*Theorem 73:* If a tangent and a diameter meet at the point of tangency, then they are perpendicular to one another.

In Figure 1*B*. According to *Theorem 73*, *m* ∠ *ABC* = 90° and *m* ∠ *ABD* = 90°.

**Figure 1 **A tangent to the circle and a diameter of the circle meeting at the point of tangency.

*Theorem 74:* If a chord is perpendicular to a tangent at the point of tangency, then it is a diameter.

**Example 1:** *Theorem 74* could be used to find the center of a circle if two tangents to the circle were known. In Figure 2*P*, is tangent to the circle at *S*. Use these facts to find the center of the circle.

**Figure 2 **Finding the center of a circle when two tangents to the circle are known.

According to *Theorem 74*, if a chord is drawn perpendicular to at *P*, it is a diameter, which means that it passes through the center of the circle.

Similarly, if a chord is drawn perpendicular to at *S*, it too would be a diameter and pass through the center of the circle. The point where these two chords intersect would then be the center of the circle. See Figure 3

**Figure 3 **Chords drawn perpendicular to tangents to help in finding the center of the circle.

*Theorem 75:* The measure of an angle formed by two chords intersecting inside a circle is equal to half the sum of the measures of the intercepted arcs associated with the angle and its vertical angle counterpart.

In Figure 4*E*.

**Figure 4** Angles formed by two chords intersecting inside a circle.

By *Theorem 75:*,

*Theorem 76:* The measure of an angle formed by a tangent and a chord meeting at the point of tangency is half the measure of the intercepted arc.

In Figure 5*R*. By *Theorem 76*, *m* ∠1 = 1/2 ( *m* ) and *m* ∠ 2 = ½ ( *m* ).

**Figure 5**A tangent to the circle and a chord meeting at the point of tangency

*Theorem 77:* The measure of an angle formed by two secants intersecting outside a circle is equal to one half the difference of the measures of the intercepted arcs.

In Figure *G.* According to *Theorem 77*, *m* ∠1 = 1/2( *m* – *m* ).

**Figure 6 **Two secants to the circle meeting outside the circle.

**Example 2:** Find *m* ∠1 in Figures 7

**Figure 7 **Angles formed by intersecting chords, secants, and/or tangents.

**Example 3:** Find the value of *y* in Figures 8

**Figure 8 **Angles formed by intersecting chords, secants, and/or tangents.