Other Polygons

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Questions 1 - 10

In a certain quadrilateral, three of the angles are $65^{\circ}$ , $120^{\circ}$ , and $34^{\circ}$ . What is the measure of the fourth angle?

$141^{\circ}$

$139^{\circ}$

$219^{\circ}$

$41^{\circ}$

$29^{\circ}$

Explanation

A quadrilateral has four angles totalling $360^{\circ}$ . So, first add up the three angles given. The sum is $219^{\circ}$ . Then, subtract that from 360. This gives you the missing angle, which is $141^{\circ}$ .

Trapezoid

The above diagram depicts trapezoid . Which is the greater quantity?

(a) $m \angle T + m \angle P$

(b) $m \angle R + m \angle A$

(a) and (b) are equal.

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

Explanation

$\overline{TR} \parallel \overline{PA}$ ; $\angle T$ and $\angle P$ are same-side interior angles, as are $\angle R$ and $\angle A$ .

The Same-Side Interior Angles Theorem states that if two parallel lines are crossed by a transversal, then the sum of the measures of a pair of same-side interior angles is always $180^{\circ}$ .

Therefore, $m \angle T + m \angle P = 180= m \angle R + m \angle A$ , making the two quantities equal.

A regular octagon has perimeter one meter. Which is the greater quantity?

(A) The length of one side

(B) 125 millimeters

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(A) is greater

(B) is greater

Explanation

A regular octagon has eight sides of equal length. The perimeter of this octagon is one meter, which is equal to 1,000 millimeters; each side, therefore, has length

$1,000 \div 8 = 125$ millimeters

making the quantities equal.

A regular octagon has perimeter one meter. Which is the greater quantity?

(A) The length of one side

(B) 125 millimeters

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(A) is greater

(B) is greater

Explanation

A regular octagon has eight sides of equal length. The perimeter of this octagon is one meter, which is equal to 1,000 millimeters; each side, therefore, has length

$1,000 \div 8 = 125$ millimeters

making the quantities equal.

Rectangle 1

The above rectangle, which has perimeter 360, is divided into squares of equal size. Give the area of the shaded portion.

Explanation

The sides of the rectangle, in total, are divided into 18 segments of equal measure, as indicated below:

Rectangle 2

The rectangle has a total perimeter of 360, so each segment - one side of a square - measures $36 0 \div 18 = 20$ . Each square has area $20 ^{2} = 400$ , so the shaded portion of the rectangle, which comprises seven squares, has area

$400 \times 7 = 2,800$ .

Which is the greater quantity?

(a) The sum of the measures of the exterior angles of a thirty-sided polygon, one per vertex

(b) The sum of the measures of the exterior angles of a forty-sided polygon, one per vertex

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

(b) is greater

Explanation

The Polygon Exterior-Angle Theorem states that the sum of the measures of the exterior angles of any polygon, one per vertex, is $360^{\circ }$ . This makes both quantities equal.

A regular pentagon has sidelength 72; the perimeter of a regular hexagon is 80% of that of the pentagon. Which is the greater quantity?

(A) The length of one side of the hexagon

(B) 50

(B) is greater

(A) is greater

It is impossible to determine which is greater from the information given

(A) and (B) are equal

Explanation

A regular pentagon has five sides of equal length; since one side is 72 units long, its perimeter is

$72 \times 5 = 360$ .

80% of this is

$0.8 \times 360 = 288$ ,

so this is the length of the hexagon, and, since all six sides are of equal length, one side measures

$288 \div 6 = 48$

(B) is greater.

Given Trapezoid , where $\overline{TR} \parallel \overline{PA}$ . Also, $m \angle A > m\angle P$

Which is the greater quantity?

(a) $m\angle T$

(b) $m \angle R$

(a) is greater

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Explanation

$\angle T$ and $\angle P$ are same-side interior angles, as are $\angle R$ and $\angle A$ .

The Same-Side Interior Angles Theorem states that if two parallel lines are crossed by a transversal, then the sum of the measures of a pair of same-side interior angles is always $180^{\circ}$ . Therefore,

$m \angle T + m \angle P = 180$ , or $m \angle P = 180 - m \angle T$

$m \angle R + m \angle A = 180$ , or $m \angle A = 180 - m \angle R$

Substitute:

$m \angle A > m\angle P$

$180 - m \angle R > 180 - m\angle T$

$180 - m \angle R -180 > 180 - m\angle T-180$

$- m \angle R > - m\angle T$

$- (- m \angle R )<-\left ( - m\angle T \right )$

$m \angle R < m\angle T$

(a) is the greater quantity

Which is the greater quantity?

(a) The sum of the measures of the exterior angles of a thirty-sided polygon, one per vertex

(b) The sum of the measures of the exterior angles of a forty-sided polygon, one per vertex

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

(b) is greater

Explanation

The Polygon Exterior-Angle Theorem states that the sum of the measures of the exterior angles of any polygon, one per vertex, is $360^{\circ }$ . This makes both quantities equal.

Trapezoid