LESSON 3–5
Proving Lines Parallel
Five-Minute Check (over Lesson 3–4)
TEKS
Then/Now
Postulate 3.4: Converse of Corresponding Angles Postulate
Postulate 3.5: Parallel Postulate
Theorems: Proving Lines Parallel
Example 1: Identify Parallel Lines
Example 2: Use Angle Relationships
Example 3: Real-World Example: Prove Lines Parallel
Over Lesson 3–4
containing the point (5, –2) in point-slope form?
A.
B.
C.
D.
Over Lesson 3–4
What is the equation of the line with slope 3
containing the point (–2, 7) in point-slope form?
A. y = 3x + 7
B. y = 3x – 2
C. y – 7 = 3x + 2
D. y – 7 = 3(x + 2)
Over Lesson 3–4
What equation represents a line with slope –3
containing the point (0, 2.5) in slope-intercept
form?
A. y = –3x + 2.5
B. y = –3x
C. y – 2.5 = –3x
D. y = –3(x + 2.5)
Over Lesson 3–4
containing the point (4, –6) in slope-intercept form?
A.
B.
C.
D.
Over Lesson 3–4
What equation represents a line containing points
(1, 5) and (3, 11)?
A. y = 3x + 2
B. y = 3x – 2
C. y – 6 = 3(x – 2)
D. y – 6 = 3x + 2
Over Lesson 3–4
A.
B.
C.
D.
Targeted TEKS
G.5(B) Construct congruent segments, congruent angles, a
segment bisector, an angle bisector, perpendicular lines, the
perpendicular bisector of a line segment, and a line parallel to a
given line through a point not on a line using a compass and
a straightedge.
G.6(A) Verify theorems about angles formed by the intersection
of lines and line segments, including vertical angles, and
angles formed by parallel lines cut by a transversal and prove
equidistance between the endpoints of a segment and points
on its perpendicular bisector and these relationships to solve
problems.
Mathematical Processes
G.1(E), G.1(F)
You found slopes of lines and used them to
identify parallel and perpendicular lines.
• Recognize angle pairs that occur with
parallel lines.
• Prove that two lines are parallel.
Identify Parallel Lines
A. Given 1  3, is it
possible to prove that any of
the lines shown are parallel?
If so, state the postulate or
theorem that justifies your
answer.
1 and 3 are corresponding angles of lines a and b.
Answer: Since 1  3, a║b by the
Converse of the
Corresponding Angles
Postulate.
Identify Parallel Lines
B. Given m1 = 103 and
m4 = 100, is it possible to
prove that any of the lines
shown are parallel? If so,
state the postulate or
theorem that justifies your
answer.
1 and 4 are alternate interior angles of lines a and c.
Answer: Since 1 is not congruent to 4, line a is
not parallel to line c by the Converse of the
Alternate Interior Angles Theorem.
A. Given 1  5, is it possible
to prove that any of the lines
shown are parallel?
A. Yes; ℓ ║ n
B. Yes; m ║ n
C. Yes; ℓ ║ m
D. It is not possible to prove
any of the lines parallel.
B. Given m4 = 105 and m5 =
70, is it possible to prove that
any of the lines shown are
parallel?
A. Yes; ℓ ║ n
B. Yes; m ║ n
C. Yes; ℓ ║ m
D. It is not possible to prove
any of the lines parallel.
Use Angle Relationships
Find mZYN so that
||
. Show your work.
Read the Item From the figure, you know that
mWXP = 11x – 25 and mZYN = 7x + 35. You are
asked to find mZYN.
Use Angle Relationships
Solve the Item WXP and ZYN are alternate exterior
angles. For line PQ to be parallel to line MN, the
alternate exterior angles must be congruent. So
mWXP = mZYN. Substitute the given angle
measures into this equation and solve for x. Once you
know the value of x, use substitution to find mZYN.
m WXP = m ZYN
Alternate exterior angles
11x – 25 = 7x + 35
Substitution
4x – 25 = 35
4x = 60
x = 15
Subtract 7x from each side.
Add 25 to each side.
Divide each side by 4.
Use Angle Relationships
Now use the value of x to find mZYN.
mZYN = 7x + 35
Original equation
= 7(15) + 35
= 140
Answer: mZYN = 140
x = 15
Simplify.
Check Verify the angle measure by using the value
of x to find mWXP.
mWXP = 11x – 25
= 11(15) – 25
= 140
Since mWXP = mZYN, WXP  ZYN
and
||
.
ALGEBRA Find x so that
A. x = 60
B. x = 9
C. x = 12
D. x = 12
||
.
Prove Lines Parallel
CONSTRUCTION In the window
shown, the diamond grid pattern is
constructed by hand. Is it possible to
ensure that the wood pieces that run
the same direction are parallel? If so,
explain how. If not, explain why not.
Answer: Measure the corresponding angles formed
by two consecutive grid lines and the
intersecting grid line traveling in the opposite
direction. If these angles are congruent, then
the grid lines that run in the same direction
are parallel by the Converse of the
Corresponding Angles Postulate.
GAMES In the game Tic-Tac-Toe, four
lines intersect to form a square with
four right angles in the middle of the
grid. Is it possible to prove any of the
lines parallel or perpendicular?
Choose the best answer.
A. The two horizontal lines are
parallel.
B. The two vertical lines are
parallel.
C. The vertical lines are
perpendicular to the horizontal
lines.
D. All of these statements are
true.
LESSON 3–5
Proving Lines Parallel