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)

simplify the answer of how you got it

Answer 1

In a right triangle, the sum of all angles is 180 degrees. Since one angle is a right angle (90 degrees), the sum of the remaining two acute angles must be 90 degrees.

Given the acute angles measure \( 2x + 6 \) degrees and \( 3x - 26 \) degrees, we can set up the equation:

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

Now, simplify the equation:

1. Combine like terms:
\[
2x + 3x + 6 - 26 = 90
\]

This simplifies to:
\[
5x - 20 = 90
\]

2. Next, add 20 to both sides:
\[
5x = 110
\]

3. Now, divide by 5:
\[
x = 22
\]

Now that we have the value of \( x \), we can find the measures of the two acute angles:

1. For the first angle:
\[
2x + 6 = 2(22) + 6 = 44 + 6 = 50 \text{ degrees}
\]

2. For the second angle:
\[
3x - 26 = 3(22) - 26 = 66 - 26 = 40 \text{ degrees}
\]

Thus, the measures of the angles in the triangle are:
- \( 50 \) degrees
- \( 40 \) degrees
- \( 90 \) degrees (the right angle)

In summary:
- The two acute angles are 50 degrees and 40 degrees. The right angle is 90 degrees.