Similarity, Right Triangles, & Trigonometry - Common Core: High School - Geometry

Card 0 of 20

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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".

Compare your answer with the correct one above

Question

In a triangle where the side opposite a $62 ^{\circ}$ has length 10 find the side opposite a $66 ^{\circ}$ angle. Round you answer to the nearest hundredth.

Answer

In order to solve this, we need to recall the law of sines.

$\frac{1}{A} \sin{\left (a \right )} = \frac{1}{B} \sin{\left (b \right )}$

Where , and are angles, and , and , are opposite side lengths.

Now let's plug in 62 for , 10 for and 66 for .
Now our equation becomes

$\left(\frac{\sin( 62 )}{ 10 }\right)= \left(\frac{\sin( 66 )}{ b }\right)$

Now we rearrange the equation to solve for

$\b = \left(\frac{ 10 \sin( 66 )}{\sin( 62 )}\right) \\b = -11.3557732263804$

Now we round our answer to the nearest tenth.

Remember if your answer is negative, multiply it by -1, because side lengths can't be negative.

Compare your answer with the correct one above

Question

In a triangle where the side opposite a $41 ^{\circ}$ has length 6 find the side opposite a $39 ^{\circ}$ angle. Round you answer to the nearest hundredth.

Answer

In order to solve this, we need to recall the law of sines.

$\frac{1}{A} \sin{\left (a \right )} = \frac{1}{B} \sin{\left (b \right )}$

Where , and are angles, and , and , are opposite side lengths.

Now let's plug in 41 for , 6 for and 39 for .
Now our equation becomes

$\left(\frac{\sin( 41 )}{ 6 }\right)= \left(\frac{\sin( 39 )}{ b }\right)$

Now we rearrange the equation to solve for

$\b = \left(\frac{ 6 \sin( 39 )}{\sin( 41 )}\right) \\b = 6.45402256283484$

Now we round our answer to the nearest tenth.

Remember if your answer is negative, multiply it by -1, because side lengths can't be negative.

Compare your answer with the correct one above

Question

In a triangle where the side opposite a $60 ^{\circ}$ has length 3 find the side opposite a $51 ^{\circ}$ angle. Round you answer to the nearest hundredth.

Answer

In order to solve this, we need to recall the law of sines.

$\frac{1}{A} \sin{\left (a \right )} = \frac{1}{B} \sin{\left (b \right )}$

Where , and are angles, and , and , are opposite side lengths.

Now let's plug in 60 for , 3 for and 51 for .
Now our equation becomes

$\left(\frac{\sin( 60 )}{ 3 }\right)= \left(\frac{\sin( 51 )}{ b }\right)$

Now we rearrange the equation to solve for b

$\b = \left(\frac{ 3 \sin( 51 )}{\sin( 60 )}\right) \\b = 1.57596947141406$

Now we round our answer to the nearest tenth.

Remember if your answer is negative, multiply it by -1, because side lengths can't be negative.

Compare your answer with the correct one above

Question

In a triangle where the side opposite a $45 ^{\circ}$ has length 13 find the side opposite a $65 ^{\circ}$ angle. Round you answer to the nearest hundredth.

Answer

In order to solve this, we need to recall the law of sines.

$\frac{1}{A} \sin{\left (a \right )} = \frac{1}{B} \sin{\left (b \right )}$

Where , and are angles, and , and , are opposite side lengths.

Now let's plug in 45 for , 13 for and 65 for .
Now our equation becomes

$\left(\frac{\sin( 45 )}{ 13 }\right)= \left(\frac{\sin( 65 )}{ b }\right)$

Now we rearrange the equation to solve for b

$\b = \left(\frac{ 13 \sin( 65 )}{\sin( 45 )}\right) \\b = -15.9863244465377$

Now we round our answer to the nearest tenth.

Remember if your answer is negative, multiply it by -1, because side lengths can't be negative.

Compare your answer with the correct one above

Question

In a triangle where the side opposite a $26 ^{\circ}$ has length 12 find the side opposite a $41 ^{\circ}$ angle. Round you answer to the nearest hundredth.

Answer

In order to solve this, we need to recall the law of sines.

$\frac{1}{A} \sin{\left (a \right )} = \frac{1}{B} \sin{\left (b \right )}$

Where , and are angles, and , and , are opposite side lengths.

Now let's plug in 26 for , 12 for and 41 for .
Now our equation becomes

$\left(\frac{\sin( 26 )}{ 12 }\right)= \left(\frac{\sin( 41 )}{ b }\right)$

Now we rearrange the equation to solve for b

$\b = \left(\frac{ 12 \sin( 41 )}{\sin( 26 )}\right) \\b = -15.5778376306642$

Now we round our answer to the nearest tenth.

Remember if your answer is negative, multiply it by -1, because side lengths can't be negative.

Compare your answer with the correct one above

Question

In a triangle where the side opposite a $36 ^{\circ}$ has length 6 find the side opposite a $58 ^{\circ}$ angle. Round you answer to the nearest hundredth.

Answer

In order to solve this, we need to recall the law of sines.

$\frac{1}{A} \sin{\left (a \right )} = \frac{1}{B} \sin{\left (b \right )}$

Where , and are angles, and , and , are opposite side lengths.

Now let's plug in 36 for , 6 for and 58 for .
Now our equation becomes

$\left(\frac{\sin( 36 )}{ 6 }\right)= \left(\frac{\sin( 58 )}{ b }\right)$

Now we rearrange the equation to solve for b

$\b = \left(\frac{ 6 \sin( 58 )}{\sin( 36 )}\right) \\b = -3.69854363983686$

Now we round our answer to the nearest tenth.

Remember if your answer is negative, multiply it by -1, because side lengths can't be negative.

Compare your answer with the correct one above

Question

In a triangle where the side opposite a $26 ^{\circ}$ has length 11 find the side opposite a $17 ^{\circ}$ angle. Round you answer to the nearest hundredth.

Answer

In order to solve this, we need to recall the law of sines.

$\frac{1}{A} \sin{\left (a \right )} = \frac{1}{B} \sin{\left (b \right )}$

Where , and are angles, and , and , are opposite side lengths.

Now let's plug in 26 for , 11 for and 17 for .
Now our equation becomes

$\left(\frac{\sin( 26 )}{ 11 }\right)= \left(\frac{\sin( 17 )}{ b }\right)$

Now we rearrange the equation to solve for b

$\b = \left(\frac{ 11 \sin( 17 )}{\sin( 26 )}\right) \\b = 2.71142149312156$

Now we round our answer to the nearest tenth.

Remember if your answer is negative, multiply it by -1, because side lengths can't be negative.

Compare your answer with the correct one above

Question

In a triangle where the side opposite a $81 ^{\circ}$ has length 11 find the side opposite a $66 ^{\circ}$ angle. Round you answer to the nearest hundredth.

Answer

In order to solve this, we need to recall the law of sines.

$\frac{1}{A} \sin{\left (a \right )} = \frac{1}{B} \sin{\left (b \right )}$

Where , and are angles, and , and , are opposite side lengths.

Now let's plug in 81 for , 11 for and 66 for .
Now our equation becomes

$\left(\frac{\sin( 81 )}{ 11 }\right)= \left(\frac{\sin( 66 )}{ b }\right)$

Now we rearrange the equation to solve for b

$\b = \left(\frac{ 11 \sin( 66 )}{\sin( 81 )}\right) \\b = 12.2250984628794$

Now we round our answer to the nearest tenth.

Remember if your answer is negative, multiply it by -1, because side lengths can't be negative.

Compare your answer with the correct one above

Tap the card to reveal the answer