To find the two acute angles of a right-angled triangle given that they are in the ratio [tex]\(1:2\)[/tex], follow these steps:
1. Understand the problem: In a right-angled triangle, one angle is always [tex]\(90^\circ\)[/tex]. Let's denote the other two acute angles as [tex]\(x\)[/tex] and [tex]\(2x\)[/tex], given that their ratio is [tex]\(1:2\)[/tex].
2. Sum of angles in a triangle: The sum of the angles in any triangle is [tex]\(180^\circ\)[/tex]. In a right-angled triangle, since one angle is [tex]\(90^\circ\)[/tex], the sum of the two acute angles must be [tex]\(90^\circ\)[/tex].
3. Set up the equation: Since the acute angles are [tex]\(x\)[/tex] and [tex]\(2x\)[/tex], their sum is [tex]\(x + 2x = 3x\)[/tex]. According to our knowledge of triangle angle sums:
[tex]\[
x + 2x + 90^\circ = 180^\circ
\][/tex]
4. Simplify the equation: Start by combining like terms:
[tex]\[
3x + 90^\circ = 180^\circ
\][/tex]
5. Solve for [tex]\(x\)[/tex]: Subtract [tex]\(90^\circ\)[/tex] from both sides of the equation:
[tex]\[
3x = 180^\circ - 90^\circ
\][/tex]
[tex]\[
3x = 90^\circ
\][/tex]
Now, divide both sides by [tex]\(3\)[/tex]:
[tex]\[
x = \frac{90^\circ}{3}
\][/tex]
[tex]\[
x = 30^\circ
\][/tex]
6. Find the two acute angles: We have one angle as [tex]\(x = 30^\circ\)[/tex]. The other angle, being twice this, is:
[tex]\[
2x = 2 \times 30^\circ = 60^\circ
\][/tex]
Therefore, the two acute angles of the right-angled triangle are [tex]\(30^\circ\)[/tex] and [tex]\(60^\circ\)[/tex].
