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ISEE Upper Level Math : How to find an angle

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Example Questions

Example Question #1 : Lines

Lines

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

Possible Answers:

75 + x

131 -x

165-x

165 +x

75-x

Correct answer:

165-x

Explanation:

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:

(y-17) + (x + 32) = 180

y + x-17 + 32= 180

y + x+15= 180

y + x+15- 15 - x = 180- 15 - x

y = 165- x

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Example Question #2 : Lines

Lines

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

Possible Answers:

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

$47 - \frac{1}{2}x$

92 - x

$184 - \frac{1}{2}x$

47 - x

Correct answer:

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

Explanation:

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 + 15) + (x - 19) = 180

2y + x+ 15 - 19 = 180

2y + x-4 = 180

2y + x-4+4 -x = 180+4 -x

2y = 184 -x

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

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

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Example Question #33 : Plane Geometry

Lines

Examine the above diagram. What is x + y ?

Possible Answers:

112

Correct answer:

Explanation:

By angle addition,

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

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

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

x + y +112 = 180

x + y +112 - 112= 180- 112

x + y = 68

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Example Question #1 : Lines

Lines

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

Possible Answers:

$\angle 2 \cong \angle 6$

$\angle 5 \cong \angle 4$

$\angle 5 \cong \angle 8$

$m \angle 4 + m \angle 7 = 180 ^{\circ }$

$m \angle 1 + m \angle 6 = 180 ^{\circ }$

Correct answer:

$\angle 5 \cong \angle 8$

Explanation:

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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Example Question #2 : Lines

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

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

What is $m \angle 3$ ?

Possible Answers:

$100^{\circ }$

$10^{\circ }$

$80^{\circ }$

$50^{\circ }$

Correct answer:

$10^{\circ }$

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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Example Question #1 : Lines

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 $m \angle 1$ ?

Possible Answers:

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

$36^{\circ}$

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

$26^{\circ}$

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

Correct answer:

$36^{\circ}$

Explanation:

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

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

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

8x+28 = 180

8x=152

x = 19

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

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Example Question #31 : Plane Geometry

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?

Possible Answers:

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

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

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

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

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

Correct answer:

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

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:

(2x+72)+(5x-125)= 180

7x - 53 = 180

7x = 233

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

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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Example Question #6 : How To Find An Angle

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?

Possible Answers:

$15\frac{1}{2}^{\circ }$

$22\frac{7}{8}^{\circ}$

$41\frac{1}{8}^{\circ}$

$68\frac{5}{8}^{\circ}$

$60\frac{1}{2} ^{\circ }$

Correct answer:

$60\frac{1}{2} ^{\circ }$

Explanation:

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

3x+14= 5x-17

14= 2x-17

2x= 31

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

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

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Example Question #5 : Lines

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 x+y .

Possible Answers:

x+y = 60

x+y = 120

x+y = 90

x+y = 45

x+y = 30

Correct answer:

x+y = 60

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,

3x+2y = 4x+21

3x+2y- 3x - 21 = 4x+21 - 3x - 21

x = 2y - 21

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

3x+2y + 2y-27 = 180

3x+4y-27 = 180

3x+4y= 207

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

3( 2y - 21)+4y= 207

6y-63+4y= 207

10y-63= 207

10y = 270

y = 27

Backsolve:

x = 2y - 21

x = 2 (27)- 21 = 54-21 = 33

x+y = 27+33 = 60 , which is the correct response.

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Example Question #1 : How To Find An Angle

Vertical_angles

Note: Figure NOT drawn to scale.

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

Possible Answers:

$120 ^{\circ }$

$144^{\circ }$

$88^{\circ }$

$96^{\circ }$

$128^{\circ }$

Correct answer:

$120 ^{\circ }$

Explanation:

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

2x+y = 2y ,

or, simplified,

2x+y - y= 2y - y

y = 2x

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

2y+ 8x = 180

Substitute for :

2(2x)+ 8x = 180

4x+8x = 180

12x= 180

x = 15

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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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : How to find an angle

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : Lines

Example Question #2 : Lines

Example Question #33 : Plane Geometry

Example Question #1 : Lines

Example Question #2 : Lines

Example Question #1 : Lines

Example Question #31 : Plane Geometry

Example Question #6 : How To Find An Angle

Example Question #5 : Lines

Example Question #1 : How To Find An Angle

All ISEE Upper Level Math Resources

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