How to find an angle

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ISEE Upper Level Quantitative Reasoning › How to find an angle

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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

CORRECT

(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

CORRECT

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.

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

$141^{\circ}$

CORRECT

$139^{\circ}$

$219^{\circ}$

$41^{\circ}$

$29^{\circ}$

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.

CORRECT

(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 hexagon has six angles with measures $(x-5)^{\circ}, x^{\circ}, (x+5)^{\circ}, (y-10)^{\circ}, y^{\circ}, (y + 10)^{\circ}.$

Which quantity is greater?

(a)

(b) 240

(a) and (b) are equal

CORRECT

(a) is greater

(b) is greater

It is impossible to tell from the information given

Explanation

The angles of a hexagon measure a total of . From the information, we know that:

$3 (x + y) \div 3 = 720 \div 3$

The quantities are equal.

Let the three angles of a triangle measure , , and .

Which of the following expressions is equal to ?

CORRECT

Explanation

The sum of the measures of the angles of a triangle is $180^{\circ }$ , so simplify and solve for in the equation:

Right_triangle

Note: Figure NOT drawn to scale.

Refer to the above figure. Which is the greater quantity?

(a)

(b)

(a) is greater.

CORRECT

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Explanation

(a) The measures of the angles of a linear pair total 180, so:

(b) The Triangle Exterior-Angle Theorem states that the measure of an exterior angle is equal to the sum of its remote interior angles. Therefore, .

Therefore (a) is the greater quantity.

Right triangle $\Delta ABC$ has right angle $\angle B$ .

$m\angle A = \left ( x + 30 \right )^ {\circ} ,m\angle C = \left ( 2x- 57\right ) ^ {\circ}$

Which is the greater quantity?

(a) $m\angle A$

(b) $m\angle C$

(a) is greater

CORRECT

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Explanation

The degree measures of the acute angles of a right triangle total 90, so we solve for in the following equation:

$\left ( x + 30 \right )+ \left ( 2x- 57\right ) = 90$

$3x \div 3 = 117 \div 3$

(a) $m\angle A = \left ( x + 30 \right )^ {\circ} = \left ( 39 + 30 \right )^ {\circ} = 69^ {\circ}$

(b) $m\angle C =\left ( 90 - 69 \right ) ^ {\circ} = 21^ {\circ}$

$m\angle A > m\angle C$

A regular polygon has interior angles that are obtuse. Which is the greater quantity?

(a) The number of sides of the polygon

(b) 4

(a) is greater

CORRECT

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Explanation

A regular four-sided polygon - a square - has four right angles; a regular triangle is equiangular and has three acute ( $60 ^{\circ }$ ) angles.

Any regular polygon with five sides or more has congruent angles that measure at least $\frac{180 \left ( 5-2 \right )}{5} = 108 ^{\circ }$ each. Therefore, any regular polygon with obtuse angles must have 5 or more sides, making (a) greater.

$\bigtriangleup ABC$ is inscribed in a circle. $\angle B$ is a right angle, and $m \overarc{ACB} = 270 ^{\circ }$ .

Which is the greater quantity?

(a) $m \angle A$

(b) $m \angle C$

(a) and (b) are equal

CORRECT

(a) is the greater quantity

(b) is the greater quantity

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

Explanation

The figure referenced is below:

Inscribed angle

$\overarc{ACB}$ has measure $270 ^{\circ }$ , so its corresponding minor arc, $\overarc{A B}$ , has measure $360 ^{\circ } - 270 ^{\circ } = 90^{\circ }$ . The inscribed angle that intercepts this arc, which is $\angle C$ , has measure half this, or $45 ^{\circ }$ . Since $\angle B$ is a right angle, the other acute angle, $\angle A$ , has measure