Angle Geometry - GED Math

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Question

Refer to the above diagram.

Which of the following is a valid alternative name for $\overrightarrow{BD}$ ?

Answer

The name of a ray includes two letters, so $\overrightarrow{BCD}$ can be eliminated.

The first letter must be the endpoint. Since $\overrightarrow{BD}$ is a name of the ray, the endpoint is , and any alternative name for the ray must begin with . This leaves only $\overrightarrow{BC}$ .

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Question

Refer to the above diagram.

$\overleftrightarrow{AE} ||\overleftrightarrow{BD}$ ; $m \angle CED = 64^{\circ }$ ; $m \angle EFC = 107 ^{\circ }$ .

What is $m \angle FCE$ ?

Answer

$\angle CED$ and $\angle ECD$ are two acute angles of a right triangle and are therefore complementary - that is,

$m \angle ECD + m \angle CED = 90^{\circ }$

$m \angle CED = 64^{\circ }$ , so

$m \angle ECD + 64^{\circ } = 90^{\circ }$

$m \angle ECD = 26^{\circ }$

$\angle ECD$ and $\angle FEC$ , being alternate interior angles formed by transversal $\overrightarrow{CE}$ across parallel lines, are congruent, so $m \angle FEC= 26^{\circ }$ .

We now look at $\Delta FEC$ , whose interior angles must have degree measures totaling $180^{\circ }$ , so

$m\angle FCE + m \angle FEC + m\angle EFC = 180^{\circ }$

$m\angle FCE + 26 ^{\circ }+107^{\circ } = 180^{\circ }$

$m\angle FCE + 133^{\circ } = 180^{\circ }$

$m\angle FCE= 47^{\circ }$

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Question

Refer to the above diagram.

Which of the following facts does not, by itself, prove that $\overleftrightarrow{AE} \parallel \overleftrightarrow{BD}$ ?

Answer

From the Parallel Postulate and its converse, as well as its various resulting theorems, two lines in a plane crossed by a transversal are parallel if any of the following happen:

Both lines are perpendicular to the same third line - this happens if $\angle FED$ is a right angle, since, from this fact and the fact that $\angle EDC$ is also right, both lines are perpendicular to $\overline{ED}$ .

Same-side interior angles are supplementary - this happens if $\angle BCE$ and $\angle FEC$ are supplementary, since they are same-side interior angles with respect to transversal $\overrightarrow{CE}$ .

Alternate interior angles are congruent - this happens if $\angle AFC \cong \angle DCF$ , since they are alternate interior angles with respect to transversal $\overleftrightarrow{FC}$ .

However, the fact that $\overrightarrow{CE}$ bisects $\angle FCD$ has no bearing on whether $\overleftrightarrow{AE} \parallel \overleftrightarrow{BD}$ is true or not, since it does not relate any two angles formed by a transversal.

" $\overrightarrow{CE}$ bisects $\angle FCD$ " is the correct choice.

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Question

In two intersecting lines, the opposite angles are and . What must be the value of ?

Answer

In an intersecting line, vertical angles are equal to each other.

Set up an equation such that both angles are equal.

Solve for . Subtract on both sides.

Add 14 on both sides.

Divide by 7 on both sides.

$\frac{7}{7}= \frac{7x}{7}$

The answer is:

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Question

Suppose a pair of opposite angles are measured and . What must the value of ?

Answer

Vertical angles are equal.

Set both angles equal and solve for x.

Subtract on both sides.

Add 8 on both sides.

Divide by 4 on both sides.

$x=\frac{5}{4}$

The answer is: $\frac{5}{4}$

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Question

Suppose two vertical angles in a pair of intersecting lines. What is the value of if one angle is and the other angle is ?

Answer

Vertical angles of intersecting lines must equal to each other.

Set up an equation such that both angle measures are equal.

Add three on both sides.

Divide by three on both sides.

$\frac{3x }{3}=\frac{36}{3}$

The answer is:

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Question

Suppose two opposite angles are measured and . What is the value of ?

Answer

Opposite angles equal. Set up an equation such that both angle values are equal.

Add 5 on both sides.

Divide by 5 on both sides.

$\frac{5x}{5} = \frac{80}{5}$

The answer is:

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Question

With a pair of intersecting lines, a set of opposite angles are measured and . What must the value of be?

Answer

Opposite angles of two intersecting lines must equal to each other. Set up an equation such that both angle are equal.

Add 9 on both sides.

Subtract on both sides.

This means that equals .

