High School: Geometry - Common Core: High School - Geometry

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Question

Given the black, green, and purple triangles below, determine which of the triangles are similar?

Answer

To determine whether triangles are similar recall what "similar" means and the AA identity. To be "similar" triangles need to have congruent angles. Also recall that all triangles have interior angles that sum to 180 degrees.

Knowing this, look at the black triangle.

Two angles are given and the third can be calculated.

$180^\circ-66^\circ-58^\circ=56^\circ$

Now, look at the green triangle.

$180^\circ-66^\circ-58^\circ=56^\circ$

Now, look at the purple triangle.

$180^\circ-66^\circ-62^\circ=52^\circ$

Since the black and green triangle have the same angle measurements, they are considered to be similar. The purple triangle only has one angle that is congruent to the other triangles thus, the purple triangle is not similar to either of the other two triangles.

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Question

Given the black, green, and purple triangles below, determine which of the triangles are similar?

Answer

To determine whether triangles are similar recall what "similar" means and the AA criterion. To be "similar" triangles need to have congruent angles. Also recall that all triangles have interior angles that sum to 180 degrees.

Knowing this, look at the black triangle.

Two angles are given and the third can be calculated.

$180^\circ-66^\circ-58^\circ=56^\circ$

Now, look at the green triangle.

$180^\circ-66^\circ-58^\circ=56^\circ$

Now, look at the purple triangle.

$180^\circ-66^\circ-56^\circ=58^\circ$

Since the black and green triangle have the same angle measurements, they are considered to be similar. The purple triangle also has the same angle measurements as the black and green triangles thus, all three triangles are similar.

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Question

The $\bigtriangleup ABC$ above has $\measuredangle A=26^\circ$ . Which of the following triangle measurements would be similar to $\bigtriangleup ABC$ .

Answer

To determine whether triangles are similar recall what "similar" means by the AA criterion. To be "similar" triangles need to have congruent angles. Also recall that all triangles have interior angles that sum to 180 degrees.

$\bigtriangleup ABC$ is given below. By the figure it is known that $\measuredangle B=118^\circ$ and by the statement, $\measuredangle A=26^\circ$ . Knowing this information, the measure of the last angle can be calculated.

$180^\circ-118^\circ-26^\circ=36^\circ$

Therefore, for a triangle to be similar to $\bigtriangleup ABC$ by the AA criterion, the triangle must have angle measurements of 26, 36, and 118 degrees. Thus, $\bigtriangleup GHI$ is a similar triangle.

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Question

The $\bigtriangleup ABC$ above has $\measuredangle A=36^\circ$ . Which of the following triangle measurements would be similar to $\bigtriangleup ABC$ .

Answer

To determine whether triangles are similar recall what "similar" means by the AA criterion. To be "similar" triangles need to have congruent angles. Also recall that all triangles have interior angles that sum to 180 degrees.

$\bigtriangleup ABC$ is given below. By the figure it is known that $\measuredangle B=118^\circ$ and by the statement, $\measuredangle A=36^\circ$ . Knowing this information, the measure of the last angle can be calculated.

$180^\circ-118^\circ-36^\circ=26^\circ$

Therefore, for a triangle to be similar to $\bigtriangleup ABC$ by the AA criterion, the triangle must have angle measurements of 26, 36, and 118 degrees. Thus, $\bigtriangleup GHI$ is a similar triangle.

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Question

The $\bigtriangleup ABC$ above has $\measuredangle C=13^\circ$ . Which of the following triangle measurements would be similar to $\bigtriangleup ABC$ .

Answer

To determine whether triangles are similar recall what "similar" means by the AA criterion. To be "similar" triangles need to have congruent angles. Also recall that all triangles have interior angles that sum to 180 degrees.

$\bigtriangleup ABC$ is given below. By the figure it is known that $\measuredangle A=17^\circ$ and by the statement, $\measuredangle C=13^\circ$ . Knowing this information, the measure of the last angle can be calculated.

$180^\circ-17^\circ-13^\circ=150^\circ$

Therefore, for a triangle to be similar to $\bigtriangleup ABC$ by the AA criterion, the triangle must have angle measurements of 17, 13, and 150 degrees. Thus, $\bigtriangleup GHI$ is a similar triangle.

