Question

With reference to the figure match the angles and arcs to their measures

Answer 1

Answer:

1. $\widehat{CE}=124^{\circ}$
2. $\widehat{CB}=114^{\circ}$
3. $\widehat{EB}=122^{\circ}$

Step-by-step explanation:

For the given circle O, AD, FA and FD are tangents.

The angle between tangent and radius is 90°.

Also $m\angle CDE=180-124=56^{\circ}$ and $m\angle CAB=180-144=66^{\circ}$

In a triangle sum of all angles is 180°, so

$m\angle EFB=180-56-66=58^{\circ}$

In a quadrilateral sum of all angle is 360°, so

$\widehat{CE}=m\angle COE=360-90-90-56=124^{\circ}$

$\widehat{CB}=m\angle COB=360-90-90-66=114^{\circ}$

$\widehat{EB}=m\angle EOB=360-90-90-58=122^{\circ}$

Answer 2

Angle BAC = (180 - 114) degrees = 66 degrees        [angle on a straight line]

Angle OCA = Angle OBA = 90 degrees                     [angle at the point where the tangent and the radius meet]

Thus, the measure of arc BC = (360 - 66 - 90 - 90) degrees = 114 degrees      [sum of interior angles of a quadrilateral]


Angle CDE = (180 - 124) degrees = 56 degrees        [angle on a straight line]

Angle OCD = Angle OED = 90 degrees                     [angle at the point where the tangent and the radius meet]

Thus, the measure of arc CE = (360 - 56 - 90 - 90) degrees = 124 degrees     [sum of interior angles of a quadrilateral]

Given that the measure of arc BC is 114 degrees and the measure of arc CE is 124 degrees, thus the measure of arc BE = (360 - 114 - 124) degrees = 122 degrees                                                      [angle at a point]

Angle OBF = Angle OEF = 90 degrees                     [angle at the point where the tangent and the radius meet]

Thus, angle BFE = (360 - 122 - 90 - 90) degrees = 58 degrees.