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. (make it short and like a 7th grader wrote it)

Answer 1

In a right triangle, the sum of all angles is 180 degrees. Since one angle is 90 degrees (the right angle), the other two angles should add up to 90 degrees.

Let's set up the equation using the given angles:

1. First angle: \(2x + 6\) degrees
2. Second angle: \(3x - 26\) degrees

Now, we add these two angles and set them equal to 90 degrees:

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

Combine like terms:

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

This simplifies to:

\[
5x - 20 = 90
\]

Next, add 20 to both sides:

\[
5x = 110
\]

Now, divide by 5:

\[
x = 22
\]

Now we can find the measures of the angles. Substitute \(x = 22\) back into the angle equations:

1. First angle:
\(2x + 6 = 2(22) + 6 = 44 + 6 = 50\) degrees

2. Second angle:
\(3x - 26 = 3(22) - 26 = 66 - 26 = 40\) degrees

So, the two acute angles are 50 degrees and 40 degrees. The measures of all angles in the triangle are:

- 90 degrees (right angle)
- 50 degrees (first acute angle)
- 40 degrees (second acute angle)

That's it!