SAT / ACT GEOMETRY
Polygons
Introduction
A polygon is a closed plane figure composed of line segments joined
together at points called vertices. A diagonal of a polygon is a line
segment joining two non-adjacent vertices.
Polygons are named according to the number of sides or angles.
A regular polygon is a polygon whose sides are equal and whose angles
are equal.
If two polygons have equal corresponding angles and equal
corresponding sides, they are said to be congruent. The symbol for
congruence is ≅.
Two polygons with equal corresponding angles and corresponding sides
in proportion are said to be similar. The symbol for similar is ~.
In similar polygons the perimeters have the same ratio as any pair of
corresponding sides.
𝑃
𝐴𝐵
𝐵𝐶
𝐸𝐴
= ′ =
=⋯=
𝑃′
𝐴 𝐵′
𝐵′𝐶′
𝐸′𝐴′
Polygons
Interior Angles Theorem
The sum of the measures of the interior angles
of any n-sided polygon is 180(n – 2) degrees.
Example 1 Find the measure of angle B.
Solution
The sum of the measures of the interior angles of a quadrilateral is 360°:
n = 4, S = 180°(n – 2) = 180°(4 – 2) = 360°
Therefore,
m (∠B) = 360° – (130° + 60° + 100°) = 360° – 290° = 70°.
Example 2 Find the measure of y.
Solution
x = 180° – 108° = 72°,
y = 360° – (72° + 125° + 55°) = 360° – 252° = 108°.
Example 3 Find the sum of the interior angles of a polygon with 12
sides.
Solution
n = 12, S = 180°(n – 2) = 180°(12 – 2) = 1800°.
Example 4 How many sides does a polygon have if the sum of its
interior angles is 1980°?
Solution
S = 180°(n – 2), S = 1980° ⇒ 180°(n – 2) = 1980°
⇒ n – 2 = 11
⇒ n = 13.
Example 5 Find the measure of each interior angle of a regular polygon
with 10 sides.
Solution
n = 8,
S = 180°(n – 2) = 180°(8 – 2) = 1080°,
𝜃=
𝑆
𝑛
=
1080°
8
= 135°.
Example 6 How many sides does a polygon have if each of its interior
angles measures 120°?
Solution
𝜃 = 120°, S = 𝜃 × n = 120°n, S = 180°(n – 2) ⇒
120n = 180(n – 2), 2n = 3(n – 2),
2n = 3n – 6, n = 6.
Example 7 The sum of the measures of the interior angles of a convex
polygon is 720°. What is the sum of the measures of the interior angles
of a second convex polygon that has two more sides than the first?
Solution
The sum of the interior angles of an n-sided polygon is
(n – 2)180° = 720°.
Divide by 180°:
n – 2 = 4, n = 6.
The second polygon has 6 + 2 = 8 sides. Sum of the measures of its
interior angles of the second convex polygon equals
180° (8 – 2) = 180° (6) = 1080°.
Polygons
Exterior Angles Theorem
The sum of the measures of the exterior
angles of any n-sided polygon is 360°.
Example 1 Find the measure of the unknown exterior angle.
Solution
y = 360° – (111° + 106°) = 360° – 217° = 143°.
Example 2 Find the measure of d.
Solution
d= 360° – (62° + 84° + 135°) = 360° – 281° = 79°.
Example 3 How many sides does a regular polygon have if each interior
angle equals 176°?
Solution
Each exterior angle = 180°– 176° = 4°
Since the sum of the exterior angles is 360°, the number of exterior
angles equals 360 ÷ 4 = 90. The polygon has 90 sides.