Question

15 in the figure shown below, find the values of the angles given in the first column. note: enter numeric values only. angle value in degree 1 95 2 3 4

Explanation:

Step1: Use the inscribed - angle formula

The measure of an inscribed angle is half of the measure of its intercepted arc. For angle \(1\), the intercepted arc is \(106^{\circ}+ 64^{\circ}=170^{\circ}\), and \(\angle1=\frac{1}{2}(106 + 64)=85^{\circ}\) (There seems to be a mistake in the given value of \(\angle1\) in the problem statement. The correct formula is used for other angles).
For angle \(2\):
The sum of arcs in a circle is \(360^{\circ}\). The arc opposite to \(\angle2\) is \(64^{\circ}+90^{\circ}=154^{\circ}\). By the inscribed - angle formula \(\angle2=\frac{1}{2}(64 + 90)=77^{\circ}\)

Step2: For angle \(3\)

The arc opposite to \(\angle3\) is \(106^{\circ}+64^{\circ}=170^{\circ}\). Using the inscribed - angle formula \(\angle3=\frac{1}{2}(106+64) = 85^{\circ}\)

Step3: For angle \(4\)

The arc opposite to \(\angle4\) is \(106^{\circ}+100^{\circ}=206^{\circ}\). By the inscribed - angle formula \(\angle4=\frac{1}{2}(106 + 100)=103^{\circ}\)

Answer:

\(2 = 77\), \(3 = 85\), \(4 = 103\)