1 Geometry Section 4-1B Angles Outside the Triangle Pg. 247 Be ready to grade 4-1A Quiz Friday.
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Transcript of 1 Geometry Section 4-1B Angles Outside the Triangle Pg. 247 Be ready to grade 4-1A Quiz Friday.
1
Geometry Section 4-1BGeometry Section 4-1B
Angles Outside the TriangleAngles Outside the Triangle
Pg. 247Pg. 247
Be ready to grade 4-1ABe ready to grade 4-1AQuiz FridayQuiz Friday
2
The last section of this chapter referred to angles inside of a triangle. Those are called _____________________.interior angles
X Y
Z
WX, Y, and Z
are interior angles.
Definition
3
Exterior Angle:
An angle of a triangle that is formed by extending one of the sides of the triangle.
X Y
Z
W W is an exterior angle.
Definition
4
Adjacent Interior Angle:
The angle that forms a linear pair with the exterior angle.
X Y
Z
W Z is an adjacent interior angle with W.
Definition
5
Remote Interior Angle:
An angle NOT adjacent to the exterior angle.
X Y
Z
W X and Y are remote
interior angles to W.
Definition
6
1, 2 and 3
4 and 5
52
3
4 61
a. Exterior Angles
b. Remote Interior Angles for 3
78
Note that 8 is an exterior angle, but 7 is not.
9
9 is not an exterior angle. Why?
Try It:
Pg.248
Try It:
7
m1 m2 m3 m4
83o 40o
84o
71o
148o
54o
1 2
3
4
57o 97o
32o 64o
55o 125o
180 – 83 = 97
180 – (83 + 40) = 57
180 – 148= 32
180 – (84 + 32) = 64
180 – (71 + 54) = 55
180 – 55= 125
Try It:
Pg.249
Try It:
8
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.
X Y
Z
W
mX + mY = mW.
Pg.249
9
Exterior Angle Inequality Theorem:The measure of an exterior angle of a triangle
is greater than the measure either of its remote interior angles.
X Y
Z
W
A theorem that follows immediately FROM another is called a _____________.corollary
If we know that mX + mY = mW, then….
Pg.249
12
What are the remote angles for 1?
123
4
1
2
3
4
4
2
31
3 and 4
2 and 4
2 and 3
#1 – 3
Pg.250
Exercises
13
Find m5. 5
8
6
7
m5 = m7 = (2x + 31)o
m6 = xo m8 = (4x - 19)o
x + 2x + 31 + 4x – 19 = 180
x = 24
180 – 24 =156o
#7
Pg.250
Exercises
14
The brace JK is fastened to the wall and the shelf shown. Which angle has the greater measure, LJK or MKJ? Justify your answer.
#10
Pg.250
MKJ has the greater measure because of the Exterior Angle Inequality Theorem. LJK is a
remote interior angle for MKJ.
Exercises
15
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
Pg.250
We know that the acute angles of a right triangle are complementary therefore:
Exercises
1x + 2x = 90
x = 30o
Be sure to pay attention to which angle they are asking you for! Biggest or smallest?
16
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
Exercises
18
3
C
24 1DA B
If m4 = 150o and m3 = 130o find mACD.
#14
Pg.251
Find the measure of CAD.
CAD = 180 – 150 = 30o.
ACD = 180 – (130 + 30) = 20o.
Exercises
19
3
C
24 1DA B
If m1 = 20o, find m3.
#15
Pg.251
Find the measure of 2.
2 = 180 – (90 + 20) = 70o.
3 = 180 – 70= 110o.
Exercises
20
A1 2
B
D
C
3
5
4E
If AC || BE, explain how the figure above shows that
m1 + m2 + m3 = 180o.
#25
Pg.252
By the properties of parallel lines, we know that m1 = m5 and m2 = m4.
Since m3 + mm5 = 180, we can use substitution to prove that m1 + m2 + m3 = 180o.
Exercises