Transcript Vertical Angles

Vertical Angles
Section 2.4
Objective
• Find the measures of angles formed by
intersecting lines.
Key Vocabulary
• Vertical angles
• Linear pair
Postulates and Theorems
• Postulate 7: Linear Pair Postulate
• Theorem 2.3: Vertical Angle Theorem
More Angle Pairs
• Previously, you learned that two angles are
adjacent if they share a common vertex and side
but have no common interior points. In this
lesson, you will study other relationships
between pairs of angles.
Pairs of Angles
Two Types
– Vertical Angles
– Linear Pair
Pairs of Angles
• Vertical Angles – two nonadjacent angles
formed by two intersecting lines
• Examples
– Vertical angles: ∢1 and ∢2; ∢3 and ∢4 are vertical angles
– NOT vertical angles: ∢AEB and ∢DEC are not vertical angles
Vertical Angles
Two angles are vertical angles if their
sides form two pairs of opposite rays.
1
2
4
3
1 and 3 are vertical angles.
2 and 4 are vertical angles.
Opposite angles
formed by
intersecting lines.
Pairs of Angles
• Linear Pair – a pair of adjacent angles whose
noncommon sides are opposite rays
• Examples
– Linear pair: ∢1 and ∢2 are a linear pair of angles (form a line)
– NOT linear pair: ∢ADB and ∢ADC are not a linear pair
Linear Pair of Angles
Two adjacent angles are a linear pair
if their noncommon sides are opposite
rays.
Two adjacent
angles that form a
line.
5
6
5 and 6 are a linear pair.
Answer the questions using the diagram.
Are 2 and 3 a linear pair?
1
2
4
SOLUTION
No. The angles are adjacent but they do not form a line.
3
Answer the questions using the diagram.
Are 2 and 3 a linear pair?
Are 3 and 4 a linear pair?
1
2
4
SOLUTION
No. The angles are adjacent but they do not form a line.
Yes. The angles are adjacent and they form a line.
3
Answer the questions using the diagram.
Are 2 and 3 a linear pair?
Are 3 and 4 a linear pair?
1
2
Are 1 and 3 vertical angles?
4
SOLUTION
No. The angles are adjacent but they do not form a line.
Yes. The angles are adjacent and they form a line.
No. They are not opposite angles formed by intersecting lines.
3
Answer the questions using the diagram.
Are 2 and 3 a linear pair?
Are 3 and 4 a linear pair?
1
2
Are 1 and 3 vertical angles?
Are 2 and 4 vertical angles?
4
3
SOLUTION
No. The angles are adjacent but their noncommon sides are not opposite rays.
Yes. The angles are adjacent and their noncommon sides are opposite rays.
No. They are not opposite angles formed by intersecting lines.
No. They are not opposite angles formed by intersecting lines.
Your Turn:
Determine whether the labeled angles are
vertical angles, a linear pair, or neither.
a.
b.
c.
SOLUTION
a. 1 and 2 are a linear pair because they are adjacent and their
noncommon sides are on the same line.
b. 3 and 4 are neither vertical angles nor a linear pair.
c. 5 and 6 are vertical angles because they are not adjacent and their
sides are formed by two intersecting lines.
Postulate 7- Linear Pair
Postulate
• If two angles form a
linear pair, then they
are supplementary.
• m∠1 + m∠2 = 180
Example
Find the measure of RSU.
SOLUTION
RSU and UST are a linear pair. By the Linear Pair
Postulate, they are supplementary. To find mRSU,
subtract mUST from 180°.
mRSU = 180° – mUST = 180° – 62° = 118°
Theorem 2.3-Vertical Angles
Theorem
• If two angles are
vertical angles,
then they are
congruent.
• m∠5 ≌ m∠7 & m∠6
≌ m∠8
5
8
66
7
Example
Find the measure of CED.
SOLUTION
AEB and CED are vertical angles. By the
Vertical Angles Theorem, CED  AEB, so
mCED = mAEB = 50°.
Example
Find m1, m2, and m3.
SOLUTION
m2 = 35°
Vertical Angles Theorem
m1 = 180° – 35° = 145°
Linear Pair Postulate
m3 = m1 = 145°
Vertical Angles Theorem
Your Turn:
Find m1, m2, and m3.
1.
ANSWER
2.
m1 = 152°; m2 = 28°;
m3 = 152°
ANSWER
m1 = 56°; m2 = 124°;
m3 = 56°
ANSWER
m1 = 113°; m2 = 67°;
m3 = 113°
3.
Example
Find the value of y.
SOLUTION
Because the two expressions are measures of vertical angles, you can
write the following equation.
(4y – 42)° = 2y°
4y – 42 – 4y = 2y – 4y
–42 = –2y
–42
–2
=
–2y
–2
21 = y
Vertical Angles Theorem
Subtract 4y from each side.
Simplify.
Divide each side by –2.
Simplify.
Your Turn
Find the value of the variable.
1.
ANSWER
43
ANSWER
16
ANSWER
5
2
3
Solve for x and y. Then find the angle measure.
( 3x + 5)˚ D
•
E
( x + 15)˚
•
( 4y – 15)˚ • B
C
A•
( y + 20)˚
SOLUTION
Use the fact that the sum of the measures of angles that form a
Use substitution to find the angle measures (x = 40, y = 35).
linear pair is 180˚.
m AED = ( 3 x + 15)˚ = (3 • 40 + 5)˚ = 125˚
m AED + m DEB = 180°
m AEC + mCEB = 180°
m
+ 15)˚
= (40 + 15)˚ = 55˚
( 3x
+ DEB
5)˚ + =
( x(+x 15)˚
= 180°
( y + 20)˚ + ( 4y – 15)˚ = 180°
m AEC = 4x
( y ++ 20
20)˚
= (35 + 20)˚ = 55˚
= 180
5y + 5 = 180
m CEB = ( 4 y –4x15)˚
= (4 • 35 – 15)˚ = 125˚
= 160
5y = 175
x = 40
y = the
35 vertical
So, the angle measures
are 125˚, 55˚, 55˚, and 125˚. Because
angles are congruent, the result is reasonable.
Joke Time
• Why don't aliens eat clowns.
• Because they taste funny.
• What do you call a fish with no eyes?
• A fsh.
Assignment
• Section 2.4, pg. 78-71: #1-65 odd