Use the image to answer the question.

Question

An illustration shows two right triangles with congruent corresponding angles. Triangle A B C is larger than triangle S R T. In triangle A B C, angle A measures 90 degrees, angle B does not have a measurement, and angle C measures 34 degrees. In triangle S R T, angle S measures 90 degrees, angle R measures 56 degrees, and angle T measures 34 degrees.

If △ABC∼△SRT
△
A
B
C
∼
△
S
R
T
, what is m∠B
m
∠
B
?

Answer 1

Since triangles $ABC$ and $SRT$ are similar (i.e., $\mathrm{△}ABC\sim \mathrm{△}SRT$), their corresponding angles are equal.

In triangle $SRT$:

• $m\mathrm{\angle }S={90}^{\circ }$
• $m\mathrm{\angle }R={56}^{\circ }$
• $m\mathrm{\angle }T={34}^{\circ }$

To find the measure of angle $B$ in triangle $ABC$, we use the fact that the sum of angles in a triangle is ${180}^{\circ }$.

Since $m\mathrm{\angle }C={34}^{\circ }$ and $m\mathrm{\angle }A={90}^{\circ }$ in triangle $ABC$:

$$m\mathrm{\angle }A+m\mathrm{\angle }B+m\mathrm{\angle }C={180}^{\circ }$$

Substituting the values we have:

$${90}^{\circ }+m\mathrm{\angle }B+{34}^{\circ }={180}^{\circ }$$

Now, combine the known angles:

$${124}^{\circ }+m\mathrm{\angle }B={180}^{\circ }$$

To isolate $m\mathrm{\angle }B$:

$$m\mathrm{\angle }B={180}^{\circ }-{124}^{\circ }={56}^{\circ }$$

Therefore, the measure of angle $B$ is:

$$\boxed{{\displaystyle {56}^{\circ }}}$$