Geometry - AA Criterion using Similarity Transformations: CCSS.Math.Content.HSG-SRT.A.3

Hsg.srt.a.3 5

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

$\\measuredangle G=17^\circ\\measuredangle H=13^\circ\ \measuredangle I=150^\circ$

$\\measuredangle G=17^\circ\\measuredangle H=32^\circ\ \measuredangle I=131^\circ$

$\\measuredangle G=24^\circ\\measuredangle H=17^\circ\ \measuredangle I=118^\circ$

$\\measuredangle G=13^\circ\\measuredangle H=39^\circ\ \measuredangle I=118^\circ$

$\\measuredangle G=17^\circ\\measuredangle H=15^\circ\ \measuredangle I=148^\circ$

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 $\measuredangle A=17^\circ$ and by the statement, $\measuredangle C=13^\circ$ . Knowing this information, the measure of the last angle can be calculated.

Hsg.srt.a.3 5

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

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.

Hsg.srt.a.3 9

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

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

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

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

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

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

Explanation

$\bigtriangleup ABC$ 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.

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

Hsg.srt.a.3 2

All triangles are similar

The black and green triangle are similar.

The green and purple triangles are similar.

The black and purple triangles are similar.

None of the triangles 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 the black triangle.

Screen shot 2016 07 15 at 2.33.19 pm

Two angles are given and the third can be calculated.

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

Now, look at the green triangle.

Screen shot 2016 07 15 at 2.33.24 pm

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

Now, look at the purple triangle.

Screen shot 2016 07 18 at 7.19.56 am

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

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.

Hsg.srt.a.3 3

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

$\\measuredangle G=26^\circ\\measuredangle H=36^\circ\ \measuredangle I=118^\circ$

$\\measuredangle G=26^\circ\\measuredangle H=32^\circ\ \measuredangle I=118^\circ$

$\\measuredangle G=24^\circ\\measuredangle H=38^\circ\ \measuredangle I=118^\circ$

$\\measuredangle G=23^\circ\\measuredangle H=39^\circ\ \measuredangle I=118^\circ$

$\\measuredangle G=36^\circ\\measuredangle H=32^\circ\ \measuredangle I=118^\circ$

Explanation

$\bigtriangleup ABC$ 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.

Hsg.srt.a.3 3

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

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.

Are triangles A and C similar?

Hsg.srt.a.3 10

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".

Given the black, green, and purple triangles below, determine which of the triangles are similar?
Hsg.srt.a.3 1

The black and green triangle are similar.

The green and purple triangle are similar.

The purple and black triangle are similar.

All triangles are similar.

None of the triangles are similar.

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.

Knowing this, look at the black triangle.

Screen shot 2016 07 15 at 2.33.19 pm

Two angles are given and the third can be calculated.

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

Now, look at the green triangle.

Screen shot 2016 07 15 at 2.33.24 pm

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

Now, look at the purple triangle.

Screen shot 2016 07 15 at 2.33.36 pm

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

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.

Hsg.srt.a.3 5

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

$\\measuredangle G=17^\circ\\measuredangle H=24^\circ\ \measuredangle I=139^\circ$

$\\measuredangle G=17^\circ\\measuredangle H=32^\circ\ \measuredangle I=139^\circ$

$\\measuredangle G=24^\circ\\measuredangle H=17^\circ\ \measuredangle I=118^\circ$

$\\measuredangle G=13^\circ\\measuredangle H=39^\circ\ \measuredangle I=118^\circ$

$\\measuredangle G=17^\circ\\measuredangle H=15^\circ\ \measuredangle I=148^\circ$

Explanation

$\bigtriangleup ABC$ 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.

Hsg.srt.a.3 5

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

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.

Hsg.srt.a.3 10

Determine which triangles are similar.

Triangles A and B are similar.

Triangles B and C are similar.

Triangles C and A are similar.

All triangles are similar.

None of the triangles are similar.

Explanation

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.

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.

More information is needed.

The triangles are similar.

The triangles are not similar.

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.

Hsg.srt.a.3 3

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

$\\measuredangle G=26^\circ\\measuredangle H=36^\circ\ \measuredangle I=118^\circ$

$\\measuredangle G=26^\circ\\measuredangle H=32^\circ\ \measuredangle I=118^\circ$

$\\measuredangle G=24^\circ\\measuredangle H=38^\circ\ \measuredangle I=118^\circ$

$\\measuredangle G=23^\circ\\measuredangle H=39^\circ\ \measuredangle I=118^\circ$

$\\measuredangle G=36^\circ\\measuredangle H=32^\circ\ \measuredangle I=118^\circ$

Explanation

$\bigtriangleup ABC$ 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.