⫺ 15.
If you know two points on a line, you will need to find the slope
of the line passing through the points and then write the
equation.
Write the equation in slope-intercept form of the line that
satisfies the given conditions.
1. m ⫽ 3, y-intercept ⫽ ⫺4
y ⴝ 3x ⴚ 4
2. m ⫽ ⫺ᎏ2ᎏ, x-intercept ⫽ 6
5
2
y ⴝ ⴚ2x ⴙ 1
5
3. passes through (⫺5, 10) and (2, 4)
4. passes through (8, 6) and (⫺3, ⫺3)
0
y ⴝ ⴚ6x ⴙ 4
7
y ⴝ 9x ⴚ 6
7
5. perpendicular to the y-axis,
passes through (⫺6, 4)
11
x ⴝ ⴚ7
8. perpendicular to the graph of
y ⫽ 4x ⫺ 1 and passes through (6, ⫺3)
y ⴝ ⴚ1x ⴚ 3
y ⴝ 3x ⴙ 18
©
Glencoe/McGraw-Hill
11
6. parallel to the y-axis,
passes through (⫺7, 3)
yⴝ4
7. m ⫽ 3 and passes through (⫺4, 6)
5
4
T24
2
Geometry: Concepts and Applications