In the diagram below, PA and PB are tangent segments to circle O from P.  If m<P = 38, find m<ABO.

embizze | Certified Educator

First, since PA and PB are tangent segments to the circle, we know that angles PAO and PBO are right angles. (Radii drawn to a point of tangency are perpendicular to the tangent.)

Then the measure of angle AOB is 142. (PAOB is a quadrilateral and thus the sum of the interior angles is 360 degrees. We have accounted for two right angles and the given 38 degrees already, so the remaining angle has measure 142. Or you could know that the central angle and the angle formed by tangents meeting at a point are supplementary.)

Now triangle AOB is isosceles. (In the same or congruent circles all radii are congruent.) So angles OAB and OBA are congruent. (The base angles of an isosceles triangle are congruent.)

Since the measure of AOB is 142, the measures of angles BAO and ABO sum to 38. Since they are congruent, each angle must have measure 19.

Alternatively -- since PA and PB are tangents drawn from a point to a circle, they are congruent. So triangle PAB is isosceles. Then angles PAB and PBA are congruent and have measure 71. Since the radii are perpendicular to the tangents, the measure of angle PBO is 90; with the measure of PBA being 71 that means angle ABO has measure 19 from the angle addition postulate.

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The measure of angle ABO is 19.