Exterior Angles in Polygons
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When the sides of a polygon are extended,
other angles are formed.
The original angles are the
interior angles
.
The angles that form linear pairs with the
interior angles are the
exterior angles
.
Angles
An exterior
angle
of a triangle…
… is equal in measure to the
sum
of the
measures of its
two remote interior angles
.
remote
interior
angles
Exterior angle
The measure of an
exterior
angle in a
triangle is the
sum
of the measures of the
2
remote interior
angles
m<1 + m<2 = m<4
m
∠
1
=
m
∠
A
+
m
∠
B
m
<G = 60 + 69
m<FHG
= 180 – 111 = 69
linear pair
x
82
°
30
°
y
Find x & y
x = 68
°
y = 112
°
y = 30 + 82
y = 112˚
180 = 30 + 82 + x
180 = 112 + x
68˚ = x
Find
2x – 5 = x + 70
x – 5 = 70
x = 75
m< JKM = 2(75) - 5
m< JKM = 150 - 5
m< JKM = 145˚
S
o
l
v
e
f
o
r
y
i
n
t
h
e
d
i
a
g
r
a
m
.
Solve on your own
before viewing the
Solution
4y + 35 = 56 + y
3y + 35 = 56
3y = 21
y= 27
Find the measure of in the diagram shown.
Solve on your own
before viewing the
Solution
40 + 3x = 5x - 10
40 = 2x - 10
50 = 2x
25 = x
m
< 1= 5x - 10
m
< 1= 5(25) - 10
m
< 1= 125 - 10
m
< 1= 115
Right Scalene triangle
Right Scalene triangle
x + 70 = 3x + 10
70 = 2x + 10
60 = 2x
30 = x
3 (30) + 10 = 100˚
Make A Triangle
Construct triangle DEF.
Make A Triangle
Construct triangle DEF.
Make A Triangle
Construct triangle DEF.
Make A Triangle
Construct triangle DEF.
Make A Triangle
Construct triangle DEF.
Q:
What’s the problem with this?
A:
The shorter segments can’t reach each other to
complete the triangle. They don’t add up.
Make A Triangle
Construct triangle DEF.
The sum of the lengths of any two sides of a
triangle is greater than the length of
the third side.
T
r
i
a
n
g
l
e
I
n
e
q
u
a
l
i
t
y
C
o
n
j
e
c
t
u
r
e
Add the two smallest sides; they MUST be
larger than the third side
for the triangle to
be formed.
Make A Triangle
Can the following lengths form a triangle?
In a triangle, if one side is longer than another
side, then angle opposite the longer side
is larger than the other.
Side-Angle Conjecture
Side-Angle
What’s the biggest side?
What’s the biggest angle?
What’s the smallest side?
What’s the smallest angle?
b
B
a
A
Side-Angle
Rank the sides from greatest to least.
b
c
a
Rank the angles from greatest to least.
C
A
B
This content explains the concept of exterior angles in polygons and the Exterior Angle Theorem. It covers how exterior angles are formed when the sides of a polygon are extended, their relationship with interior angles, and how to calculate their measures using the Exterior Angle Theorem. Various examples are provided to demonstrate the application of the theorem in solving for unknown angle measures.
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Presentation Transcript
Angles When the sides of a polygon are extended, other angles are formed. The original angles are the interior angles. The angles that form linear pairs with the interior angles are the exterior angles.
EXTERIOR ANGLE THEOREM An exterior angle of a triangle is equal in measure to the sum measures of its two remote interior angles. sum of the remote remote interior interior angles angles Exterior angle Exterior angle
EXTERIOR ANGLE THEOREM (YOUR NEW BEST FRIEND) The measure of an exterior angle in a triangle is the sum sum of the measures of the 2 remote interior remote interior angles exterior exterior angle angle remote remote interior interior angles angles 2 2 1 1 3 3 4 4 m<1 + m<2 = m<4 m<1 + m<2 = m<4
EXTERIOR ANGLE THEOREM m 1 = m A + m B
EXAMPLES m<FHG = 180 111 = 69 linear pair m<G = 60 + 69
EXAMPLES Find x & y y = 30 + 82 y = 112 82 30 x y 180 = 30 + 82 + x 180 = 112 + x 68 = x x = 68 y = 112
EXAMPLES Find m JKM 2x 5 = x + 70 x 5 = 70 x = 75 m< JKM = 2(75) - 5 m< JKM = 150 - 5 m< JKM = 145
EXAMPLES Solve for y in the diagram. Solve on your own Solve on your own before viewing the before viewing the Solution
SOLUTION 4y + 35 = 56 + y 3y + 35 = 56 3y = 21 y= 27 y= 27
EXAMPLES Find the measure of in the diagram shown. 1 Solve on your own Solve on your own before viewing the before viewing the Solution
SOLUTION 40 + 3x = 5x - 10 40 = 2x - 10 50 = 2x 25 = x 25 = x m < 1= 5x - 10 m < 1= 5(25) - 10 m < 1= 125 - 10 m m < 1= 115 < 1= 115
SOLUTION x + 70 = 3x + 10 70 = 2x + 10 60 = 2x 30 = x Right Scalene triangle 3 (30) + 10 = 100
Make A Triangle Construct triangle DEF. D E D F F E
Make A Triangle Construct triangle DEF. D E D F F E
Make A Triangle Construct triangle DEF. D E
Make A Triangle Construct triangle DEF. D E
Make A Triangle Construct triangle DEF. D E
Make A Triangle Construct triangle DEF. 5 3 D E 13 Q: What s the problem with this? A: The shorter segments can t reach each other to complete the triangle. They don t add up.
Triangle Inequality Conjecture The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Add the two smallest sides; they MUST be larger than the third side larger than the third side for the triangle to be formed.
Make A Triangle Can the following lengths form a triangle? 2.2 ft 2 ft 9 ft 9 ft 13 ft 13 ft 1.4 mm 4 mm 5 mm 5 mm 10 mm 10 mm 3.5 cm 5 cm cm 4 cm 4 cm 4.7 7 ft ft 15 15 ft ft ft ft ?? cm ? 6.7 7 ft ft 7 7 ft ft ft ft 5.10 mm 10 mm 3 mm 3 mm 6 mm 6 mm 7.10 mm 10 mm 13 mm 13 mm mm mm ? 8.8 m 8 m 7 m 7 m 1 m 1 m ? 10. 12. 9. 11. 1 mm 1 mm 5 mm 5 mm 3 mm 3 mm 9 mm 9 mm 2 mm 2 mm 1 mm 1 mm 12 mm 12 mm 22 mm 22 mm mm mm ?? 7 mm 7 mm 8 mm 8 mm mm mm ??
Side-Angle Conjecture In a triangle, if one side is longer than another side, then angle opposite the longer side is larger than the other.
Side-Angle C b a A B c What s the biggest side? What s the biggest angle? b B a A What s the smallest side? What s the smallest angle?
Side-Angle Rank the sides from greatest to least. 46 b b c a a 42 92 c Rank the angles from greatest to least. A C A B 7 4 B C 5
