Geometry - 8th Grade Math

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Question

The image provided contains a set of parallel lines, $\overline{AB}$ and $\overline{CD}$ , and a transversal line, $\overline{EF}$ . If angle $\angle{\ a}$ is equal to $135^\circ$ , then which of the other angles is equal to $135^\circ?$

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Answer

First, we need to define some key terms:

Parallel Lines: Parallel lines are lines that will never intersect with each other.

Transversal Line: A transversal line is a line that crosses two parallel lines.

In the the image provided, lines $\overline{AB}$ and $\overline{CD}$ are parallel lines and line $\overline{EF}$ is a transversal line because it crosses the two parallel lines.

It is important to know that transversal lines create angle relationships:

Vertical angles are congruent

Corresponding angles are congruent

Alternate interior angles are congruent

Alternate exterior angles are congruent

Let's look at angle $\angle{\ a}$ in the image provided below to demonstrate our relationships.

Angle $\angle{\ a}$ and $\angle{\ c}$ are vertical angles.

Angle $\angle{\ a}$ and $\angle{\ e}$ are corresponding angles.

Angle $\angle{\ a}$ and $\angle{\ g}$ are exterior angles.

Angle $\angle{\ a}$ is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs $\angle{\ e}$ and $\angle{\ c}$ as well as angle $\angle{\ d}$ and $\angle{\ f}$ .

For this problem, we want to find the angle that is congruent to angle $\angle{\ a}$ . Based on our answer choices, angle $\angle{\ a}$ and $\angle{\ c}$ are vertical angles; thus, both angle $\angle{\ a}$ and $\angle{\ c}$ are congruent and equal $135^\circ$

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Question

Calculate the length of the missing side of the provided triangle. Round the answer to the nearest whole number.

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Answer

The provided triangle is a right triangle. We know this because the angle marker in the left corner of the triangle indicates that the triangle possesses a right or $90^\circ$ angle. When a triangle includes a right angle, the triangle is said to be a "right triangle."

We can use the Pythagorean Theorem to help us solve this problem.

The Pythagorean Theorem states that for right triangles, the square of the hypotenuse is equal to the sum of the square of the other two sides. In other terms:

We can use the formula and substitute the known side lengths from the problem to solve for the missing side length:

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Question

Calculate the volume of the cone provided. Round the answer to the nearest hundredth.

Tap to reveal answer

Answer

In order to solve this problem, we need to recall the formula used to calculate the volume of a cone:

$V=\pi $r^2$\left ( $\frac{h}{3}$ \right )$

Now that we have this formula, we can substitute in the given values and solve:

$V=\pi9(2)$

$V=\pi18$

$V=56.55\textup{ $in}^3$$

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Question

The image provided contains a set of parallel lines, $\overline{AB}$ and $\overline{CD}$ , and a transversal line, $\overline{EF}$ . If angle $\angle{\ a}$ is equal to $135^\circ$ , then which of the other angles is equal to $135^\circ?$

Tap to reveal answer

Answer

First, we need to define some key terms:

Parallel Lines: Parallel lines are lines that will never intersect with each other.

Transversal Line: A transversal line is a line that crosses two parallel lines.

In the the image provided, lines $\overline{AB}$ and $\overline{CD}$ are parallel lines and line $\overline{EF}$ is a transversal line because it crosses the two parallel lines.

It is important to know that transversal lines create angle relationships:

Vertical angles are congruent

Corresponding angles are congruent

Alternate interior angles are congruent

Alternate exterior angles are congruent

Let's look at angle $\angle{\ a}$ in the image provided below to demonstrate our relationships.

Angle $\angle{\ a}$ and $\angle{\ c}$ are vertical angles.

Angle $\angle{\ a}$ and $\angle{\ e}$ are corresponding angles.

Angle $\angle{\ a}$ and $\angle{\ g}$ are exterior angles.

Angle $\angle{\ a}$ is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs $\angle{\ e}$ and $\angle{\ c}$ as well as angle $\angle{\ d}$ and $\angle{\ f}$ .

For this problem, we want to find the angle that is congruent to angle $\angle{\ a}$ . Based on our answer choices, angle $\angle{\ a}$ and $\angle{\ c}$ are vertical angles; thus, both angle $\angle{\ a}$ and $\angle{\ c}$ are congruent and equal $135^\circ$

← Didn't Know|Knew It →

Question

Calculate the length of the missing side of the provided triangle. Round the answer to the nearest whole number.

Tap to reveal answer

Answer

The provided triangle is a right triangle. We know this because the angle marker in the left corner of the triangle indicates that the triangle possesses a right or $90^\circ$ angle. When a triangle includes a right angle, the triangle is said to be a "right triangle."

We can use the Pythagorean Theorem to help us solve this problem.

The Pythagorean Theorem states that for right triangles, the square of the hypotenuse is equal to the sum of the square of the other two sides. In other terms:

We can use the formula and substitute the known side lengths from the problem to solve for the missing side length:

← Didn't Know|Knew It →

Question

Calculate the volume of the cone provided. Round the answer to the nearest hundredth.

Tap to reveal answer

Answer

In order to solve this problem, we need to recall the formula used to calculate the volume of a cone:

$V=\pi $r^2$\left ( $\frac{h}{3}$ \right )$

Now that we have this formula, we can substitute in the given values and solve:

$V=\pi9(2)$

$V=\pi18$

$V=56.55\textup{ $in}^3$$

← Didn't Know|Knew It →

Question

The image provided contains a set of parallel lines, $\overline{AB}$ and $\overline{CD}$ , and a transversal line, $\overline{EF}$ . If angle $\angle{\ a}$ is equal to $135^\circ$ , then which of the other angles is equal to $135^\circ?$

Tap to reveal answer

Answer

First, we need to define some key terms:

Parallel Lines: Parallel lines are lines that will never intersect with each other.

