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

Study concepts, example questions & explanations for Common Core: High School - Geometry

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Example Questions

Example Question #7 : Aa Criterion Using Similarity Transformations: Ccss.Math.Content.Hsg Srt.A.3

Determine whether the statement is true or false.

Hsg.srt.a.3 7

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.

Possible Answers:

True

False

Correct answer:

True

Explanation:

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.

Hsg.srt.a.3 7

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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Example Question #8 : Aa Criterion Using Similarity Transformations: Ccss.Math.Content.Hsg Srt.A.3

Hsg.srt.a.3 9

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

Possible Answers:

$\\\measuredangle G=92^\circ \\\measuredangle H=38^\circ \\\measuredangle I=50^\circ$

$\\\measuredangle G=92^\circ \\\measuredangle H=40^\circ \\\measuredangle I=48^\circ$

$\\\measuredangle G=92^\circ \\\measuredangle H=42^\circ \\\measuredangle I=46^\circ$

$\\\measuredangle G=92^\circ \\\measuredangle H=38^\circ \\\measuredangle I=42^\circ$

$\\\measuredangle G=92^\circ \\\measuredangle H=42^\circ \\\measuredangle I=45^\circ$

Correct answer:

$\\\measuredangle G=92^\circ \\\measuredangle H=42^\circ \\\measuredangle I=46^\circ$

Explanation:

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.

Hsg.srt.a.3 9

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

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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Example Question #141 : High School: Geometry

Hsg.srt.a.3 10

Determine which triangles are similar.

Possible Answers:

Triangles C and A are similar.

None of the triangles are similar.

All triangles are similar.

Triangles A and B are similar.

Triangles B and C are similar.

Correct answer:

Triangles A and B are similar.

Explanation:

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.

Screen shot 2016 07 19 at 6.10.09 am

Two angles are given and the third can be calculated.

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

Now, look at triangle B.

Screen shot 2016 07 19 at 6.10.15 am

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

Now, look at triangle C.

Screen shot 2016 07 19 at 6.10.25 am

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

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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Example Question #10 : Aa Criterion Using Similarity Transformations: Ccss.Math.Content.Hsg Srt.A.3

Hsg.srt.a.3 10

Determine which of the triangles are similar.

Possible Answers:

All the triangles are similar.

Triangles A and B are similar.

Triangles B and C are similar.

None of the triangles are similar.

Triangles C and A are similar.

Correct answer:

None of the triangles are similar.

Explanation:

Knowing this, look at triangle A.

Screen shot 2016 07 19 at 6.19.58 am

Two angles are given and the third can be calculated.

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

Now, look at triangle B.

Screen shot 2016 07 19 at 6.20.20 am

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

Now, look at triangle C.

Screen shot 2016 07 19 at 6.20.27 am

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

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

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Example Question #31 : Similarity, Right Triangles, & Trigonometry

Are triangles A and C similar?

Hsg.srt.a.3 10

Possible Answers:

Yes

Correct answer:

Yes

Explanation:

Knowing this, look at triangle A.

Screen shot 2016 07 19 at 6.30.23 am

Two angles are given and the third can be calculated.

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

Now, look at triangle C.

Screen shot 2016 07 19 at 6.30.28 am

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

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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Example Question #12 : Aa Criterion Using Similarity Transformations: Ccss.Math.Content.Hsg Srt.A.3

Determine whether the triangles are similar.

Hsg.srt.a.3 7

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

Possible Answers:

More information is needed.

The triangles are similar.

The triangles are not similar.

Correct answer:

More information is needed.

Explanation:

Hsg.srt.a.3 7

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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Example Question #31 : Similarity, Right Triangles, & Trigonometry

True or False: The sine of any acute angle is equal to the cosine of its complement.

Possible Answers:

False

True

Correct answer:

True

Explanation:

Consider the triangle below

First we must understand the definition of complementary angles. Complementary angles are a pair of angles that add up to 90 degrees. Looking at the triangle above, we see that $\angle BAC$ and $\angle ACB$ add up to 90, so they are complementary.

