ISEE Upper Level Quantitative Reasoning - Lines

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.

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.

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 $141^{\circ}$ .

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

(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.

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

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

Which of the following expressions is equal to ?

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 $\Delta ABC$ 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

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

$m\angle A > m\angle C$

Right_triangle

Note: Figure NOT drawn to scale.

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

(a)

(b)

(a) is greater.

(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.

$\angle 1$ and $\angle 2$ are complementary; $m \angle 1 =2 m \angle 2 - 50$ .

Which is the greater quantity?

(A) $m \angle 1$

(B) $m \angle 2$

(B) is greater

(A) is greater

(A) and (B) are equal

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

Explanation

Two angles are complementary if their degree measures total 90. Therefore,

$m \angle 1 + m \angle 2 = 90$

Since $m \angle 1 =2 m \angle 2 - 50$ , we can substitute, and we can solve for $m\angle2$ :

$(2 m \angle 2 - 50)+ m \angle 2 = 90$

$3 m \angle 2 - 50 = 90$

$3 m \angle 2 - 50 + 50 = 90 + 50$

$3 m \angle 2 = 140$

$3 m \angle 2 \div 3 = 140 \div 3$

$m \angle 2 = 46 \frac{2}{3}$

$m \angle 1 = 90 - m \angle 2 = 90 - 46 \frac{2}{3} = 43 \frac{1}{3}$

$m \angle 2 > m \angle 1$ , making (B) the greater quantity.

$\angle 1$ and $\angle 2$ are supplementary; $\angle 1$ and $\angle 3$ are complementary.

$m \angle 2 = 100 ^{\circ }$ .

What is $m \angle 3$ ?

$10^{\circ }$

$80^{\circ }$

$100^{\circ }$

$50^{\circ }$

Explanation

Supplementary angles and complementary angles have measures totaling $180^{\circ }$ and $90^{\circ }$ , respectively.

$m \angle 2 = 100 ^{\circ }$ , so its supplement $\angle 1$ has measure

$m \angle 1 = 180^{\circ } - m \angle 2 = 180^{\circ } - 100 ^{\circ } = 80^{\circ }$

$\angle 3$ , the complement of $\angle 1$ , has measure

$m \angle 3 = 90^{\circ } - m \angle 1 = 90^{\circ } - 80 ^{\circ } = 10^{\circ }$

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