To find the measures of all three angles in a triangle given the conditions, we can proceed with the following steps:
1. Assign Variables:
- Let the smallest angle be denoted as \(\( x \)\) degrees.
2. Express the Other Angles:
- According to the problem, one angle measures five degrees more than twice the smallest angle. This angle can be written as:
\(\[
2x + 5
\]\)
- The largest angle measures eleven degrees less than three times the measure of the smallest angle. This angle can be written as:
\(\[
3x - 11
\]\)
3. Set Up the Triangle Angle Sum Equation:
- The sum of the angles in any triangle is always 180 degrees. Hence, we can set up the equation:
\(\[
x + (2x + 5) + (3x - 11) = 180
\]\)
4. Combine Like Terms and Solve for \(\( x \)\):
\(\[
x + 2x + 5 + 3x - 11 = 180
\]\)
\(\[
6x - 6 = 180
\]\)
\(\[
6x = 186
\]\)
\(\[
x = 31
\]\)
5. Find the Measures of All Angles:
- The smallest angle \(\( x \)\) is:
\(\[
x = 31 \text{ degrees}
\]\)
- The second angle \(\( (2x + 5) \)\) is:
\(\[
2(31) + 5 = 62 + 5 = 67 \text{ degrees}
\]\)
- The largest angle \(\( (3x - 11) \)\) is:
\(\[
3(31) - 11 = 93 - 11 = 82 \text{ degrees}
\]\)
6. Verify the Sum of the Angles:
- Sum of the angles:
\(\[
31 + 67 + 82 = 180 \text{ degrees}
\]\)
- This confirms that the angles add up to 180 degrees as required for a triangle.
So, the measures of the three angles in the triangle are:
- 31 degrees (smallest angle)
- 67 degrees (second angle)
- 82 degrees (largest angle)
