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High School Math : Angles

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Example Questions

Example Question #1 : Angles

Solve for $\displaystyle x$ and $\displaystyle y$.

(Figure not drawn to scale).

Possible Answers:

$\small x=15^o;\ y=7.5^o$ $\displaystyle \small x=15^o;\ y=7.5^o$

$\small x=7.5^o;\ y=18.75^o$ $\displaystyle \small x=7.5^o;\ y=18.75^o$

$\small x=15^o;\ y=37.5^o$ $\displaystyle \small x=15^o;\ y=37.5^o$

$\small x=15^o;\ y=52.5^o$ $\displaystyle \small x=15^o;\ y=52.5^o$

Correct answer:

$\small x=15^o;\ y=37.5^o$ $\displaystyle \small x=15^o;\ y=37.5^o$

Explanation:

The angles containing the variable $\displaystyle x$ all reside along one line, therefore, their sum must be 180^o $\displaystyle 180^o$.

$\small 5x+4x+3x=180^o$ $\displaystyle \small 5x+4x+3x=180^o$

$\small 12x=180^o$ $\displaystyle \small 12x=180^o$

$\small x=15^o$ $\displaystyle \small x=15^o$

Because $\displaystyle 2y$ and $\displaystyle 5x$ are opposite angles, they must be equal.

$\small 2y=5x$ $\displaystyle \small 2y=5x$

$\small x=15^o$ $\displaystyle \small x=15^o$

$\small 2y=5(15^o)=75^o$ $\displaystyle \small 2y=5(15^o)=75^o$

$\small y=\frac{75^o}{2}=37.5^o$ $\displaystyle \small y=\frac{75^o}{2}=37.5^o$

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Example Question #1 : Understanding Complementary And Suplmentary Angles

Are $73^{\circ}$ $\displaystyle 73^{\circ}$ and $17^{\circ}$ $\displaystyle 17^{\circ}$ complementary angles?

Possible Answers:

Yes

Maybe

Not enough information

Correct answer:

Yes

Explanation:

Complementary angles add up to $90^{\circ}$ $\displaystyle 90^{\circ}$. Therefore, these angles are complementary.

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Example Question #2 : Angles

What angle is complementary to $34^\circ$ $\displaystyle 34^\circ$?

Possible Answers:

$56^\circ$ $\displaystyle 56^\circ$

$156^\circ$ $\displaystyle 156^\circ$

$34^\circ$ $\displaystyle 34^\circ$

$66^\circ$ $\displaystyle 66^\circ$

$146^\circ$ $\displaystyle 146^\circ$

Correct answer:

$56^\circ$ $\displaystyle 56^\circ$

Explanation:

Two complementary angles add up to $90^\circ$ $\displaystyle 90^\circ$.

Therefore, $34^\circ+x^\circ=90^\circ$ $\displaystyle 34^\circ+x^\circ=90^\circ$.

$x^\circ=90^\circ-34^\circ$ $\displaystyle x^\circ=90^\circ-34^\circ$

$x^\circ=56^\circ$ $\displaystyle x^\circ=56^\circ$

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Example Question #3 : Angles

Which of the following angles is supplementary to $15^\circ$ $\displaystyle 15^\circ$?

Possible Answers:

$12^\circ$ $\displaystyle 12^\circ$

$75^\circ$ $\displaystyle 75^\circ$

$85^\circ$ $\displaystyle 85^\circ$

$175^\circ$ $\displaystyle 175^\circ$

$165^\circ$ $\displaystyle 165^\circ$

Correct answer:

$165^\circ$ $\displaystyle 165^\circ$

Explanation:

When two angles are supplementary, they add up to $180^\circ$ $\displaystyle 180^\circ$.

For this problem, we can set up an equation and solve for the supplementary angle:

$x^\circ+15^\circ=180^\circ$ $\displaystyle x^\circ+15^\circ=180^\circ$

$x^\circ=180^\circ-15^\circ$ $\displaystyle x^\circ=180^\circ-15^\circ$

$x^\circ=165^\circ$ $\displaystyle x^\circ=165^\circ$

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Example Question #4 : Angles

What angle is supplementary to $131^\circ$ $\displaystyle 131^\circ$?

Possible Answers:

$41^\circ$ $\displaystyle 41^\circ$

$229^\circ$ $\displaystyle 229^\circ$

$49^\circ$ $\displaystyle 49^\circ$

$140^\circ$ $\displaystyle 140^\circ$

$-41^\circ$ $\displaystyle -41^\circ$

Correct answer:

$49^\circ$ $\displaystyle 49^\circ$

Explanation:

Supplementary angles add up to $180^\circ$ $\displaystyle 180^\circ$. That means:

$x+131^\circ=180^\circ$ $\displaystyle x+131^\circ=180^\circ$

$x=180^\circ-131^\circ$ $\displaystyle x=180^\circ-131^\circ$

$x=49^\circ$ $\displaystyle x=49^\circ$

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Example Question #1 : Graphing Functions

Solve for $\displaystyle n$.

