Download Connections The Geometry of Lines and Angles

The Geometry of
Lines and Angles
Connections
Have you ever . . .
• Looked at the patterns of streets on a map?
• Marked out the location of a fence?
• Measured parallel lines to hang two pictures on the wall?
Lines and angles are all around us in everyday shapes. To understand distances, shapes, and forms, geometry looks at lines and
the angles they form wherever they meet.
A line in geometry is a two-dimensional figure on a
plane. Lines can be parallel (;) or intersecting. Parallel
lines never meet. Intersecting lines meet like two streets
at an intersection.
• Perpendicular (=) intersecting lines form four
right angles.
Intersecting Lines
Perpendicular
b
c
a
1
2
4
3
d
• Intersecting lines form congruent (equal sized)
opposite angles: E1 , E3 and E2 , E4
• The sum of two angles on a line is 180°: E1 + E2 =180c
Transversal Lines
Perpendicular
p
m
n
t
r
5 6
7 8
9 10 s
11 12
A line crossing two parallel lines is called a transversal.
• Perpendicular (=) transversals form eight
right angles.
• Transversals form two congruent sets of angles:
E5 , E8 , E9 , E12 and E6 , E7 , E10 , E11
• If you draw a transversal at an angle, you can
see which angles appear congruent.
203
Essential Math Skills
Learn
It!
Intersecting Lines: Find All the Angles
Use your knowledge of intersecting lines to deduce unknown information.
• Find angles that you know.
• Describe other angles based on their relationship to known angles.
The diagram shows the intersections of
four streets.
10 = 40°
u
t
2
1
• A Street (line a) ; B Street (line b).
4
3
• Tarry Lane (line t) and Uptown Drive
(line u) form transversals.
a
5 6
7 8
10
11
9
• Uptown Drive is = to both A Street and
B Street.
b
12
14
13
An architect needs to know the measure of
E1 to plan a building on that lot.
Mark Angles You Know
Find all the angles that you know. Remember, perpendicular (=) lines give you information
about angles. Mark angles that are given and angles formed by perpendicular lines.
?
1. What are the measurements of all the angles that you know? Mark them on
the diagram.
You are given the measure of one angle (E10 = 40c).
Because Uptown Drive is perpendicular to two parallel streets, you know it forms right angles:
E5 = 90c
E7 = 90c
E11 = 90c
E14 = 90c
E6 = 90c
E8 = 90c
E12/13 = 90c
E9/10 = 90c
10 = 40°
u
t
2
1
3
4
90˚ 90˚
90˚ 90˚
90˚
10
90˚
9
90˚
b
12
13
Mark Angles You Can Find by Subtraction
Mark all the angles that you can find by subtracting a portion of a bigger angle.
204
a
90˚
The Geometry of Lines and Angles
?
2. What are the measurements of angles you can find by subtraction? Mark them on
the diagram.
Since E9/10 is a right angle, you can find E9 by subtraction.
90c - 40c = 50c
Mark Opposite Angles
When two lines intersect, opposite angles (angles across from each other) are congruent.
Identify and mark the measurements of opposite angles.
?
3. What are the measurements of all opposite angles? Mark them on the diagram.
You can mark the angles opposite to E9 and E10: E13 = 40c and E12 = 50c
Mark Congruent Angles on a Parallel Line
When a transversal crosses two parallel lines, it forms two congruent sets of angles. For
example, E1, E2, E3, and E4 are congruent to E9, E10/11, E12, and E13/14.
?
4. What are the measurements of E1, E2, E3, and E4? Mark them
on the diagram.
Using
Un P A C
E1 and E4 are congruent, and they are the same as E9 and E12. E2
and E3 are congruent, and they are the same as E10/11 and E13/14.
• E1 = 50c
• E2 = 40c + 90c = 130c
• E3 = 50c
• E4 = 40c + 90c = 130c
To understand
a problem with
lines, look for lines
that cross each
other, forming
angles. Identify
opposite angles,
perpendicular lines,
and parallel lines.
Find the Solution
derstand
Identify what you need to know to solve the problem. In this case,
the architect wants to know the measurement of E1.
lan
The answer is 50°.
ttack
heck
205
Essential Math Skills
e
ic
Pract
It!
Solve each problem using relationships of lines and angles.
1. Scaffolds like this one are used for construction
work. A scaffold is created using intersecting bars
for support.
An engineer is working on a design that has bars meeting at 65°
angles. Find the value of EA, EB, and EC in the engineer's diagram.
B
C
65°
A
a. What are the measurements of angles you can find by subtraction?
b. What are the measurements of opposite angles?
2. When homes are built, the contractor creates a frame for
each room using parallel wood beams and supports that form
transversals. An electrician wants to know the measurement
of EX in the diagram so he can plan the wiring.
206
a. Mark angles you can find by subtraction, opposite angles,
and congruent angles on the parallel lines. Explain how
you know the measurements of these angles.
b. What is the value of EX?
X
135°
The Geometry of Lines and Angles
3. Delores is an engineer who works with the local railway system. She is designing a
transition track for trains to move onto when they are out of service. The transition
track will veer off at a 50° angle. When the train returns to service going southbound on
this track, at what angle will the train’s turn be? Explain your reasoning.
50°
X
4. This map shows a bike path that crosses a road running
parallel to a river. If EB is 77.2°, what is EH? Explain
your reasoning.
A B
C D
E F
G H
Review &
Practice
Get more practice
with lines and
angles on pages
335–336.
207
Essential Math Skills
5. Using your knowledge that two angles on a straight line equal 180°, explain why
EA = EC in this diagram.
