Question

unit 6a test name:
1 find ( m angle b ) if ( m overparen{a b}=120^{circ} ) and ( m angle a=42^{circ} ).

a. ( 78^{circ} )
b. ( 156^{circ} )
c. ( 18^{circ} )
d. ( 60^{circ} )
2 if ( m overparen{a b}=30^{circ} ) and ( m overparen{c d}=110^{circ} ), find ( m angle e ).

a. ( 70^{circ} )
b. ( 15^{circ} )
c. ( 30^{circ} )
d. ( 40^{circ} )
3 find the value of each variable.

a. ( x=75^{circ} ) and ( y=78^{circ} )
b. ( x=78^{circ} ) and ( y=65^{circ} )
c. ( x=102^{circ} ) and ( y=115^{circ} )
d. ( x=71.5^{circ} ) and ( y=71.5^{circ} )

Explanation:

Step1: Use the property of inscribed quadrilaterals

In a circle, the sum of opposite angles of an inscribed quadrilateral is \(180^{\circ}\).

Step2: Analyze the given options

For question 1:
The measure of an inscribed angle is half the measure of its intercepted arc. Let's assume the relevant circle - angle relationship. If we consider the general formula for angles related to arcs in a circle. But since it's not clear from the problem description (the first figure is not fully described in terms of angle - arc relationships in a standard way, but if we assume a basic triangle - like situation in a circle).
For question 2:
The measure of an angle formed by two secants \(m\angle E=\frac{1}{2}(m\widehat{CD}-m\widehat{AB})\)
Substitute \(m\widehat{AB} = 30^{\circ}\) and \(m\widehat{CD}=110^{\circ}\)
\(m\angle E=\frac{1}{2}(110 - 30)\)
\(m\angle E=\frac{1}{2}\times80=40^{\circ}\)
For question 3:
In an inscribed quadrilateral, if we assume one pair of opposite angles. Let's say if \(x\) and \(y\) are angles of an inscribed quadrilateral. If we assume a standard inscribed quadrilateral where \(x + y= 180^{\circ}\) (opposite angles of an inscribed quadrilateral sum to \(180^{\circ}\)). But since the figure is not clear, if we go by the property that in an inscribed quadrilateral, the angle is half the sum or difference of arcs. But if we assume a basic inscribed quadrilateral property. If we consider the options:
For an inscribed quadrilateral, opposite angles are supplementary. If we assume \(x = 78^{\circ}\), then its opposite angle (if we assume a relationship) and \(y=102^{\circ}\) (not matching). But if we consider another property. Wait, if we consider the first problem (assuming it's a triangle - like in a circle, but no). For the second problem:
Using the formula \(m\angle E=\frac{1}{2}(m\widehat{CD}-m\widehat{AB})\)
\(m\angle E=\frac{1}{2}(110 - 30)=40^{\circ}\)

Answer:

1. (No clear solution as figure 1 description is insufficient).
2. D. \(40^{\circ}\)
3. (No clear solution as figure 3 description is insufficient).