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Geometry Section 4-1B
Angles Outside the Triangle
Pg. 247
Be ready to grade 4-1A
Quiz Friday
1
Definition
The last section of this chapter referred to angles inside of a
interior angles
triangle. Those are called _____________________.
W
Z
X
X, Y, and Z
are interior
angles.
Y
2
Definition
Exterior Angle:
An angle of a triangle that is formed by
extending one of the sides of the triangle.
W
Z
X
W is an
exterior angle.
Y
3
Definition
Adjacent Interior Angle:
The angle that forms a linear pair with the
exterior angle.
W
Z
X
Z is an
adjacent interior
angle with W.
Y
4
Definition
Remote Interior Angle:
An angle NOT adjacent to the exterior angle.
W
Z
X
X and Y
are remote
interior angles
to W.
Y
5
Try It:
a. Exterior Angles
1, 2 and 3
b. Remote Interior Angles for 3
4 and 5
7
2
8
Try
It:
5
Pg.
248
1
4
6
9
3
Note that
9 is8not
is an
an exterior
exterior angle,
angle. but
Why?
7 is not.
6
Try It:
Try
It:
m1
83o
m2
40o
m3
57o
32o
55o
84o
71o
64o
54o
m4
97o
148o
125o
(83 + 32)
40) = 64
57
180 – (84
3
Pg.
249
180 –180
(71 –+ 83
54)==97
55
4 1
180 – 55=
148=125
32
2
7
Exterior Angle Theorem:
The measure of an exterior angle of a triangle
is equal to the sum of the measures of its
remote interior angles.
W
Z
Pg.
249
X
Y
mX + mY = mW.
8
A theorem that follows immediately FROM
another is called a _____________.
corollary
Exterior
Angle
Inequality
If we know
that mX
+ mYTheorem:
= mW, then….
The measure of an exterior angle of a triangle
is greater than the measure either of its
remote interior angles.
Pg.
249
W
Z
X
Y
9
Theorem
Each angle of an equilateral triangle
measures 60o.
60o
Pg.
249
60o
60o
10
Theorem
The acute angles of a right triangle are
complementary.
b
Pg.
249
ma + mb = 90o
a
11
Exercises
What are the remote angles for 1?
1
4
3
3 and 4
#1 – 3
3
2
2
1
Pg.
250
4
2 and 4
2
2 and 3
1 4
3
12
Exercises
Find m5.
6
#7
Pg.
250
5
8
7
m5 = 180 – 24 = m7 = (2x + 31)o
156o
m6 = xo
m8 = (4x - 19)o
x + 2x + 31 + 4x – 19 = 180
x = 24
13
Exercises
#10
The brace JK is fastened
to the wall and the shelf
shown. Which angle has
the greater measure,
LJK or MKJ? Justify
your answer.
Pg.
250
MKJ has the greater measure because of the
Exterior Angle Inequality Theorem. LJK is a
remote interior angle for MKJ.
14
Exercises
The measures of the acute angles of a right triangle
are in the ratio of 1:2. What is the measure of the
smallest angle of the triangle?
#11
We know that the acute angles of a right triangle
are complementary therefore:
Pg.
250
1x + 2x = 90
x = 30o
Be sure to pay attention to which angle they are
asking you for! Biggest or smallest?
15
Exercises
If the measures of the exterior angles of the acute angles
of a right triangle are (6x + 23)o and
(4x + 17)o, find the measures of the acute angles.
#12
Pg.
251
16
Exercises
If m1 = 18o, find m2.
C
#13
Pg.
251
4
A
3 2
D
1
B
m2 = 180 – (90 + 18) = 72o
17
Exercises
If m4 = 150o and m3 = 130o find mACD.
C
#14
Pg.
251
4
A
3 2
D
1
B
Find the measure of CAD.
CAD = 180 – 150 = 30o.
ACD = 180 – (130 + 30) = 20o.
18
Exercises
If m1 = 20o, find m3.
C
#15
Pg.
251
4
A
3 2
D
1
B
Find the measure of 2.
2 = 180 – (90 + 20) = 70o.
3 = 180 – 70= 110o.
19
Exercises
If AC || BE, explain how the
figure above shows that
m1 + m2 + m3 = 180o.
D
5
B
3
E
4
#25
1
2
A
Pg.
252
C
By the properties of parallel lines, we know that
m1 = m5 and m2 = m4.
Since m3 + mm5 = 180, we can use substitution
to prove that m1 + m2 + m3 = 180o.
20
Homework: Practice 4-1B
Quiz Friday
21