To find the measures of the angles in a triangle where one of the acute angles of a right triangle is [tex]\(63^{\circ}\)[/tex], follow these steps:
1. Understanding the Problem:
- In any triangle, the sum of the angles is [tex]\(180^{\circ}\)[/tex].
- In a right triangle, one of the angles is always [tex]\(90^{\circ}\)[/tex].
- We are given one of the acute angles as [tex]\(63^{\circ}\)[/tex].
2. Identify the Known Angles:
- We have a right angle: [tex]\[ 90^{\circ}. \][/tex]
- We are given one acute angle: [tex]\[ 63^{\circ}. \][/tex]
3. Write the Equation for the Sum of Angles in a Triangle:
- The sum of the angles in the triangle must equal [tex]\(180^{\circ}\)[/tex]. So, we can set up the equation:
[tex]\[
\text{Right Angle} + \text{First Acute Angle} + \text{Second Acute Angle} = 180^{\circ}.
\][/tex]
4. Insert the Known Values:
[tex]\[
90^{\circ} + 63^{\circ} + \text{Second Acute Angle} = 180^{\circ}.
\][/tex]
5. Solve for the Second Acute Angle:
- Combine the known angles:
[tex]\[
90^{\circ} + 63^{\circ} = 153^{\circ}.
\][/tex]
- Subtract from [tex]\(180^{\circ}\)[/tex] to find the second acute angle:
[tex]\[
180^{\circ} - 153^{\circ} = 27^{\circ}.
\][/tex]
6. Conclusion:
- So, the measures of the angles in the triangle are:
- The right angle: [tex]\(90^{\circ}\)[/tex].
- The given acute angle: [tex]\(63^{\circ}\)[/tex].
- The other acute angle: [tex]\(27^{\circ}\)[/tex].
Therefore, the measures of the angles are [tex]\( 63^{\circ}, 27^{\circ}, \)[/tex] and [tex]\( 90^{\circ} \)[/tex].
