Let's solve the problem step-by-step.
Given: A triangle [tex]\(ABC\)[/tex] is inscribed in a circle such that [tex]\(\overline{AB}\)[/tex] is a diameter. The measure of the arc [tex]\( \overparen{BC} \)[/tex] is [tex]\(134^\circ\)[/tex].
We need to find the measures of the angles of the triangle [tex]\(ABC\)[/tex].
1. Understanding the Information:
- Since [tex]\(\overline{AB}\)[/tex] is a diameter of the circle, angle [tex]\( \angle ACB \)[/tex] subtended by this diameter will be a right angle ([tex]\(90^\circ\)[/tex]). This is due to the fact that any angle inscribed in a semicircle is a right angle.
[tex]$ \angle ACB = 90^\circ $[/tex]
2. Using the Arc Measure:
- The given arc [tex]\( \overparen{BC} \)[/tex] has an arc measure of [tex]\(134^\circ\)[/tex].
- The angle subtended by this arc at the circumference of the circle (which would be [tex]\(\angle BAC\)[/tex] or [tex]\(\angle BCA\)[/tex]) is half of the arc measure.
- Thus, the angle [tex]\( \angle ABC \)[/tex], which is subtended by the arc [tex]\( \overparen{BC} \)[/tex], will be half of [tex]\(134^\circ\)[/tex]:
[tex]$ \angle ABC = \frac{134^\circ}{2} = 67^\circ $[/tex]
3. Finding the Third Angle:
- Knowing that the sum of angles in a triangle is [tex]\(180^\circ\)[/tex], we can find the third angle [tex]\( \angle BAC \)[/tex]:
[tex]$ \angle BAC = 180^\circ - \angle ABC - \angle ACB $[/tex]
Substituting the known values:
[tex]$ \angle BAC = 180^\circ - 67^\circ - 90^\circ $[/tex]
[tex]$ \angle BAC = 23^\circ $[/tex]
4. Summary of Angle Measures:
- Therefore, the measures of the angles in the triangle [tex]\(ABC\)[/tex] are:
[tex]\[
\angle ACB = 90^\circ, \quad \angle ABC = 67^\circ, \quad \angle BAC = 23^\circ
\][/tex]
So the final answer is [tex]\( \boxed{90^\circ, 67^\circ, 23^\circ} \)[/tex].
