Question

A right triangle has acute angles measuring 2x+6
degrees and 3x−26
degrees. Use the Triangle Angle Sum Theorem to find the measures of the missing angles of the triangle.(4 points)

Answer 1

In a right triangle, the sum of the angles is always 180 degrees. Since one angle is a right angle (90 degrees), we can set up the equation for the two acute angles.

Let the two acute angles be given as:

- First angle: \( A = 2x + 6 \)
- Second angle: \( B = 3x - 26 \)

According to the Triangle Angle Sum Theorem, we have:

\[
A + B + 90 = 180
\]

This simplifies to:

\[
A + B = 90
\]

Now, substituting the expressions for angles A and B:

\[
(2x + 6) + (3x - 26) = 90
\]

Now combine like terms:

\[
2x + 3x + 6 - 26 = 90
\]

\[
5x - 20 = 90
\]

Now, add 20 to both sides:

\[
5x = 110
\]

Dividing both sides by 5 gives:

\[
x = 22
\]

Now we can find the measures of the acute angles by substituting \( x \) back into the expressions for the angles:

1. **First angle:**

\[
A = 2x + 6 = 2(22) + 6 = 44 + 6 = 50 \text{ degrees}
\]

2. **Second angle:**

\[
B = 3x - 26 = 3(22) - 26 = 66 - 26 = 40 \text{ degrees}
\]

Thus, the measures of the angles in the triangle are:

- First acute angle: \( 50 \) degrees
- Second acute angle: \( 40 \) degrees
- Right angle: \( 90 \) degrees

In summary, the angles of the triangle are \( 50 \) degrees, \( 40 \) degrees, and \( 90 \) degrees.