Interior Intersections
When chords, secants, and tangents intersect in a circle, special relationships exist between the angle and arc measures formed.
Figure 1 (inside the circle)
Find each measure:
- m∠AED = 77°
- If two secants or chords intersect inside a circle, then the measure of the angle formed is equal to half the sum of the measures of the intercepted arcs.
- m∠ZYW = 128°
- 620 = (43+81) / 2
- 128 = (13+x) * 73
- X = 183°
Figure 2 (on the circle)
Find each measure:
- m∠STR = 149°
- If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is equal to half the measure of its intercepted arc.
- m∠LK = 50°
- 181 - 62 = (x+74)
- 124 = x +74
- X = 50°
- m∠ỆT = 107°
- 101 = 1/(x+95)
- 202 = x+95
- X = 107°
Figure 3 (outside the circle)
Find each measure:
- m∠2DEG = 154°
- If a secant and a tangent intersect in the exterior of a circle, the measure of the angle formed is equal to half the measure of the intercepted arc.
- m∠XY = 58°
- m∠XZY = 2(157) = 302°
- m∠XY = 58°
m∠KLM = 74°
- 140 - 66 = (x+74)
- 74 = x
- X = 74°
m∠BCD = 24°
- (67-19) = 48
- 48 = 24°
Exterior Intersections
m∠KNM = 314°
- If secants and/or tangents intersect on the exterior of a circle, then the measure of the angle formed is equal to half the difference of the intercepted arcs.
m∠PQR = 132°
- (313-47) = 266
- 266 = 132°
- m∠QU = 116°
- 35 = 2(x-46)
- 70 = X-46
- X = 116⁰
- m∠ABC = 74°
- (210-62) = 148
- 148 = 74°
- m∠MK = 48°
- 56 = 1/2 (160-x)
- 112 = 160-X
- X = 48°
- m∠CDE = 74°
- (254-106) = (148)
- 148 = 74°
In summary, it is essential to understand the different measures and formulas to calculate the intersection of chords, secants, and tangents within and outside a circle. These relationships help in solving a variety of geometry problems and are crucial for understanding the properties of circles.
