https://file.aiquickdraw.com/mathbot/file/1728579162557q54gupo0.jpg

The problem in the image shows a right triangle where:

- \( \angle D \) is the right angle, so \( m \angle D = 90^\circ \).
- \( m \angle B = (4x + 2)^\circ \).
- \( m \angle C = (3x - 10)^\circ \).

Since the sum of the angles in any triangle is \( 180^\circ \), we can set up the equation:

\[
m \angle D + m \angle B + m \angle C = 180^\circ
\]

Substituting the known values:

\[
90^\circ + (4x + 2)^\circ + (3x - 10)^\circ = 180^\circ
\]

Now, simplify and solve for \( x \):

\[
90 + 4x + 2 + 3x - 10 = 180
\]
\[
7x + 82 = 180
\]
\[
7x = 180 - 82
\]
\[
7x = 98
\]
\[
x = \frac{98}{7} = 14
\]

Now, substitute \( x = 14 \) back into the expressions for \( m \angle B \) and \( m \angle C \):

\[
m \angle B = 4(14) + 2 = 56 + 2 = 58^\circ
\]
\[
m \angle C = 3(14) - 10 = 42 - 10 = 32^\circ
\]

So, the measures of the angles are:

- \( m \angle D = 90^\circ \)
- \( m \angle B = 58^\circ \)
- \( m \angle C = 32^\circ \)

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the sum of the interior angles in a triangle?
2. If one angle of a triangle is \( 45^\circ \) and another is \( 65^\circ \), what is the third angle?
3. How do you solve for unknown variables in angle equations?
4. What defines a right triangle, and how can you identify it?
5. What other types of triangles exist besides right triangles?

**Tip:** When solving for unknowns in a triangle, always remember the angle sum property of a triangle is \( 180^\circ \).

In the triangle below, with right angle ∠D, suppose that m∠B = (4x + 2)° and m∠C = (3x - 10)°. Find the degree measure of each angle in the triangle.

Find the measures of angles B and C in a right triangle where m∠B = (4x + 2)° and m∠C = (3x - 10)°. Learn to apply the angle sum property to solve for unknowns in a triangle. Suitable for students in Grades 8-10.

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