To solve for \( x \) and the two unknown angles inside a \( 90^\circ \) angle, we start by setting up an equation based on the information provided.
The equation will be:
\[ (x + 4) + (3x + 2) = 90 \]
Now, simplify and combine like terms:
\[ x + 4 + 3x + 2 = 90 \] \[ 4x + 6 = 90 \]
Next, subtract 6 from both sides to isolate the term with \( x \):
\[ 4x = 90 - 6 \] \[ 4x = 84 \]
Now, divide both sides by 4 to solve for \( x \):
\[ x = \frac{84}{4} = 21 \]
Now that we have \( x \), we can find the measurements of the two angles.
- The first angle is:
\[ x + 4 = 21 + 4 = 25^\circ \]
- The second angle is:
\[ 3x + 2 = 3(21) + 2 = 63 + 2 = 65^\circ \]
Thus, the solution is:
- \( x = 21 \)
- The two angles are \( 25^\circ \) and \( 65^\circ \).