To determine the measure of the largest angle in a triangle where the angles are in the ratio [tex]\(6:1:5\)[/tex], we need to follow a series of steps.
1. Understand the Ratio:
The given ratio is [tex]\(6:1:5\)[/tex]. This means when compared to a common proportion, the angles in the triangle can be thought of as [tex]\(6x\)[/tex], [tex]\(1x\)[/tex], and [tex]\(5x\)[/tex], where [tex]\(x\)[/tex] is some positive constant that scales these ratios into actual angle measures.
2. Sum of Angles in a Triangle:
In any triangle, the sum of the internal angles is always [tex]\(180^\circ\)[/tex].
Therefore, we can set up the equation:
[tex]\[
6x + 1x + 5x = 180^\circ
\][/tex]
3. Combine Like Terms and Solve for [tex]\(x\)[/tex]:
Combining the terms on the left side, we get:
[tex]\[
12x = 180^\circ
\][/tex]
To find [tex]\(x\)[/tex], divide both sides of the equation by 12:
[tex]\[
x = \frac{180^\circ}{12} = 15^\circ
\][/tex]
4. Find Each Angle:
Now that we have [tex]\(x = 15^\circ\)[/tex], we can find the actual measures of each angle by multiplying [tex]\(x\)[/tex] by the respective ratio numbers:
- The first angle, [tex]\(6x\)[/tex], is:
[tex]\[
6 \times 15^\circ = 90^\circ
\][/tex]
- The second angle, [tex]\(1x\)[/tex], is:
[tex]\[
1 \times 15^\circ = 15^\circ
\][/tex]
- The third angle, [tex]\(5x\)[/tex], is:
[tex]\[
5 \times 15^\circ = 75^\circ
\][/tex]
5. Determine the Largest Angle:
Comparing the three angles, [tex]\(90^\circ\)[/tex], [tex]\(15^\circ\)[/tex], and [tex]\(75^\circ\)[/tex], it is evident that the largest angle is [tex]\(90^\circ\)[/tex].
Thus, the measure of the largest angle in the triangle is [tex]\(\boxed{90^\circ}\)[/tex].
