To find the size of each angle in a triangle where the angles are in the ratios [tex]\(1: 2: 3\)[/tex], we can follow these steps:
1. Express the angles in terms of a common variable:
- Let the angles be [tex]\(x\)[/tex], [tex]\(2x\)[/tex], and [tex]\(3x\)[/tex].
2. Sum of angles in a triangle:
- The sum of angles in a triangle is always [tex]\(180^\circ\)[/tex].
3. Set up the equation:
- Therefore, [tex]\(x + 2x + 3x = 180^\circ\)[/tex].
4. Simplify the equation:
- Combine the terms on the left-hand side: [tex]\(6x = 180^\circ\)[/tex].
5. Solve for [tex]\(x\)[/tex]:
- Divide both sides of the equation by 6:
[tex]\[
x = \frac{180^\circ}{6} = 30^\circ
\][/tex]
6. Determine each angle:
- The first angle is [tex]\(x\)[/tex]:
[tex]\[
\text{Angle 1} = 30^\circ
\][/tex]
- The second angle is [tex]\(2x\)[/tex]:
[tex]\[
\text{Angle 2} = 2 \times 30^\circ = 60^\circ
\][/tex]
- The third angle is [tex]\(3x\)[/tex]:
[tex]\[
\text{Angle 3} = 3 \times 30^\circ = 90^\circ
\][/tex]
Thus, the three angles of the triangle are [tex]\(30^\circ\)[/tex], [tex]\(60^\circ\)[/tex], and [tex]\(90^\circ\)[/tex].
