Of course! Let's solve this step by step.
Given:
- The measures of the two acute angles in a right-angled triangle are 2x° and 3x° respectively.
Step-by-Step Solution:
1. Understanding the Context:
- In a right-angled triangle, one of the angles is always 90 degrees.
- The sum of all the angles in any triangle is always 180 degrees.
2. Calculate the Sum of Acute Angles:
- Since one angle is 90 degrees, the sum of the other two acute angles must be:
[tex]\[
180^\circ - 90^\circ = 90^\circ
\][/tex]
- Therefore, the sum of the two acute angles [tex]\( 2x \)[/tex] and [tex]\( 3x \)[/tex] is 90 degrees:
[tex]\[
2x + 3x = 90^\circ
\][/tex]
3. Solve for x:
- Combine the terms on the left side:
[tex]\[
5x = 90^\circ
\][/tex]
- To isolate [tex]\( x \)[/tex], divide both sides by 5:
[tex]\[
x = \frac{90^\circ}{5} = 18^\circ
\][/tex]
4. Determine Each Acute Angle:
- The measure of the first angle is [tex]\( 2x \)[/tex]:
[tex]\[
2x = 2 \times 18^\circ = 36^\circ
\][/tex]
- The measure of the second angle is [tex]\( 3x \)[/tex]:
[tex]\[
3x = 3 \times 18^\circ = 54^\circ
\][/tex]
Therefore, the two acute angles in the right-angled triangle are:
- The first angle: [tex]\( 36^\circ \)[/tex]
- The second angle: [tex]\( 54^\circ \)[/tex]
So, the sizes of the acute angles are 36 degrees and 54 degrees, respectively.