Transversal Line: A transversal line is a line that crosses two parallel lines.

In the the image provided, lines $\overline{AB}$ and $\overline{CD}$ are parallel lines and line $\overline{EF}$ is a transversal line because it crosses the two parallel lines.

It is important to know that transversal lines create angle relationships:

Vertical angles are congruent

Corresponding angles are congruent

Alternate interior angles are congruent

Alternate exterior angles are congruent

Let's look at angle $\angle{\ a}$ in the image provided below to demonstrate our relationships.

Angle $\angle{\ a}$ and $\angle{\ c}$ are vertical angles.

Angle $\angle{\ a}$ and $\angle{\ e}$ are corresponding angles.

Angle $\angle{\ a}$ and $\angle{\ g}$ are exterior angles.

Angle $\angle{\ a}$ is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs $\angle{\ e}$ and $\angle{\ c}$ as well as angle $\angle{\ d}$ and $\angle{\ f}$ .

For this problem, we want to find the angle that is congruent to angle $\angle{\ a}$ . Based on our answer choices, angle $\angle{\ a}$ and $\angle{\ c}$ are vertical angles; thus, both angle $\angle{\ a}$ and $\angle{\ c}$ are congruent and equal $135^\circ$

← Didn't Know|Knew It →

Question

Calculate the length of the missing side of the provided triangle. Round the answer to the nearest whole number.

Tap to reveal answer

Answer

The provided triangle is a right triangle. We know this because the angle marker in the left corner of the triangle indicates that the triangle possesses a right or $90^\circ$ angle. When a triangle includes a right angle, the triangle is said to be a "right triangle."

We can use the Pythagorean Theorem to help us solve this problem.

The Pythagorean Theorem states that for right triangles, the square of the hypotenuse is equal to the sum of the square of the other two sides. In other terms:

We can use the formula and substitute the known side lengths from the problem to solve for the missing side length:

← Didn't Know|Knew It →

Question

Calculate the volume of the cone provided. Round the answer to the nearest hundredth.

Tap to reveal answer

Answer

In order to solve this problem, we need to recall the formula used to calculate the volume of a cone:

$V=\pi $r^2$\left ( $\frac{h}{3}$ \right )$

Now that we have this formula, we can substitute in the given values and solve:

$V=\pi9(2)$

$V=\pi18$

$V=56.55\textup{ $in}^3$$

← Didn't Know|Knew It →

Question

The image provided contains a set of parallel lines, $\overline{AB}$ and $\overline{CD}$ , and a transversal line, $\overline{EF}$ . If angle $\angle{\ a}$ is equal to $135^\circ$ , then which of the other angles is equal to $135^\circ?$

Tap to reveal answer

Answer

First, we need to define some key terms:

Parallel Lines: Parallel lines are lines that will never intersect with each other.

Transversal Line: A transversal line is a line that crosses two parallel lines.

In the the image provided, lines $\overline{AB}$ and $\overline{CD}$ are parallel lines and line $\overline{EF}$ is a transversal line because it crosses the two parallel lines.

It is important to know that transversal lines create angle relationships:

Vertical angles are congruent

Corresponding angles are congruent

Alternate interior angles are congruent

Alternate exterior angles are congruent

Let's look at angle $\angle{\ a}$ in the image provided below to demonstrate our relationships.

Angle $\angle{\ a}$ and $\angle{\ c}$ are vertical angles.

Angle $\angle{\ a}$ and $\angle{\ e}$ are corresponding angles.

Angle $\angle{\ a}$ and $\angle{\ g}$ are exterior angles.

Angle $\angle{\ a}$ is an exterior angle; therefore, it does not have an alternate interior angle. In this image, the alternate interior angles are the angle pairs $\angle{\ e}$ and $\angle{\ c}$ as well as angle $\angle{\ d}$ and $\angle{\ f}$ .

For this problem, we want to find the angle that is congruent to angle $\angle{\ a}$ . Based on our answer choices, angle $\angle{\ a}$ and $\angle{\ c}$ are vertical angles; thus, both angle $\angle{\ a}$ and $\angle{\ c}$ are congruent and equal $135^\circ$

← Didn't Know|Knew It →

Question

Calculate the length of the missing side of the provided triangle. Round the answer to the nearest whole number.

Tap to reveal answer

Answer

The provided triangle is a right triangle. We know this because the angle marker in the left corner of the triangle indicates that the triangle possesses a right or $90^\circ$ angle. When a triangle includes a right angle, the triangle is said to be a "right triangle."

We can use the Pythagorean Theorem to help us solve this problem.

The Pythagorean Theorem states that for right triangles, the square of the hypotenuse is equal to the sum of the square of the other two sides. In other terms:

We can use the formula and substitute the known side lengths from the problem to solve for the missing side length:

← Didn't Know|Knew It →

Question

Calculate the volume of the cone provided. Round the answer to the nearest hundredth.

Tap to reveal answer

Answer

In order to solve this problem, we need to recall the formula used to calculate the volume of a cone:

$V=\pi $r^2$\left ( $\frac{h}{3}$ \right )$

Now that we have this formula, we can substitute in the given values and solve:

$V=\pi9(2)$

$V=\pi18$

$V=56.55\textup{ $in}^3$$

← Didn't Know|Knew It →

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