(Figure not drawn to scale).

Possible Answers:

8^o $\displaystyle 8^o$

12^o $\displaystyle 12^o$

14^o $\displaystyle 14^o$

10^o $\displaystyle 10^o$

Correct answer:

14^o $\displaystyle 14^o$

Explanation:

The angles are supplementary, therefore, the sum of the angles must equal 180^o $\displaystyle 180^o$.

$\small (4n+22^o)+(8n-10^o)=180^o$ $\displaystyle \small (4n+22^o)+(8n-10^o)=180^o$

$\small 4n+22^o+8n-10^o=180^o$ $\displaystyle \small 4n+22^o+8n-10^o=180^o$

$\small 12n+12^o=180^o$ $\displaystyle \small 12n+12^o=180^o$

$\small 12n=168^o$ $\displaystyle \small 12n=168^o$

$\small n=14^o$ $\displaystyle \small n=14^o$

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Example Question #2 : Angles

Are $129^{\circ}$ $\displaystyle 129^{\circ}$ and $51^{\circ}$ $\displaystyle 51^{\circ}$ supplementary angles?

Possible Answers:

Yes

Not enough information

Correct answer:

Yes

Explanation:

Since supplementary angles must add up to $180^{\circ}$ $\displaystyle 180^{\circ}$, the given angles are indeed supplementary.

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Example Question #1 : Understanding Complementary And Suplmentary Angles

Which of the following angles is complementary to $32^\circ$ $\displaystyle 32^\circ$?

Possible Answers:

$148^\circ$ $\displaystyle 148^\circ$

$58^\circ$ $\displaystyle 58^\circ$

$158^\circ$ $\displaystyle 158^\circ$

$48^\circ$ $\displaystyle 48^\circ$

$68^\circ$ $\displaystyle 68^\circ$

Correct answer:

$58^\circ$ $\displaystyle 58^\circ$

Explanation:

Two complementary angles add up to $90^\circ$ $\displaystyle 90^\circ$.

$90^\circ=32^\circ+x^\circ$ $\displaystyle 90^\circ=32^\circ+x^\circ$

$90^\circ-32^\circ=x^\circ$ $\displaystyle 90^\circ-32^\circ=x^\circ$

$58^\circ=x^\circ$ $\displaystyle 58^\circ=x^\circ$

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Example Question #1 : Angles

What angle is supplementary to $142^\circ$ $\displaystyle 142^\circ$?

Possible Answers:

$218^\circ$ $\displaystyle 218^\circ$

$38^\circ$ $\displaystyle 38^\circ$

$48^\circ$ $\displaystyle 48^\circ$

$-52^\circ$ $\displaystyle -52^\circ$

$308^\circ$ $\displaystyle 308^\circ$

Correct answer:

$38^\circ$ $\displaystyle 38^\circ$

Explanation:

When two angles are supplementary, they add up to $180^\circ$ $\displaystyle 180^\circ$.

$142^\circ+x=180^\circ$ $\displaystyle 142^\circ+x=180^\circ$

Solve for $\displaystyle x$:

$142^\circ+x=180^\circ$ $\displaystyle 142^\circ+x=180^\circ$

$x=180^\circ-142^\circ$ $\displaystyle x=180^\circ-142^\circ$

$x=38^\circ$ $\displaystyle x=38^\circ$

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Example Question #3 : Angles

Which of the following angles is coterminal with $\frac{2\pi}{7}$ $\displaystyle \frac{2\pi}{7}$ ?

Possible Answers:

$-\frac{2\pi}{7}$ $\displaystyle -\frac{2\pi}{7}$

$-\frac{12\pi}{7}$ $\displaystyle -\frac{12\pi}{7}$

$-\frac{9\pi}{7}$ $\displaystyle -\frac{9\pi}{7}$

$\frac{9\pi}{7}$ $\displaystyle \frac{9\pi}{7}$

$\frac{12\pi}{7}$ $\displaystyle \frac{12\pi}{7}$

Correct answer:

$-\frac{12\pi}{7}$ $\displaystyle -\frac{12\pi}{7}$

Explanation:

For an angle to be coterminal with $\theta$ $\displaystyle \theta$, that angle must be of the form $\theta + 2\pi N$ $\displaystyle \theta + 2\pi N$ for some integer $\displaystyle N$ - or, equivalently, the difference of the angle measures multiplied by $\frac{1}{2\pi}$ $\displaystyle \frac{1}{2\pi}$must be an integer. We apply this test to all five choices.

