To solve for the measure of the third angle in the given triangle, we will follow a step-by-step approach.
1. Understand the Relationship Between the Angles:
- Let the measure of the first angle be [tex]\( x \)[/tex].
- The second angle is twice the measure of the first angle, so it is [tex]\( 2x \)[/tex].
- The third angle is three times the measure of the second angle, so it is [tex]\( 3 \times 2x = 6x \)[/tex].
2. Sum of the Angles in a Triangle:
- The sum of the angles in any triangle is always [tex]\( 180^\circ \)[/tex].
3. Set Up the Equation:
- We know that the sum of the first angle, the second angle, and the third angle is [tex]\( 180^\circ \)[/tex].
- Therefore, the equation is:
[tex]\[
x + 2x + 6x = 180^\circ
\][/tex]
4. Combine Like Terms:
- Combine the terms involving [tex]\( x \)[/tex]:
[tex]\[
9x = 180^\circ
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
- Divide both sides of the equation by 9 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{180^\circ}{9} = 20^\circ
\][/tex]
6. Find the Measure of the Third Angle:
- The third angle is given as [tex]\( 6x \)[/tex].
- Substitute [tex]\( x = 20^\circ \)[/tex] to find the measure of the third angle:
[tex]\[
6x = 6 \times 20^\circ = 120^\circ
\][/tex]
Therefore, the measure of the third angle is [tex]\( 120^\circ \)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{120^\circ} \][/tex]