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Question

The $\bigtriangleup ABC$ above has $\measuredangle B=139^\circ$ . Which of the following triangle measurements would be similar to $\bigtriangleup ABC$ .

Answer

To determine whether triangles are similar recall what "similar" means by the AA criterion. To be "similar" triangles need to have congruent angles. Also recall that all triangles have interior angles that sum to 180 degrees.

$\bigtriangleup ABC$ is given below. By the figure it is known that $\measuredangle A=17^\circ$ and by the statement, $\measuredangle B=139^\circ$ . Knowing this information, the measure of the last angle can be calculated.

$180^\circ-17^\circ-139^\circ=24^\circ$

Therefore, for a triangle to be similar to $\bigtriangleup ABC$ by the AA criterion, the triangle must have angle measurements of 17, 24, and 139 degrees. Thus, $\bigtriangleup GHI$ is a similar triangle.

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Question

Determine whether the statement is true or false.

In $\bigtriangleup ABC$ $\measuredangle A=73^\circ$ , $\measuredangle B=73^\circ$ , and in $\bigtriangleup JKL$ the $\measuredangle K=34^\circ$ and $\measuredangle L=73^\circ$ . $\bigtriangleup ABC$ and $\bigtriangleup JKL$ are similar by the AA criterion.

Answer

To determine whether triangles are similar recall what "similar" means and the AA identity. To be "similar" triangles need to have congruent angles. Also recall that all triangles have interior angles that sum to 180 degrees.

Looking at the given triangles and their characteristics, similarity can be identified.

In $\bigtriangleup ABC$ $\measuredangle A=73^\circ$ , $\measuredangle B=73^\circ$ , and in $\bigtriangleup JKL$ the $\measuredangle K=34^\circ$ and $\measuredangle L=73^\circ$ .

First calculate the measurement of angle C.

$180^\circ-73^\circ-73^\circ=35^\circ$

Therefore, $\bigtriangleup ABC$ and $\bigtriangleup JKL$ are similar by the AA criterion.

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Question

Determine whether the triangles are similar.

In triangle ABC, angle A measures 73 degrees. In triangle JKL, angle K measures 34 degrees.

Answer

To determine whether triangles are similar recall what "similar" means and the AA identity. To be "similar" triangles need to have congruent angles. Also recall that all triangles have interior angles that sum to 180 degrees.

In $\bigtriangleup ABC$ $\measuredangle A=73^\circ$ , and in $\bigtriangleup JKL$ the $\measuredangle K=34^\circ$ .

Since only one angle is known from each triangle there is not enough information to determine whether these two triangles are similar by the AA criterion.

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Question

The $\bigtriangleup ABC$ above has $\measuredangle B=42^\circ$ . Which of the following triangle measurements would be similar to $\bigtriangleup ABC$ .

Answer

To determine whether triangles are similar recall what "similar" means by the AA criterion. To be "similar" triangles need to have congruent angles. Also recall that all triangles have interior angles that sum to 180 degrees.

$\bigtriangleup ABC$ is given below. By the figure it is known that $\measuredangle A=92^\circ$ and by the statement, $\measuredangle B=42^\circ$ . Knowing this information, the measure of the last angle can be calculated.

$180^\circ-92^\circ-42^\circ=46^\circ$

Therefore, for a triangle to be similar to $\bigtriangleup ABC$ by the AA criterion, the triangle must have angle measurements of 42, 92, and 46 degrees. Thus, $\bigtriangleup GHI$ is a similar triangle.

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Question

Determine which triangles are similar.

Answer

To determine whether triangles are similar recall what "similar" means and the AA criterion. To be "similar" triangles need to have congruent angles. Also recall that all triangles have interior angles that sum to 180 degrees.

Knowing this, look at triangle A.

Two angles are given and the third can be calculated.

$180^\circ-61^\circ-59^\circ=60^\circ$

Now, look at triangle B.

$180^\circ-60^\circ-59^\circ=61^\circ$

Now, look at triangle C.

