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ISEE Upper Level Quantitative Reasoning › Lines

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Lines

Examine the above diagram. What is ?

Explanation

By angle addition,

$\left ( x - 13\right )+ 100 + (y + 25) = 180$

$x - 13\right+ 100 + y + 25 = 180$

$x + y - 13\right+ 100+ 25 = 180$

$\angle 1$ 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

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

$\angle 1$ 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

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

Lines

Examine the above diagram. What is ?

Explanation

By angle addition,

$\left ( x - 13\right )+ 100 + (y + 25) = 180$

$x - 13\right+ 100 + y + 25 = 180$

$x + y - 13\right+ 100+ 25 = 180$

Thingy

Note: Figure NOT drawn to scale.

In the above figure, $m \angle 1=\left ( x+17 \right )^{\circ}$ and $m \angle 2= \left (7x+11 \right )^{\circ}$ . Which of the following is equal to $m \angle 1$ ?

$36^{\circ}$

$26^{\circ}$

$7\frac{3}{4}^{\circ }$

$29\frac{5}{7}^{\circ}$

$21\frac{5}{7}^{\circ}$

Explanation

$\angle 1$ and $\angle 2$ form a linear pair, so their angle measures total $180^{\circ}$ . Set up and solve the following equation:

$m \angle 1+ m \angle 2 = 180^{\circ}$

$\left ( x+17 \right ) + \left (7x+11 \right ) = 180$

$m \angle 1=\left ( x+17 \right )^{\circ} = \left ( 19+17 \right )^{\circ} = 36^{\circ}$

Thingy

Note: Figure NOT drawn to scale.

In the above figure, $m \angle 1=\left ( x+17 \right )^{\circ}$ and $m \angle 2= \left (7x+11 \right )^{\circ}$ . Which of the following is equal to $m \angle 1$ ?

$36^{\circ}$

$26^{\circ}$

$7\frac{3}{4}^{\circ }$

$29\frac{5}{7}^{\circ}$

$21\frac{5}{7}^{\circ}$

Explanation

$\angle 1$ and $\angle 2$ form a linear pair, so their angle measures total $180^{\circ}$ . Set up and solve the following equation:

$m \angle 1+ m \angle 2 = 180^{\circ}$

$\left ( x+17 \right ) + \left (7x+11 \right ) = 180$

$m \angle 1=\left ( x+17 \right )^{\circ} = \left ( 19+17 \right )^{\circ} = 36^{\circ}$

Two angles which form a linear pair have measures $(2x+72)^{\circ}$ and $(5x-125)^{\circ}$ . Which is the lesser of the measures (or the common measure) of the two angles?

$41\frac{3}{7}^{\circ}$

$22\frac{6}{7}^{\circ}$

$81\frac{1}{2} ^{\circ }$

$65\frac{2}{3 } ^{\circ }$

$33\frac{2}{7}^{\circ }$

Explanation

Two angles that form a linear pair are supplementary - that is, they have measures that total $180^{\circ}$ . Therefore, we set and solve for in this equation:

$x = \frac{233}{7}= 33 \frac{2}{7}$

The two angles have measure

$2 \cdot 33 \frac{2}{7} +72= 138 \frac{4}{7} ^{\circ}$

and

$5 \cdot 33 \frac{2}{7} -125 = 41\frac{3}{7}^{\circ}$

$41\frac{3}{7}^{\circ}$ is the lesser of the two measures and is the correct choice.

Two angles which form a linear pair have measures $(2x+72)^{\circ}$ and $(5x-125)^{\circ}$ . Which is the lesser of the measures (or the common measure) of the two angles?

$41\frac{3}{7}^{\circ}$

$22\frac{6}{7}^{\circ}$

$81\frac{1}{2} ^{\circ }$

$65\frac{2}{3 } ^{\circ }$

$33\frac{2}{7}^{\circ }$

Explanation

Two angles that form a linear pair are supplementary - that is, they have measures that total $180^{\circ}$ . Therefore, we set and solve for in this equation:

$x = \frac{233}{7}= 33 \frac{2}{7}$

The two angles have measure

$2 \cdot 33 \frac{2}{7} +72= 138 \frac{4}{7} ^{\circ}$

and

$5 \cdot 33 \frac{2}{7} -125 = 41\frac{3}{7}^{\circ}$

$41\frac{3}{7}^{\circ}$ is the lesser of the two measures and is the correct choice.

A line intersects parallel lines and . $\angle 1$ and $\angle 2$ are corresponding angles; $\angle 1$ and $\angle 3$ are same side interior angles.

$m \angle 1 = \left (3x+2y \right )^{\circ }$

$m \angle 2 = \left ( 4x+21\right )^{\circ }$

$m \angle 3 = \left ( 2y-27\right )^{\circ}$

Evaluate .

Explanation

When a transversal such as crosses two parallel lines, two corresponding angles - angles in the same relative position to their respective lines - are congruent. Therefore,

Two same-side interior angles are supplementary - that is, their angle measures total 180 - so

We can solve this system by the substitution method as follows:

Backsolve:

, which is the correct response.

A line intersects parallel lines and . $\angle 1$ and $\angle 2$ are corresponding angles; $\angle 1$ and $\angle 3$ are same side interior angles.

$m \angle 1 = \left (3x+2y \right )^{\circ }$

$m \angle 2 = \left ( 4x+21\right )^{\circ }$

$m \angle 3 = \left ( 2y-27\right )^{\circ}$

Evaluate .

Explanation

When a transversal such as crosses two parallel lines, two corresponding angles - angles in the same relative position to their respective lines - are congruent. Therefore,

Two same-side interior angles are supplementary - that is, their angle measures total 180 - so

We can solve this system by the substitution method as follows:

Backsolve:

, which is the correct response.

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