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Geometry - Circles - Other Angles
Other Angles
Theorem 73: If a tangent and a diameter meet at the point of tangency, then they are perpendicular to one another.
In Figure 1 , diameter AB meets tangent at B. According to Theorem 73, AB ⊥ which means that 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 , is tangent to the circle at 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 , chords AC and BD intersect inside the circle at 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 , chord QR and tangent meet at 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 6 , secants and intersect at 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 (a) through (d).
Figure 7
Angles formed by intersecting chords, secants, and/or tangents.
Example 3: Find the value of y in Figures 8 (a) through (d).
Figure 8
Angles formed by intersecting chords, secants, and/or tangents
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