Question

find the measure of ∠xtu
u
91°
124°
y
45°
t
70°
30°
v
x
w

a. 45°
b. 91°
c. 136°
d. 125°

Explanation:

Step1: Recall circle - angle sum property

The sum of angles around a point is $360^{\circ}$. In a circle with center $T$, the sum of all central - angles is $360^{\circ}$. We want to find $\angle XTU$.

Step2: Calculate the sum of known angles

The known angles are $45^{\circ},70^{\circ},30^{\circ},124^{\circ},91^{\circ}$.

Step3: Find $\angle XTU$

We know that the sum of all angles around point $T$ is $360^{\circ}$. Let $\angle XTU = x$. Then $x+45^{\circ}+70^{\circ}+30^{\circ}+124^{\circ}+91^{\circ}=360^{\circ}$.
First, find the sum of the known angles: $45 + 70+30 + 124+91=360^{\circ}-x$.
$45+70 = 115$, $115+30 = 145$, $145+124 = 269$, $269+91 = 360$.
So, $\angle XTU=360-(45 + 70+30 + 124+91)=0^{\circ}$ (This is wrong. Let's use another way. Since the angles are non - overlapping central angles of a circle, we can also calculate $\angle XTU$ by adding the non - adjacent angles to it in the circle).
We know that the sum of angles around a point is $360^{\circ}$.
The angles that are not part of $\angle XTU$ are $124^{\circ}$ and $91^{\circ}$.
$\angle XTU=360-(124 + 91+45+30 + 70)$.
First, $124+91 = 215$, $215+45 = 260$, $260+30 = 290$, $290+70 = 360$.
We can also calculate $\angle XTU$ as follows:
The sum of the angles in the circle is $360^{\circ}$.
We know that $\angle XTY = 45^{\circ}$, $\angle YTU=91^{\circ}$.
$\angle XTU=\angle XTY+\angle YTU$.
$\angle XTU = 45^{\circ}+91^{\circ}=136^{\circ}$.

Answer:

C. $136^{\circ}$