A
B
D
C
6. The sum of the angles of a triangle equal 180°. Based on this information and what
you know about intersecting lines, mark the measures of all 12 angles in these three
intersections. Explain how you calculated the measurements.
Build Your
Math Skills
61.5°
112°
Extend rays, the
lines that form the
sides of an angle,
out beyond the
vertex, the point
where the lines
meet, to form
intersecting lines.
ray
vertex
208
The Geometry of Lines and Angles
Check Your Skills
Use your knowledge of lines and angles to answer the following questions.
1. This diagram shows studs in a wall frame. The carpenter
wants to know if the studs are parallel. How could the
carpenter do this?
a. Check that EA and EI are congruent
b. Check that EA, EE, and EJ are congruent
A
c. Check that EA, EH, and EL are congruent
C
d. Check that EA, ED, and EL are congruent
2.
D
a.128.7°
b.137.7°
c.142.7°
d.148.7°
K
H
G
A builder is creating the stair railing shown in
the diagram. What is the measure of EA?
3.
F
E
B
I
J
L
A
42.3°
Two streets cross at the angle shown in
the diagram. If EA measures 77.1°, what
does EB measure?
a.77.1°
b.99.9°
c.101.1°
d.102.9°
4. Tycho is planning to cut custom wood pieces to
inlay in a tabletop. Each wood piece is shaped as a
parallelogram, formed by two sets of parallel lines
as shown in the diagram. Tycho needs to measure
the angles to set the blades on his saw. If EA is 112°,
what does EB measure?
A
B
D
A
C
B
209
Essential Math Skills
5.
In her afternoon football game, Alexa runs
across the 40 yard line at an angle of 134.2°.
She turns left and crosses the 20 yard line at 138.5°
as shown in the diagram. At what angle did
she turn?
a.86.7°
b.87.3°
c.89.8°
d.92.2°
134.2°
138.5°
X°
Use this diagram to answer questions 6 through 8.
A
C
E
G
B
D
F
H
6. In this diagram of a transversal crossing two parallel lines, how many angle
measurements do you need to know to find all eight angle measurements?
7. In this diagram of a transversal crossing two parallel lines, if
EB is 31°, which of the following is true?
a. EH = 31c
b. EE = 149c
c. EC = 149c
d. EA = 31c
8. In this diagram of a transversal crossing two parallel lines,
which set of angles have the same measures?
210
a. ED, EF, EA
b. EB, EH, EG
c. EC, EG, EF
d. EF, EB, EA
Remember
the Concept
• Opposite angles are
congruent.
• Two angles that form
a straight line add up
to 180°.
• A transversal through
two parallel lines
creates two sets of
congruent angles.
Answers and Explanations
By substitution, EC + 180c - EA = 180c.
The Geometry of Lines
and Angles
page 203
Intersecting Lines: Find All the Angles
Practice It!
pages 206–208
EC - EA = 0c
EC = EA
6.
50.5°
1a.EC = 180c - 65c = 115c
129.5°
You can find this angle by subtraction because the
sum of two angles that make a straight line is 180°.
1b.EB = 65c
129.5°
50.5°
112° 68°
112°
68°
EA = 115c
2a.180c - 135c = 45c
118.5°
61.5°
61.5°
118.5°
Transversals form congruent sets of angles across
parallel lines.
45°
135°
135°
45° 45°
135°
135°
45° 45°
135°
135°
45°
2b.45°
3.
130°
The railroad track forms a straight line. The transition track leaves at a 50° angle. The 50° angle plus the
angle the train returns at will add up to 180°.
50c + X = 180c
X = 180c - 50c = 130c
4.
If EB is 77.2°, EH is 102.8°.
EB and ED form a straight line, so they add up to 180°.
180c - 77.2c = 102.8c
Since EH is in the same position in a group of angles
formed by the transversal crossing a parallel line, EH
is congruent to ED. Therefore, EH is 102.8°.
5.
Since two angles that form a straight line must equal
180°, EA + EB = 180c.
For the same reason, EC + EB = 180c.
Since EA + EB = 180c, EB = 180c - EA.
The angles adjacent to the angles marked 112c and
61.5c form straight lines with those angles, so subtract the angle from 180c to find the adjacent angles.
Angles adjacent to the 112c angle are 68c, and angles
adjacent to the 61.5c angle are 118.5c.
The opposite angles from the angles marked 112c and
61.5c measure the same as those angles.
This gives you two angles of the triangle in the center.
Since the sum of the three angles of the triangle will
be 180c:
68c + 61.5c + x = 180c
129.5c + x = 180c
x = 50.5c
The angles adjacent to the 50.5c angle form straight
lines with that angle, so they measure 129.5c.
The angle opposite to the 50.5c angle is congruent to
that angle, so it is 50.5c.
Check Your Skills
pages 209–210
1.
c. Check that EA, EH, and EL are congruent
2.
b. 137.7°
3.
d. 102.9°
4.
68°
If EA is 112°, EB is 180° minus 112°, or 68°.
5.
b. 87.3°
i
Essential Math Skills
By extending the two lines that show where Alexa
runs, you get two transversals that intersect and
form a triangle with the 20 yard line. You can use
your knowledge of transversals and triangles to find
the angles.
134.2°
138.5°
87.3°
92.7°
134.2°
41.5°
45.8°
6.
One
You can find every angle from only one angle. All
angles can be calculated by subtraction, by identifying
opposite angles, and by identifying angles formed by
the transversal intersecting the parallel line.
7.
b. EE = 149c
8.
c. EC, EG, EF
ii