$\frac{9\pi}{7}$ $\displaystyle \frac{9\pi}{7}$:

$\left ( \frac{1}{2\pi } \right ) \left ( \frac{2\pi}{7} - \frac{9\pi}{7} \right )$ $\displaystyle \left ( \frac{1}{2\pi } \right ) \left ( \frac{2\pi}{7} - \frac{9\pi}{7} \right )$

$=\left ( \frac{1}{2\pi } \right )\left (- \frac{7\pi}{7} \right ) =\left ( \frac{1}{2\pi } \right ) \pi = \frac{1}{2 }$ $\displaystyle =\left ( \frac{1}{2\pi } \right )\left (- \frac{7\pi}{7} \right ) =\left ( \frac{1}{2\pi } \right ) \pi = \frac{1}{2 }$

$\frac{12\pi}{7}$ $\displaystyle \frac{12\pi}{7}$:

$\left ( \frac{1}{2\pi } \right ) \left ( \frac{2\pi}{7} - \frac{12\pi}{7} \right )$ $\displaystyle \left ( \frac{1}{2\pi } \right ) \left ( \frac{2\pi}{7} - \frac{12\pi}{7} \right )$

$=\left ( \frac{1}{2\pi } \right )\left (- \frac{10\pi}{7} \right ) =- \frac{5}{7}$ $\displaystyle =\left ( \frac{1}{2\pi } \right )\left (- \frac{10\pi}{7} \right ) =- \frac{5}{7}$

$-\frac{2\pi}{7}$ $\displaystyle -\frac{2\pi}{7}$:

$\left ( \frac{1}{2\pi } \right ) \left [\frac{2\pi}{7} -\left ( -\frac{2\pi}{7} \right ) \right ]$ $\displaystyle \left ( \frac{1}{2\pi } \right ) \left [\frac{2\pi}{7} -\left ( -\frac{2\pi}{7} \right ) \right ]$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{2\pi}{7} + \frac{2\pi}{7 }\right )$ $\displaystyle =\left ( \frac{1}{2\pi } \right )\left ( \frac{2\pi}{7} + \frac{2\pi}{7 }\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{4\pi}{7} \right ) = \frac{2}{7}$ $\displaystyle =\left ( \frac{1}{2\pi } \right )\left ( \frac{4\pi}{7} \right ) = \frac{2}{7}$

$-\frac{9\pi}{7}$ $\displaystyle -\frac{9\pi}{7}$:

$\left ( \frac{1}{2\pi } \right ) \left [\frac{2\pi}{7} -\left ( -\frac{9\pi}{7} \right ) \right ]$ $\displaystyle \left ( \frac{1}{2\pi } \right ) \left [\frac{2\pi}{7} -\left ( -\frac{9\pi}{7} \right ) \right ]$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{2\pi}{7} + \frac{9\pi}{7 }\right )$ $\displaystyle =\left ( \frac{1}{2\pi } \right )\left ( \frac{2\pi}{7} + \frac{9\pi}{7 }\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{11\pi}{7} \right ) = \frac{11}{14 }$ $\displaystyle =\left ( \frac{1}{2\pi } \right )\left ( \frac{11\pi}{7} \right ) = \frac{11}{14 }$

$-\frac{12\pi}{7}$ $\displaystyle -\frac{12\pi}{7}$:

$\left ( \frac{1}{2\pi } \right ) \left [\frac{2\pi}{7} -\left ( -\frac{12\pi}{7} \right ) \right ]$ $\displaystyle \left ( \frac{1}{2\pi } \right ) \left [\frac{2\pi}{7} -\left ( -\frac{12\pi}{7} \right ) \right ]$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{2\pi}{7} + \frac{12\pi}{7 }\right )$ $\displaystyle =\left ( \frac{1}{2\pi } \right )\left ( \frac{2\pi}{7} + \frac{12\pi}{7 }\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{14\pi}{7} \right ) = \left ( \frac{1}{2\pi } \right ) \cdot2\pi = 1$ $\displaystyle =\left ( \frac{1}{2\pi } \right )\left ( \frac{14\pi}{7} \right ) = \left ( \frac{1}{2\pi } \right ) \cdot2\pi = 1$

$-\frac{12\pi}{7}$ $\displaystyle -\frac{12\pi}{7}$ is the correct choice, since only that choice passes our test.

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High School Math : Angles

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Angles

Example Question #1 : Understanding Complementary And Suplmentary Angles

Example Question #2 : Angles

Example Question #3 : Angles

Example Question #4 : Angles

Example Question #1 : Graphing Functions

Example Question #2 : Angles

Example Question #1 : Understanding Complementary And Suplmentary Angles

Example Question #1 : Angles

Example Question #3 : Angles

All High School Math Resources

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