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Question

In the figure above, $a\parallel b$ . If the measure of $\measuredangle 1=6x^{\circ}$ and $\measuredangle 6=4x^{\circ}$ , what is the measure of $\measuredangle8$ ?

Answer

Since we have two parallel lines, we know that $\measuredangle1=\measuredangle4$ since they are opposite angle.

We also know that $\measuredangle 4 \text{ and }\measuredangle 6$ are supplementary because they are consecutive interior angles. Thus, we know that $\measuredangle 1$ is also supplementary to $\measuredangle6$ .

We can then set up the following equation to solve for .

Thus, $\measuredangle 1=108^{\circ}$ and $\measuredangle6=72^{\circ}$ .

Now, notice that $\measuredangle 8=\measuredangle 4$ because they are corresponding angles. Thus, $\measuredangle 8=\measuredangle 1=108^{\circ}$ .

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Question

Find the value of $\angle3$ .

Assume the two horizontal lines are parallel.

Answer

Start by noticing that the two angles with the values of and are supplementary.

Thus, we can write the following equation and solve for .

Since $\angle3$ and are vertical angles, they must also have the same value.

Thus, $\angle3=4(26)-5=99^{\circ}$

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Question

Refer to the above diagram.

$\overleftrightarrow{AE} ||\overleftrightarrow{BD}$ ; $m \angle BCE = 136^{\circ }$ . Evaluate $m \angle CED$ .

Answer

$\angle BCE$ and $\angle ECD$ form a linear pair, so they are supplementary - that is, their degree measures total $180^{\circ }$ , so

$m \angle BCE + m\angle ECD =180^{\circ }$

$136^{\circ } + m\angle ECD =180^{\circ }$

$m\angle ECD =180^{\circ } - 136 ^{\circ } = 44^{\circ }$

$m \angle CED$ and $\angle ECD$ are acute angles of right triangle $\Delta ECD$ , so they are complementary - that is, their degree measures total $90^{\circ }$ , so

$m \angle CED + m\angle ECD =90^{\circ }$

$m \angle CED + 44 ^{\circ } =90^{\circ }$

$m \angle CED =90^{\circ } - 44 ^{\circ } = 46 ^{\circ }$

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Question

Angles A and B are supplementary. The measure of angle A is . The measure of Angle B is . Find the value of .

Answer

Since angles A and B are supplementary, thier measurements add up to equal 180 degrees. Therefore we can set up our equation like such:

-or-

Combine like terms and solve for :

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Question

Angles A, B, and C are supplementary. The measure of angle A is . The measure of angle B is . The measure for angle C is . Find the value of .

Answer

Since angles A, B, and C are supplementary, their measures add up to equal 180 degrees. Therefore we can set up the equation as such:

-or-

Combine like terms and solve for :

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Question

Angles A, B, and C are supplementary. The measure of angle A is . The measure of angle B is . The measure for angle C = . What are the measure for the three angles?

Answer

Since angles A, B, and C are supplementary, their measures add up to equal 180 degrees. Therefore we can set up an equation as such:

-or-

Combine like terms and solve for x:

Plug back into the three angle measurements:

4

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Question

Find the measure of angle B if it is the supplement to angle A:

$m\angle A=8^{o}$

Answer

If two angles are supplementary, that means the sum of their degrees of measure will add up to 180. In order to find the measure of angle B, subtract angle A from 180 like shown:

$m\angle B=180^{o}-8^{o}=172^{o}$

This gives us a final answer of 172 degrees for angle B.

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Question

If a set of angles are supplementary, what is the other angle if one angle is degrees?

Answer

Two angles that are supplementary must add up to 180 degrees.

To find the other angle, subtract 101 from 180.

The answer is:

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Question

What angle is supplementary to 54 degrees?

Answer

Supplementary angles must add up to 180 degrees.

To find the other angle, we will need to subtract 54 from 180.

The answer is:

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Question

If and are supplementary angles, what must be a possible angle?

Answer

The sum of the two angles supplement to each other will add up to 180 degrees.

Set up the equation.

Solve for .

Divide by 10 on both sides.

$\frac{10x }{10}= \frac{180}{10}$

Substitute for and , and we have 36 and 144, which add up to 180.

The answer is:

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Question

If the angles and are supplementary, what is the value of ?

Answer

Supplementary angles sum to 180 degrees.

Set up an equation to solve for .

Substitute this value to .

The answer is:

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Question

Suppose there are two angles. If a given angle is , and both angles are supplementary, what must be the other angle?

Answer

Supplementary angles add up to 180 degrees.

This means we will need to subtract the known angle quantity from 180.

Distribute the negative.

The answer is:

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