Section 6.3: Arcs and Angles
Inscribed angles are angles whose vertices are on a circle and that intersect an arc on the circle. The measure of an inscribed angle is half of the measure of the intercepted arc and half the measure of the central angle intersecting the same arc. Inscribed angles that intercept the same arc are congruent.
If an angle is inscribed in a semicircle, it will be half the measure of a semicircle (180 degrees), therefore measuring 90 degrees. Angles in semicircle is one way of finding missing missing angles and lengths. Pythagorean's theorem can be used to find missing lengths (remember that the diameter is the hypotenuse). Also, the measure of an angle formed by a chord to a tangent is half the intercepted arc.
A cyclic quadrilateral has vertices on the same circle and is inscribed in the circle. The opposite angles have the same endpoints (the other vertices) and together their intercepted arcs include the entire circle. Since the measure of an inscribed angle is half the intercepted arc, the sum of the opposite angles must be 180 degrees.
A secant is a line, ray, or line segment that intersects a circle in two places. Three points are covered: (1) secants that intersect in a circle which divide each other proportionally, (2) the angle formed by secants which intersects in a circle and is half the sum of the intercepted arcs, and (3) two secants drawn from the same point outside a circle that form an anglewhose measure is half the difference of the intercepted arcs.
If an angle is inscribed in a semicircle, it will be half the measure of a semicircle (180 degrees), therefore measuring 90 degrees. Angles in semicircle is one way of finding missing missing angles and lengths. Pythagorean's theorem can be used to find missing lengths (remember that the diameter is the hypotenuse). Also, the measure of an angle formed by a chord to a tangent is half the intercepted arc.
A cyclic quadrilateral has vertices on the same circle and is inscribed in the circle. The opposite angles have the same endpoints (the other vertices) and together their intercepted arcs include the entire circle. Since the measure of an inscribed angle is half the intercepted arc, the sum of the opposite angles must be 180 degrees.
A secant is a line, ray, or line segment that intersects a circle in two places. Three points are covered: (1) secants that intersect in a circle which divide each other proportionally, (2) the angle formed by secants which intersects in a circle and is half the sum of the intercepted arcs, and (3) two secants drawn from the same point outside a circle that form an anglewhose measure is half the difference of the intercepted arcs.
Notes and Powerpoints
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