Question

1. given the following diagram use angle relationships and reasoning to find the missing values.

a) segments ab and cb are tangent to circle p.
find the measure of ∠abc.

Explanation:

Step1: Recall tangent - radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. So, \( \angle PAB = 90^{\circ} \) and \( \angle PCB=90^{\circ} \) because \( AB \) and \( CB \) are tangents to circle \( P \) at points \( A \) and \( C \) respectively, and \( PA \), \( PC \) are radii.

Step2: Use the sum of interior angles of a quadrilateral

The sum of the interior angles of a quadrilateral is \( 360^{\circ} \). In quadrilateral \( PABC \), we know \( \angle PAB = 90^{\circ} \), \( \angle PCB = 90^{\circ} \), and \( \angle APC=120^{\circ} \) (given). Let \( \angle ABC=x \). Then, by the angle - sum property of a quadrilateral:
\( \angle PAB+\angle ABC + \angle PCB+\angle APC=360^{\circ} \)
Substitute the known values:
\( 90^{\circ}+x + 90^{\circ}+120^{\circ}=360^{\circ} \)

Step3: Solve for \( x \)

First, simplify the left - hand side of the equation:
\( 90 + 90+120+x=360 \)
\( 300 + x=360 \)
Subtract 300 from both sides:
\( x=360 - 300 \)
\( x = 60^{\circ} \)

Answer:

The measure of \( \angle ABC \) is \( 60^{\circ} \)