Download Angle and Side Length Relationships

Angle and Side Length Relationships
Complementary Angles
Angles in which the sum of their measures is 90o.
Supplementary Angles
Angles in which the sum of their measures is 180o.
Congruent Angles
Angles that have the same measure.
Example
Solve for x if angle A and angle B are complementary.
(8x – 15)o
(3x – 5)o
Since the two angles are complementary, the sum of their measures should equal
90o.
Steps
1. Set sum of both angles = 90
2. Simplify both sides
3. Add 20 to both sides
4. Divide both sides by 11
3x – 5 + 8x – 15 = 90
11x – 20 = 90
11x = 110
x = 10
Angle and Side Relationship in One Triangle
The largest angle opens to the longest side
The smallest angle opens to the shortest side
Example
B
45o
15
6
115o C
A
30o
11
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Example
Find the perimeter of ∆YXZ given ∆ABC
B
Z
12 m
10 m
C
~ ∆YXZ.
4m
Y
X
8m
A
Answer:
Side AC of the blue triangle corresponds to side YZ of the green triangle. Since it
is given that the triangles are similar and the length of AC is twice the length of
the YZ, we can conclude that the lengths of each of the sides of the blue triangle
is twice the length of its corresponding green sides. Since AB corresponds with
YX, we can conclude that YX has a length of 5 m (half of 10 m). Since BC
corresponds with XZ, we can conclude that XZ has a length of 6 m (half of 12 m).
To find the perimeter, we add the lengths of the three sides together (4 + 5 + 6).
So, the perimeter of ∆YXZ is 15 m.
Interior and Exterior Angles
interior
angle
exterior
angle
Below is a list of formulas that relate the interior and exterior angles to the
number of sides a polygon has.
Sum of Interior Angles
Sum of the measures of the interior angles of a convex polygon with
n sides = (n – 2)180o.
Sum of Exterior Angles
Sum of the measures of the exterior angles of a convex polygon with
n sides = 360o.
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Example
n: 5
Interior sum: (5 – 2)180 = 540o
Exterior sum: 360o
Interior Angle of Regular Polygon
Interior Angle Measure
a=
180(n − 2)
n
Example
Find the measure of the interior angle of
a regular hexagon.
Answer:
Since a hexagon has 6 sides, substitute 6 in for n in the formula
180(n − 2) 180(6 − 2) 180(4) 720
=
=
=
= 120 .
n
6
6
6
So, the measure of an interior angle of a regular hexagon is 120o.
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