Transcript Angles and Parallel Lines

Parallel Lines and
Transversals
Lesson 3.4
r || w
Objective:
• Find the congruent
angles formed when a
transversal cuts
parallel lines.
Key Vocabulary
• None
Postulates
• 8 - Corresponding Angles
Theorems
• 3.5 Alternate Interior Angles
• 3.6 Alternate Exterior Angles
• 3.7 Same-Side Interior Angles
Parallel Lines and Angle Pairs
• Line 𝓏 is a transversal of parallel lines 𝓍
and 𝓎.
• Since lines 𝓍 and 𝓎 are parallel, there are
special relationships between specific
pairs of angles.
Review:
Parallel Lines
Transversal
• Two or more lines are parallel if and
only if they are in the same plane and
they do not intersect. (line w and r)
r
w
r || w
• A line intersecting
two or more coplanar
lines. (lines r and w)
r
w
r || w
Postulate 8
Corresponding ’s Postulate
• If 2 parallel lines are cut by a
transversal, then each
pair of corresponding ’s is .
1
2
l
• i.e. If l m, then 12.
m
Corresponding Angles
• Look for angles in an F shape to help you
find corresponding angles.
• If two parallel lines are cut by a transversal, then
Corresponding angles are congruent.
1
r
2
3
4
5
6
w
7
r || w
8
Example 1
Find the measure of the numbered angle.
a.
b.
c.
b. m5 = 135°
c. m2 = 90°
SOLUTION
a. m6 = 60°
Your Turn:
Find the measure of the numbered angle.
1.
ANSWER
120°
ANSWER
145°
ANSWER
45°
2.
3.
Example 2:
In the figure
and
Find
Corresponding Angles Postulate
Vertical Angles Postulate
Transitive Property
Definition of congruent angles
Substitution
Answer:
Your Turn:
In the figure
Answer:
and
Find
Theorem 3.5
Alternate Interior ’s Theorem
• If 2 parallel lines are cut by a transversal, then each pair
of alternate interior ’s is .
l
1
2
m
• i.e. If l m, then 12.
Alternate Interior Angles
• Look for angles inside a Z or N shape to
find alternate interior angles.
• If two parallel lines are cut by a transversal, then
Alternate Interior angles are congruent.
r
3
4
5
w
r || w
6
Example 3
Find the measure of PQR.
a.
b.
c.
b. mPQR = 120°
c. mPQR = 70°
SOLUTION
a. mPQR = 35°
Your Turn:
Find the measure of the numbered angle.
1.
ANSWER
90°
ANSWER
65°
ANSWER
100°
2.
3.
Theorem 3.6
Alternate Exterior ’s Theorem
• If 2 parallel lines are cut by a transversal, then the pairs
of alternate exterior ’s are .
l
m
1
2
• i.e. If l m, then 12.
• If two parallel lines are cut by a transversal, then
Alternate Exterior Angles are congruent.
1
2
r
w
7
r || w
8
Example 4
Find the measures of 1 and 2.
SOLUTION
The measure of 2 is 75° because alternate exterior angles are
congruent. The measure of 2 can be used to find the measure of
1.
m1 + m2 = 180°
m1 + 75° = 180°
m1 + 75° – 75° = 180° – 75°
m1 = 105°
Linear Pair Postulate
Substitute 75° for m2.
Subtract 75° from each side.
Simplify.
Your Turn:
Find the measure of the numbered angle.
1.
ANSWER
130°
ANSWER
42°
ANSWER
90°
2.
3.
Your Turn:
Use the diagram. Tell whether the angles are
congruent or not congruent. Explain.
4.
1 and 8
ANSWER
congruent by the Alternate
Exterior Angles Theorem
5.
3 and 4
ANSWER
Not congruent; the angles are
a linear pair.
6.
4 and 2
ANSWER
Not congruent; the angles are a
linear pair.
Your Turn:
Use the diagram. Tell whether the angles are
congruent or not congruent. Explain.
7.
2 and 7
ANSWER
congruent by the Alternate
Exterior Angles Theorem
8.
3 and 7
ANSWER
congruent by the
Corresponding Angles
Postulate
9.
3 and 8
ANSWER
Not congruent; there is no
special relationship between
these angles.
Theorem 3.7
Same-Side Interior ’s Theorem
• If 2 parallel lines are cut by a transversal, then each pair
of same-side interior ’s is supplementary.
l
1
m
2
• i.e. If l m, then 1 & 2 are supplementary or m1 +
m2 = 180°.
Same-Side Interior Angles
• Look for angles inside a C shape to find
same-side interior angles.
• If two parallel lines are cut by a transversal, then each
pair of Same-Side Interior Angles is supplementary.
r
3
4
5
w
r || w
6
Example 5
Find the measure of the numbered angle.
a.
b.
SOLUTION
a. m5 + 80° = 180°
m5 = 100°
b.
m6 + 130° = 180°
m6 = 50°
Example 6
Find the value of x.
SOLUTION
(x + 15)° = 125°
x = 110
Corresponding Angles Postulate
Subtract 15 from each side.
Your Turn:
Find the value of x.
1.
ANSWER
85
ANSWER
104
ANSWER
40
2.
3.
• If two parallel lines
are cut by a
transversal, then ……
– Corresponding
angles are
congruent,
– Alternate Interior
angles are
congruent, . . . . And .
...
– Alternate Exterior
angles are
congruent.
11 1
lr
2
3
4 4
5 5
lw
6
7
88
58
58 ˚˚
r
2
3
458
˚
58˚
585 ˚
w
6
7
8 58˚
r || w
If angle 1 = 58 ˚ then angle 5 = 58 ˚ because they are corresponding
angles, which are congruent to each other
Since angle 1 = 58 ˚ then angle 4 = 58 ˚ and since angle 5 = 58 ˚ then
angle 8 = 58 ˚ because they are vertical angles, which are congruent to
each other
58 ˚
r
122˚
2
3122
˚
4
5
w
6 ˚
122
7
8
122
˚
r || w
If angle 1 = 58 ˚ then angle 2 = 122 ˚ because the two angles form a line,
which is equal to 180 ˚
Since angle 2 = 122 ˚ then angle 7 = 122 ˚ because they are Alternate
Exterior angles, which are congruent to each other.
Since angle 3 = 122 ˚ then angle 6 = 122 ˚ because they are Alternate
Interior angles, which are congruent to each other.
Example 7:
What is the measure of RTV?
Example 7:
Alternate Interior Angles Theorem
Definition of congruent angles
Substitution
Example 7:
Alternate Interior Angles
Theorem
Definition of congruent
angles
Substitution
Angle Addition
Postulate
Answer: RTV = 125°
Example 8:
ALGEBRA If
and
find x and y.
Find x.
by the Corresponding Angles
Postulate.
Example 8:
Definition of congruent angles
Substitution
Subtract x from each side and
add 10 to each side.
Find y.
by the Alternate Exterior Angles
Theorem.
Definition of congruent angles
Substitution
Example 8:
Simplify.
Add 100 to each side.
Divide each side by 4.
Answer:
Your Turn:
and
ALGEBRA If
find x and y.
Answer:
Example 9:
1
125o
Find:
m1 =
m2 =
m3 =
m4 =
m5 =
m6 =
x=
55°
125°
55°
2
3
5
4
125°
55°
125°
40°
x+15o
6
Joke Time
• What flower grows between your nose and
your chin?
• Tulips
• How many sides are there to a circle?
• 2 – inside and outside.
• What do you get when you cross an
elephant and Darth Vader?
• An elevader.
Assignment
• Section 3.4, pg. 132-135: #1-12 all, 15-55 odd