Solve Simple Equations for an Unknown Angle in a Figure: CCSS.Math.Content.7.G.B.5 - Common Core: 7th Grade Math

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Question

Two angles are supplementary and have a ratio of 1:4. What is the size of the smaller angle?

Answer

Since the angles are supplementary, their sum is 180 degrees. Because they are in a ratio of 1:4, the following expression could be written:

$x+4x=180$

$5x=180$

$x=36^{\circ}$

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Question

One angle of an isosceles triangle has measure $110^{\circ }$ . What are the measures of the other two angles?

Answer

An isosceles triangle not only has two sides of equal measure, it has two angles of equal measure. This means one of two things, which we examine separately:

Case 1: It has another $110^{\circ }$ angle. This is impossible, since a triangle cannot have two obtuse angles.

Case 2: Its other two angles are the ones that are of equal measure. If we let be their common measure, then, since the sum of the measures of a triangle is $180^{\circ }$ ,

$2x \div 2= 70 \div 2$

Both angles measure $35^{\circ}$

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Question

One of the base angles of an isosceles triangle is $42^{\circ}$ . Give the measure of the vertex angle.

Answer

The base angles of an isosceles triangle are always equal. Therefore both base angles are $42^{\circ}$ .

Let the measure of the third angle. Since the sum of the angles of a triangle is $180^{\circ}$ , we can solve accordingly:

$42+42+x=180\Rightarrow x=180-84\Rightarrow x=96^{\circ}$

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Question

Note: Figure NOT drawn to scale.

What is the measure of angle

Answer

The two angles at bottom are marked as congruent. One forms a linear pair with a $152 ^{\circ }$ angle, so it is supplementary to that angle, making its measure $(180-152)^{\circ } = 28^{\circ }$ . Therefore, each marked angle measures $28^{\circ }$ .

The sum of the measures of the interior angles of a triangle is $180^{\circ }$ , so:

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Question

What angle is complementary to $65$^{\circ}$$ ?

Answer

To find the other angle, subtract the given angle from $90$^{\circ}$$ since complementary angles add up to $90$^{\circ}$$ .

The complementary is:

$\angle 90-\angle65 = \angle25$

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Question

What is the supplementary angle to $100$^{\circ}$$ ?

Answer

Supplementary angles add up to $180$^{\circ}$$ . In order to find the correct angle, take the known angle and subtract that from $180$^{\circ}$$ .

$180$^{\circ}$ -100$^{\circ}$ = 80$^{\circ}$$

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Question

What angle is complement to $\frac{\pi}{6}$ ?

Answer

The complement to an angle is ninety degrees subtract the angle since two angles must add up to 90. In this case, since we are given the angle in radians, we are subtracting from $\frac{\pi}{2}$ instead to find the complement. The conversion between radians and degrees is: $\pi \textup{ radians } = 180 \textup{ degrees}$

$\frac{\pi}{2}-\frac{\pi}{6}$

Reconvert the fractions to the least common denominator.

$\frac{\pi}{2}-\frac{\pi}{6} = \frac{3\pi}{6}-\frac{\pi}{6} = \frac{2\pi}{6}$

Reduce the fraction.

$\frac{2\pi}{6} = \frac{\pi}{3}$

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Question

The figure represents a set of complementary angles, solve for .

Answer

Complementary angles are defined as two angles that when added together equal $90^\circ$

From the question, we know that the two angles are complimentary, and thus equal $90^\circ$ , so we can set up the following equation:

Next we can solve for :

$\frac{\begin{array}[b]{r}x+45=90\ -45-45\end{array}}{x= 45}$

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Question

The figure represents a set of complementary angles, solve for .

Answer

Complementary angles are defined as two angles that when added together equal $90^\circ$

From the question, we know that the two angles are complimentary, and thus equal $90^\circ$ , so we can set up the following equation:

Next we can solve for :

$\frac{\begin{array}[b]{r}x+34=90\ -34-34\end{array}}{x= 56}$

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Question

The figure represents a set of complementary angles, solve for .

