Download Solving for Interior Angles of Triangles with Equations (Day 2)

Solving for Interior & Exterior Angles
of Triangles and Polygons with
Equations
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.

Theorem 4.1 Triangle Sum Theorem:
The sum of the measures of the interior angles of a
triangle is 180o.

Theorem 4.2 Exterior Angle Theorem:
The measure of an exterior angle of a triangle is equal
to the sum of the measures of the two nonadjacent
interior angles.
Solve for the variable using an Equation :
4x 19 41 180 4x  60  180
Demonstrate your steps mathematically :
19˚
4x˚
41˚
Solve for the missing measure:
4  30   120
4x  60  180
4x  60  60  180  60
4x  120
4 x 120

4
4
x  30
Check:
4  30   60  180
120  60  180
180  180
Solve for the variable using an Equation :
Solve for the missing measure:
6 8  48
6x 91 41 180 6x  132  180
Demonstrate your steps mathematically :
6x˚
91˚
41˚
6x  132  180
6x 132 132  180 132
6x  48
6 x 48

6
6
x 8
Check:
6 8  132  180
48  132  180
180  180
Solve for the variable using an Equation :
2x 2x 40 180 4x  40  180
40˚
2x˚
Demonstrate your steps mathematically :
2x˚
4x  40  180
4x  40  40  180  40
4x  140
4 x 140

4
4
x  35
Check:
Solve for the missing measure:
2  35  70
4  35  40  180
140  40  180
180  180
Ex.4: Find
mJKM.
Ex.5: Find the measure of 1 in the diagram shown.
Find Angle Measures in Polygons
Diagonal: Connects two nonconsecutive vertices
A
E
B
Divides the shape into
triangles.
C
How many triangles in the pentagon?
D
• How many triangles in a hexagon,
quadrilateral?
• 4, 2
• What is the pattern?
• The number of triangles equals the # of
sides minus 2
triangles = (n – 2)
• How many triangles in a 15-gon?
• 13
•
•
•
•
•
•
•
So how many degrees in a pentagon?
Number of triangles?
3
Number of degree in each triangle?
180
Total number of degrees in the pentagon?
540 = 3 * 180
• Degrees = (n – 2) * 180, where n = # of sides
Find the measure of the interior angles of
the indicated convex polygon
•
•
•
•
•
•
•
octagon
octagon = 8
(n-2) *180 = degrees
n=8
(8-2) * 180
6 * 180
1080 = degrees
Find the measure of the interior angles of
the indicated convex polygon
•
•
•
•
•
•
13-gon
(n-2) * 180 = degrees
n=13
(13-2) * 180
11* 180
1980 = degrees
• How can we find the number of sides a
shape has based on the sum of the interior
angles?
• D = (n-2) * 180
• n = (D/180) + 2
• So how many sides is the figure with 1620
degrees?
• n = (1620/180) + 2
• n=9+2
• n = 11
• The sum of the measures of the interior
angles of a convex polygon is 1440.
Classify the polygon by the number of
sides.
• n = (Interior Angles/180) + 2
• n = (1440/180) + 2
• n = (8) +2
• n=10
Exterior Angles
• The sum of exterior
angles is always equal
to 360˚
1
A
5
2
E
B
Interior and Exterior
angles always add to
180
C
D
3
4
Why Exterior Angles equal 360
• What are the measure
of the interior angles?
180*4 = 720
720 / 6 = 120
• What are the exterior
angles?
• 180 – 120 = 60
• 60 * 6 = 360
• Regular Hexagon
Find an unkown interior angle measure
ALGEBRA
Find the value of x in the diagram shown.
SOLUTION
The polygon is a quadrilateral. Use the
Corollary to the Polygon Interior Angles
Theorem to write an equation involving
x. Then solve the equation.
x° + 108° + 121° + 59° =
360°
x + 288 =
360
x = 72
ANSWER
Combine like terms.
Subtract 288 from each side.
The value of x is 72.
Example
3.
Use the diagram at the right. Find m
and m T.
ANSWER
4.
S
103°, 103°
The measures of three of the interior angles of a quadrilateral are
89°, 110°, and 46°. Find the measure of the fourth interior angle.
ANSWER
115°
Standardized Test Practice
SOLUTION
Use the Polygon Exterior Angles Theorem to write and
solve an equation.
x° + 2x° + 89° + 67° =
3x + 156 =
360°
Polygon Exterior Angles Theorem
360
Combine like terms.
x = 68
ANSWER
Solve for x.
The correct answer is B.
What is the value of x in the
diagram shown?
a = 136°, b = 35°, c = 126°
136 + 35 + 126 + x = 360
297 + x = 360
x = 360 – 297
x = 63
Example
5.
A convex hexagon has exterior angles with measures 34°, 49°, 58°,
67°, and 75°. What is the measure of an exterior angle at the sixth
vertex?
ANSWER
77°