Answer:
a. 30°
b. 40°
c. 105°
d. 15°
Explanation:
Given various chords of a circle that form inscribed angles, you want to find the measures of the angles.
Relevant relations
The relations you need to use to answer this question are ...
- The measure of the arc of a circle is equal to the measure of the central angle it subtends.
- The measures of all arcs of a circle total 360°.
- The measures of the angles of a triangle total 180°
- The measure of an inscribed angle is half the measure of the arc it subtends. (All inscribed angles that subtend the same arc are congruent.)
Application
a.
The inscribed angle marked 'a' subtends the same arc as the inscribed angle marked 30°.
a = 30°
b.
The inscribed angle marked 'b' subtends the same arc as the inscribed angle marked 40°.
b = 40°
c.
The inscribed angle marked 'c' intercepts the circle at two points that divide the circle into a larger arc and a smaller one. The smaller arc subtends the central angle marked 150°, so has that measure. The larger arc measures 360° -150° = 210°. The angle marked 'c' has half that measure:
c = 210°/2
c = 105°
d.
The angle marked 'd' is a base angle of the isosceles triangle whose a.pex angle is 150°.
d = (180° -150°)/2
d = 15°
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Additional comment
The shaded quadrilateral that includes point O and angle 'c' could be redrawn as an inscribed quadrilateral by moving the vertex from point O to somewhere on the 210° arc of the circle. Then the inscribed angle at that vertex would be 150°/2 = 75°. (This does not change the angle at 'c'.)
The opposite angles of an inscribed quadrilateral are supplementary, so c = 180° -75° = 105°.