Question

right triangle has acute angles 2x+6 degrees and 3x-26 degrees use triangle angle sum theorem to find the measures of the missing angle

Answer 1

In a right triangle, one angle is always \(90^\circ\), and the sum of all angles in any triangle is \(180^\circ\). Thus, for the given right triangle, we can use the Triangle Angle Sum Theorem, which states:

\[
\text{Sum of angles} = 180^\circ
\]

Let the acute angles be \(A = 2x + 6^\circ\) and \(B = 3x - 26^\circ\). The third angle, the right angle, is \(90^\circ\).

According to the angle sum theorem for this triangle, we have:

\[
A + B + 90^\circ = 180^\circ
\]

By substituting for \(A\) and \(B\), we get:

\[
(2x + 6) + (3x - 26) + 90 = 180
\]

Now simplify the equation:

\[
2x + 6 + 3x - 26 + 90 = 180
\]

Combine like terms:

\[
(2x + 3x) + (6 - 26 + 90) = 180
\]

This results in:

\[
5x + 70 = 180
\]

Now isolate \(x\) by subtracting \(70\) from both sides:

\[
5x = 180 - 70
\]

\[
5x = 110
\]

Now, divide by \(5\):

\[
x = 22
\]

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

For angle \(A\):

\[
A = 2x + 6 = 2(22) + 6 = 44 + 6 = 50^\circ
\]

For angle \(B\):

\[
B = 3x - 26 = 3(22) - 26 = 66 - 26 = 40^\circ
\]

Thus, the angles of the triangle are:

- Angle \(A\): \(50^\circ\)
- Angle \(B\): \(40^\circ\)
- Right angle: \(90^\circ\)

The missing angles \(A\) and \(B\) are \(50^\circ\) and \(40^\circ\), respectively.