To solve the given right triangle, we’ll need to find:

1. $m \angle C$ (the measure of angle $C$),
2. $AB$ (the hypotenuse), and
3. $BC$ (the adjacent side to angle $A$).

Given:

• $\angle A = 62^\circ$,
• $AC = 10$ (opposite to $\angle A$),
• $\angle B = 90^\circ$.

Step 1: Find $m \angle C$

Since $\angle A + \angle B + \angle C = 180^\circ$ and $\angle B = 90^\circ$: $\angle C = 180^\circ - 90^\circ - 62^\circ = 28^\circ$

Thus, $m \angle C \approx 28^\circ$

Step 2: Find $AB$ (Hypotenuse)

Using the sine function with angle $A$: $\sin(62^\circ) = \frac{AC}{AB}$ Rearrange to solve for $AB$: $AB = \frac{AC}{\sin(62^\circ)} = \frac{10}{\sin(62^\circ)}$ Using $\sin(62^\circ) \approx 0.8829$: $AB \approx \frac{10}{0.8829} \approx 11.3$

Step 3: Find $BC$ (Adjacent side to $\angle A$)

Using the tangent function with angle $A$: $\tan(62^\circ) = \frac{AC}{BC}$ Rearrange to solve for $BC$: $BC = \frac{AC}{\tan(62^\circ)} = \frac{10}{\tan(62^\circ)}$ Using $\tan(62^\circ) \approx 1.8807$: $BC \approx \frac{10}{1.8807} \approx 5.3$

Summary of the Results

• $m \angle C \approx 28^\circ$
• $AB \approx 11.3$
• $BC \approx 5.3$

Would you like further details or have any questions about these steps?

Here are 5 related questions to deepen understanding:

1. How would you calculate the length of $AC$ if only $AB$ and $\angle A$ were given?
2. Can you use the Pythagorean theorem to verify the results for $AB$ and $BC$?
3. What trigonometric function would you use to find $BC$ if $\angle C$ and $AB$ were given?
4. How does rounding affect the accuracy of these results?
5. What is the relationship between the side lengths and angles in a right triangle?

Tip: Remember that trigonometric functions are very useful in solving right triangles when one angle and one side are known.