How to find an angle - ISEE Upper Level Mathematics Achievement

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Question

Examine the above diagram. If $l \parallel m$ , give in terms of .

Answer

The two marked angles are same-side exterior angles of parallel lines, which are supplementary - that is, their measures have sum 180. We can solve for in this equation:

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Question

Examine the above diagram. If $l \parallel m$ , give in terms of .

Answer

The two marked angles are same-side interior angles of parallel lines, which are supplementary - that is, their measures have sum 180. We can solve for in this equation:

$2y \div 2 = \left ( 184 -x \right )\div 2$

$y= 92 - \frac{1}{2}x$

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Question

Examine the above diagram. Which of the following statements must be true whether or not and are parallel?

Answer

Four statements can be eliminated by the various parallel theorems and postulates. Congruence of alternate interior angles or corresponding angles forces the lines to be parallel, so

$\angle 5 \cong \angle 4 \Rightarrow l \parallel m$ and

$\angle 2 \cong \angle 6 \Rightarrow l \parallel m$ .

Also, if same-side interior angles or same-side exterior angles are supplementary, the lines are parallel, so

$m \angle 4 + m \angle 7 = 180 ^{\circ } \Rightarrow l \parallel m$ and

$m \angle 1 + m \angle 6 = 180 ^{\circ } \Rightarrow l \parallel m$ .

However, $\angle 5 \cong \angle 8$ whether or not $l \parallel m$ since they are vertical angles, which are always congruent.

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Question

Examine the above diagram. What is ?

Answer

By angle addition,

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

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

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

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Question

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

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

What is $m \angle 3$ ?

Answer

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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Question

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 $m \angle 1$ ?

Answer

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

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

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

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

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Question

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?

Answer

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.

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Question

Two vertical angles have measures $(3x+14)^{\circ }$ and $\left ( 5x-17\right )^{\circ }$ . Which is the lesser of the measures (or the common measure) of the two angles?

Answer

Two vertical angles - angles which share a vertex and whose union is a pair of lines - have the same measure. Therefore, we set up and solve the equation

$x= 15\frac{1}{2}$

$3x+14 = 3 \cdot 15\frac{1}{2} + 14 = 46\frac{1}{2} + 14= 60\frac{1}{2} ^{\circ }$

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Question

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 .

Answer

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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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the measure of $\angle 1$ .

Answer

The top and bottom angles, being vertical angles - angles which share a vertex and whose union is a pair of lines - have the same measure, so

,

or, simplified,

The right and bottom angles form a linear pair, so their degree measures total 180. That is,

Substitute for :

The left and right angles, being vertical angles, have the same measure, so, since the right angle measures $\left (8x \right )^{\circ } =\left (8 \cdot 15 \right )^{\circ } = 120 ^{\circ }$ , this is also the measure of the left angle, $\angle 1$ .

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Question

Figure NOT drawn to scale

The above figure shows Trapezoid , with $\overline{RT}$ and $\overline{RA }$ tangent to the circle. $m \angle T = 106 ^{\circ }$ ; evaluate $m \angle A$ .

Answer

By the Same-Side Interior Angle Theorem, since $\overline{TR} || \overline{AP}$ , $\angle T$ and $\angle P$ are supplementary - that is, their degree measures total $180 ^{\circ }$ . Therefore,

$m \angle P + m \angle T = 180 ^{\circ }$

$m \angle P = 180 ^{\circ } - m \angle T$

$m \angle P = 180 ^{\circ } - 106 ^{\circ } = 74^{\circ }$

$\angle P$ is an inscribed angle, so the arc it intercepts, $\overarc{TA}$ , has twice its degree measure;

$m \overarc{TA} = 2 \cdot m \angle P =2 \cdot 74^{\circ } = 148 ^{\circ }$ .

The corresponding major arc, $\overarc{TPA}$ , has as its measure

$m \overarc{TPA} = 360 ^{\circ }- m \overarc{TA} = 360 ^{\circ }- 148 ^{\circ }= 212 ^{\circ }$

The measure of an angle formed by two tangents to a circle is equal to half the difference of those of its intercepted arcs:

$m \angle R = \frac{1}{2} \left (m \overarc{TPA} - m \overarc{TA} \right )$

$m \angle R = \frac{1}{2} \left (212 ^{\circ } -148 ^{\circ } \right )= \frac{1}{2} \cdot 64 ^{\circ } = 32^{\circ }$

Again, by the Same-Side Interior Angles Theorem, $\angle R$ and $\angle A$ are supplementary, so

$m \angle A + m \angle R= 180 ^{\circ }$

$m \angle A = 180 ^{\circ } - m \angle R$

$m \angle A = 180 ^{\circ } - 32 ^{\circ } = 148^{\circ }$

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