Answer

Complementary angles are defined as two angles that when added together equal $90^\circ$

From the question, we know that the two angles are complimentary, and thus equal $90^\circ$ , so we can set up the following equation:

Next we can solve for :

$\frac{\begin{array}[b]{r}x+25=90\ -25-25\end{array}}{x= 65}$

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Question

The figure represents a set of complementary angles, solve for .

Answer

Complementary angles are defined as two angles that when added together equal $90^\circ$

From the question, we know that the two angles are complimentary, and thus equal $90^\circ$ , so we can set up the following equation:

Next we can solve for :

$\frac{\begin{array}[b]{r}x+20=90\ -20-20\end{array}}{x= 70}$

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Question

The figure represents a set of complementary angles, solve for .

Answer

Complementary angles are defined as two angles that when added together equal $90^\circ$

From the question, we know that the two angles are complimentary, and thus equal $90^\circ$ , so we can set up the following equation:

Next we can solve for :

$\frac{\begin{array}[b]{r}x+12=90\ -12-12\end{array}}{x= 78}$

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Question

The figure represents a set of complementary angles, solve for .

Answer

Complementary angles are defined as two angles that when added together equal $90^\circ$

From the question, we know that the two angles are complimentary, and thus equal $90^\circ$ , so we can set up the following equation:

Next we can solve for :

$\frac{\begin{array}[b]{r}x+5=90\ -5-5\end{array}}{x= 85}$

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Question

The figure represents a set of complementary angles, solve for .

Answer

Complementary angles are defined as two angles that when added together equal $90^\circ$

From the question, we know that the two angles are complimentary, and thus equal $90^\circ$ , so we can set up the following equation:

Next we can solve for :

$\frac{\begin{array}[b]{r}x+52=90\ -52-52\end{array}}{x= 38}$

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Question

The figure represents a set of complementary angles, solve for .

Answer

Complementary angles are defined as two angles that when added together equal $90^\circ$

From the question, we know that the two angles are complimentary, and thus equal $90^\circ$ , so we can set up the following equation:

Next we can solve for :

$\frac{\begin{array}[b]{r}x+61=90\ -61-61\end{array}}{x= 29}$

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Question

The figure represents a set of complementary angles, solve for .

Answer

Complementary angles are defined as two angles that when added together equal $90^\circ$

From the question, we know that the two angles are complimentary, and thus equal $90^\circ$ , so we can set up the following equation:

Next we can solve for :

$\frac{\begin{array}[b]{r}x+67=90\ -67-67\end{array}}{x= 23}$

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Question

The figure represents a set of complementary angles, solve for .

Answer

Complementary angles are defined as two angles that when added together equal $90^\circ$

From the question, we know that the two angles are complimentary, and thus equal $90^\circ$ , so we can set up the following equation:

Next we can solve for :

$\frac{\begin{array}[b]{r}x+76=90\ -76-76\end{array}}{x= 14}$

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Question

The figure represents a set of complementary angles, solve for .

Answer

Complementary angles are defined as two angles that when added together equal $90^\circ$

From the question, we know that the two angles are complimentary, and thus equal $90^\circ$ , so we can set up the following equation:

Next we can solve for :

$\frac{\begin{array}[b]{r}x+80=90\ -80-80\end{array}}{x= 10}$

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Question

The figure represents a set of complementary angles, solve for .

Answer

Complementary angles are defined as two angles that when added together equal $90^\circ$

From the question, we know that the two angles are complimentary, and thus equal $90^\circ$ , so we can set up the following equation:

Next we can solve for :

$\frac{\begin{array}[b]{r}x+84=90\ -84-84\end{array}}{x= 6}$

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Question

The figure represents a set of supplementary angles, solve for .

Answer

Supplementary angles are defined as two angles that when added together equal $180^\circ$

From the question, we know that the two angles are supplementary, and thus equal $180^\circ$ , so we can set up the following equation:

Next we can solve for :

$\frac{\begin{array}[b]{r}x+90=180\ -90-90\end{array}}{x= 90}$

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