$180^\circ-60^\circ-60^\circ=60^\circ$

Since triangles A and B have the same angle measurements, they are considered to be similar. Triangle C only has one angle that is congruent to the other triangles thus, triangle C is not similar to either of the other two triangles.

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Question

Determine which of the triangles are similar.

Answer

To determine whether triangles are similar recall what "similar" means and the AA criterion. To be "similar" triangles need to have congruent angles. Also recall that all triangles have interior angles that sum to 180 degrees.

Knowing this, look at triangle A.

Two angles are given and the third can be calculated.

$180^\circ-64^\circ-59^\circ=57^\circ$

Now, look at triangle B.

$180^\circ-60^\circ-59^\circ=61^\circ$

Now, look at triangle C.

$180^\circ-60^\circ-64^\circ=56^\circ$

Since triangles A, B, and C do not have any angles that are congruent, none of these triangles are similar.

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Question

Are triangles A and C similar?

Answer

To determine whether triangles are similar recall what "similar" means and the AA criterion. To be "similar" triangles need to have congruent angles. Also recall that all triangles have interior angles that sum to 180 degrees.

Knowing this, look at triangle A.

Two angles are given and the third can be calculated.

$180^\circ-62^\circ-56^\circ=62^\circ$

Now, look at triangle C.

$180^\circ-62^\circ-62^\circ=56^\circ$

Since triangles A and C have the same angle measurements, they are considered to be similar. Therefore, the answer to the question is "yes".

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Question

A truck is traveling down a hill, which of the following statements is/are true?

Answer

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of parallel and perpendicular lines, since those terms are among the answer selections.

Parallel lines: In a plane, parallel lines are lines that will never intersect. This means they have the same slope but different intercepts.

Perpendicular lines: In a plane, perpendicular lines are lines that intersect by creating a $90^\circ$ degree angle. This also means they have opposite sign, reciprocal slopes.

Now, look at the aspects of this particular problem.

"A truck is traveling down a hill"

From this statement, it cannot be assumed that the hill is a straight line nor can it be assumed that the hill goes on forever. Therefore, the truck and the hill will never be parallel. Also, for these same reasons it is known that the truck will never be perpendicular to the hill. The relation of the truck's tires to the hill will never be parallel since they constantly touch.

Thus, the correct answer choice is,

"The body of the truck is not perpendicular to the hill."

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Question

The seat of a swing on a swing set is attached to the top horizontal bar by two chains that are exactly inches from each other and of equal length. The seat of the swing is also inches. Which of the following statements describes the geometric relationship between the two chains?

Answer

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of parallel and perpendicular lines, since those terms are among the answer selections.

Parallel lines: In a plane, parallel lines are lines that will never intersect. This means they have the same slope but different intercepts.

Perpendicular lines: In a plane, perpendicular lines are lines that intersect by creating a $90^\circ$ angle. This also means they have opposite sign, reciprocal slopes.

Now, look at the aspects of this particular problem.

"The seat of a swing on a swing set is attached to the top horizontal bar by two chains that are exactly inches from each other and of equal length. The seat of the swing is also inches."

The question is asking to define the relationship between the two chains that hold the swing to the swing set. Since the two chains are exactly inches apart from one another and attached to the pole which is horizontal from the swing and the swing seat itself is inches, it is concluded that the two chains are parallel to one another.

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Question

The seat of a swing on a swing set is attached to the top horizontal bar by two chains that are exactly inches from each other and of equal length. The seat of the swing is also inches. Which of the following statements describes the geometric relationship between one of the chains and the horizontal bar it is attached to?

Answer

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of parallel and perpendicular lines, since those terms are among the answer selections.

Parallel lines: In a plane, parallel lines are lines that will never intersect. This means they have the same slope but different intercepts.

Perpendicular lines: In a plane, perpendicular lines are lines that intersect by creating a $90^\circ$ angle. This also means they have opposite sign, reciprocal slopes.

Now, look at the aspects of this particular problem.

"The seat of a swing on a swing set is attached to the top horizontal bar by two chains that are exactly inches from each other and of equal length. The seat of the swing is also inches."

