Polygons

High School Math 1 · Unit 14 · polygons-angle-theorems · Teacher edition

polygons-angle-theorems

Name ______________________________Date ______________

1.
What is the sum of the interior angles of a hexagon, in degrees?
Answer: ______________
2.
What is the measure of each interior angle of a regular $12$ -gon, in degrees?
Answer: ______________
3.
Each interior angle of a regular polygon measures $156^\circ$ . How many sides does the polygon have?
Answer: ______________
4.
Which statement about quadrilaterals is true?
1. (A)
  Every square is a rhombus
2. (B)
  Every rectangle is a rhombus
3. (C)
  No rhombus is a rectangle
4. (D)
  Every parallelogram is a rectangle
5.
The interior angles of a convex hexagon, in degrees, form an arithmetic sequence with common difference $10$ . What is the measure of the largest angle, in degrees?
1. (A)
  $115$
2. (B)
  $125$
3. (C)
  $135$
4. (D)
  $145$
5. (E)
  $155$

Answer key — Polygons

1.
720polygons-angle-theorems-01
$S = (6 - 2) \cdot 180^\circ = 4 \cdot 180^\circ = \mathbf{720}$ degrees.
2.
150polygons-angle-theorems-02
$\dfrac{(12 - 2) \cdot 180^\circ}{12} = \dfrac{1800^\circ}{12} = \mathbf{150}$ degrees. Check: exterior angle $360 \div 12 = 30$ , and $180 - 30 = 150$ . ✓
3.
15polygons-angle-theorems-03
Exterior angle $= 180 - 156 = 24^\circ$ , so $n = \frac{360}{24} = \mathbf{15}$ sides. Check: $\frac{(15-2) \cdot 180}{15} = \frac{2340}{15} = 156$ . ✓
4.
(A)
Every square is a rhombus
polygons-angle-theorems-04
A square has four equal sides, so it satisfies the definition of a rhombus: every square is a rhombus (and every square is a rectangle too). (A $2 \times 5$ rectangle has unequal sides, so not every rectangle is a rhombus; a square is a rhombus that is also a rectangle, so “no rhombus is a rectangle” fails; a tilted parallelogram has no right angles.)
5.
(D)
$145$
polygons-angle-theorems-05
$6a + 150 = 720 \Rightarrow a = 95$ . The angles are $95, 105, 115, 125, 135, 145$ , so the largest is $\mathbf{145}$ degrees. Check: $95 + 145 = 240$ , and three such pairs give $720$ . ✓ ( $155$ uses six jumps instead of five; $135$ is the fifth angle.)