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 [tex]\( x \)[/tex] 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:
[tex]\[
2x + 5
\][/tex]
- The largest angle measures eleven degrees less than three times the measure of the smallest angle. This angle can be written as:
[tex]\[
3x - 11
\][/tex]
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:
[tex]\[
x + (2x + 5) + (3x - 11) = 180
\][/tex]
4. Combine Like Terms and Solve for [tex]\( x \)[/tex]:
[tex]\[
x + 2x + 5 + 3x - 11 = 180
\][/tex]
[tex]\[
6x - 6 = 180
\][/tex]
[tex]\[
6x = 186
\][/tex]
[tex]\[
x = 31
\][/tex]
5. Find the Measures of All Angles:
- The smallest angle [tex]\( x \)[/tex] is:
[tex]\[
x = 31 \text{ degrees}
\][/tex]
- The second angle [tex]\( (2x + 5) \)[/tex] is:
[tex]\[
2(31) + 5 = 62 + 5 = 67 \text{ degrees}
\][/tex]
- The largest angle [tex]\( (3x - 11) \)[/tex] is:
[tex]\[
3(31) - 11 = 93 - 11 = 82 \text{ degrees}
\][/tex]
6. Verify the Sum of the Angles:
- Sum of the angles:
[tex]\[
31 + 67 + 82 = 180 \text{ degrees}
\][/tex]
- 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)