The question is asking to define the relationship between one of the chains and the horizontal bar it is attached to. Since the swing will hang directly down from the two chains and the bar is horizontal to ground it can be assumed that the chain and the bar form a $90^\circ$ angle and thus, they are perpendicular to one another.

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Question

The seat of a swing on a swing set is attached to the top horizontal bar by two chains that are exactly inches from each other and of equal length. The seat of the swing is also inches. Which of the following statements describes the geometric relationship between the horizontal bar and the swing?

Answer

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of parallel and perpendicular lines, since those terms are among the answer selections.

Parallel lines: In a plane, parallel lines are lines that will never intersect. This means they have the same slope but different intercepts.

Perpendicular lines: In a plane, perpendicular lines are lines that intersect by creating a $90^\circ$ angle. This also means they have opposite sign, reciprocal slopes.

Now, look at the aspects of this particular problem.

"The seat of a swing on a swing set is attached to the top horizontal bar by two chains that are exactly inches from each other and of equal length. The seat of the swing is also inches."

The question is asking to define the relationship between the horizontal bar and the swing seat. Since the two chains are exactly inches apart from one another and of equal length and attached to the pole which is horizontal from the swing and the swing seat itself is inches, it is concluded that the seat and the horizontal bar are parallel to one another.

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Question

A circular pizza is cut into equal slices. Which of the following is an accurate mathematical description of one of the pizza slices?

Answer

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of circles and corresponding angles.

A central angle is known as the angle of a circle where the vertex of the angle is located at the center of the circle.

A circle is composed of 360 degrees.

Knowing these characteristics, solve for the central angle of one slice of pizza.

$\360^\circ=8\cdot \text{slice} \\\frac{360^\circ}{8}=\text{silce} \\45^\circ=\text{slice}$

Therefore, the correct answer is

"The central angle of one pizza slice is degrees."

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Question

A circular pizza that has a radius of inches and is cut into equal slices. Which of the following is an accurate mathematical description of one of the pizza slices?

Answer

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of circles and corresponding angles.

A central angle is known as the angle of a circle where the vertex of the angle is located at the center of the circle.

A circle is composed of 360 degrees.

The circumference of a circle is the length around the circle and the radius is the length from the center of the circle to any point on the circle's edge.

For this particular question, calculate the circumference and then calculate the arc length of each slice pizza slice.

$\C=2 \pi \codt r \C=2\cdot 6\pi \C=12\pi$

Since there are 8 equal slices, divide the circumference by 8.

$C=\frac{12\pi}{8}=\frac{3\cdot 4\pi}{2\cdot 4}=\frac{3\pi}{2}$

Therefore, the correct answer is

"The arc length of one slice of pizza is $\frac{3}{2}\pi$ inches."

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Question

Looking at the given clock where the radius is inches, which of the following statements accurately describes the space between the hour and minute hand?

Answer

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of circles and corresponding angles.

A central angle is known as the angle of a circle where the vertex of the angle is located at the center of the circle.

A circle is composed of 360 degrees.

The area of a circle is found by using the formula $A=\pi r^2$ .

For this particular problem first calculate the area of the clock.

$\A=\pi r^2 \A=\pi(4^2) \A=16\pi$

Now, since the clock reads 4:50, the distance between the hour and minute hands is $\frac{5}{12}$ of the total clock. From here, calculate the area between the two hands.

$\{16\pi}\times\frac{5}{12} \\\frac{80\pi }{12}$

Therefore, the correct answer is

The area between the hour and minute hand is $\frac{80\pi}{12}$ .

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Question

Looking at the given clock where the radius is inches, which of the following statements accurately describes the space between the hour and minute hand (Going clockwise)?

Answer

This question is trying to put a mathematical definition to a real life situation. First, recall the definitions of circles and corresponding angles.

A central angle is known as the angle of a circle where the vertex of the angle is located at the center of the circle.

A circle is composed of 360 degrees.

Also recall that a straight line measures 180 degrees.

Looking at the given clock, it is seen that a straight line can be created by connecting the 12 and 6 on the clock. Since the clock reads 11:35 the angle between the hour and minute hand is greater than 180 degrees because the hour hand is behind the 12 and the minute hand is behind the 6 on the clock.

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