Recall that the definition of sine is $sin = \frac{opposite}{hypotenuse}$ and cosine is $cos = \frac{adjacent}{hypotenuse}$ .

Next, list out the sine and cosine of $\angle BAC$ and $\angle ACB$ .

$sin(\angle BAC) = \frac{10.83}{11.39}$

$sin(\angle ACB) = \frac{3.52}{11.39}$

$cos(\angle BAC) = \frac{3.52}{11.39}$

$cos(\angle ACB) = \frac{10.83}{11.39}$

Notice that $sin(\angle BAC) = \frac{10.83}{11.39} = cos(\angle ACB)$

And $sin(\angle ACB) = \frac{3.52}{11.39} = cos(\angle BAC)$

We have just shown that the sine of the two angles is equal to the cosines of their complements. This is true for any acute angles.

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Example Question #32 : Similarity, Right Triangles, & Trigonometry

You have two complementary angles, $\angle A$ and $\angle B$ . You know that $\angle A = 30$ . Using your sine and cosine relationships, what is $\angle B$ in degrees?

Possible Answers:

$\angle B = 45$

$\angle B = 30$

$\angle B = 60$

$\angle B = 90$

Correct answer:

$\angle B = 60$

Explanation:

You know that $\angle A = 30$ . We use this information to set up our work in the following way:

$sin(\angle A) = cos(\angle B)$ $sin(\angle A) = cos(\angle B)$

$\implies sin(30) = cos(\angle B)$

$\implies \frac{1}{2} = cos(\angle B)$

$\implies cos^{-1}(\frac{1}{2}) = \angle B$

$\implies 60 = \angle B$

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Example Question #33 : Similarity, Right Triangles, & Trigonometry

You have two complementary angles, $\angle A = 2x + 5$ and $\angle B = 4x - 12$ . Solve for , round to the second decimal place.

Possible Answers:

x = 23.59

x = 16.17

x = 4.32

x = 17.20

Correct answer:

x = 16.17

Explanation:

To solve for , we need to use the definition of complementary. Both of these angles should add up to be 90, so we can set up this problem by setting the sum of the two angles equal to 90.

$\angle A + \angle B = 90$

$\implies (2x + 5) + (4x -12) = 90$

$\implies 6x - 7 = 90$

$\implies 6x = 97$

$\implies x = 16.17$

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Example Question #1 : Sine And Cosine Relationship Of Complementary Angles: Ccss.Math.Content.Hsg Srt.C.7

True or False: The sine and cosine relationship of complementary angles is only true for the acute angles of a right triangle.

Possible Answers:

True

False

Correct answer:

False

Explanation:

This relationship is true for any pair of angles that are complementary (i.e. any two angles that add up to 90). We can see this in obtuse triangles that have two acute angles that add up to 90, we can see it in acute triangles, and equilateral triangles as well. Beyond triangles this relationship is also true.

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All Common Core: High School - Geometry Resources

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

Study concepts, example questions & explanations for Common Core: High School - Geometry

All Common Core: High School - Geometry Resources

Example Questions

Example Question #7 : Aa Criterion Using Similarity Transformations: Ccss.Math.Content.Hsg Srt.A.3

Example Question #8 : Aa Criterion Using Similarity Transformations: Ccss.Math.Content.Hsg Srt.A.3

Example Question #141 : High School: Geometry

Example Question #10 : Aa Criterion Using Similarity Transformations: Ccss.Math.Content.Hsg Srt.A.3

Example Question #31 : Similarity, Right Triangles, & Trigonometry

Example Question #12 : Aa Criterion Using Similarity Transformations: Ccss.Math.Content.Hsg Srt.A.3

Example Question #31 : Similarity, Right Triangles, & Trigonometry

Example Question #32 : Similarity, Right Triangles, & Trigonometry

Example Question #33 : Similarity, Right Triangles, & Trigonometry

Example Question #1 : Sine And Cosine Relationship Of Complementary Angles: Ccss.Math.Content.Hsg Srt.C.7

All Common Core: High School - Geometry Resources

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