The Inscribed Angle Theorem is a fundamental concept in circle geometry. An inscribed angle is formed by two chords that intersect on the circle. The theorem states that the measure of an inscribed angle is exactly half the measure of its intercepted arc. In this diagram, angle alpha at point A is an inscribed angle. The central angle at point O that subtends the same arc is twice the measure of the inscribed angle, which is two alpha. This relationship holds true for any inscribed angle in a circle.
The Inscribed Angle Theorem holds true in three different cases, depending on the position of the center of the circle relative to the inscribed angle. In Case 1, the center O is inside the angle. In Case 2, the center O is on one of the sides of the angle. And in Case 3, the center O is outside the angle. In all three cases, the fundamental relationship remains the same: the measure of the inscribed angle is always half the measure of its intercepted arc. This can be written as angle A equals one-half times arc BC.
Let's prove the Inscribed Angle Theorem for Case 1, where the center O is inside the angle. First, we draw radii from the center O to points A, B, and C. Since all radii of a circle are equal, triangle AOB is isosceles with OA equals OB. Similarly, triangle AOC is isosceles with OA equals OC. In an isosceles triangle, the base angles are equal. Using the exterior angle property of triangles, we can show that the inscribed angle alpha equals half the sum of the central angles beta and gamma. Since the sum of these central angles equals the measure of arc BC, we conclude that the inscribed angle alpha equals half the measure of its intercepted arc BC.
The Inscribed Angle Theorem has several important applications in geometry. First, inscribed angles that intercept the same arc are equal. In this example, angles BAC and BDC intercept the same arc BC, so they are equal. Second, any angle inscribed in a semicircle is a right angle. This is because the intercepted arc is a semicircle measuring 180 degrees, so the inscribed angle measures half of that, which is 90 degrees. Third, in a cyclic quadrilateral, opposite angles are supplementary, meaning they sum to 180 degrees. This is because each pair of opposite angles intercepts arcs that together make up the entire circle, which measures 360 degrees. These applications make the Inscribed Angle Theorem a powerful tool in geometric proofs and constructions.
To summarize what we've learned about the Inscribed Angle Theorem: First, the measure of an inscribed angle equals half the measure of its intercepted arc. This fundamental relationship holds true in all three cases: when the center is inside the angle, when the center is on one of the sides of the angle, and when the center is outside the angle. The theorem leads to several important applications: inscribed angles that intercept the same arc are equal; any angle inscribed in a semicircle is a right angle; and opposite angles in a cyclic quadrilateral are supplementary, meaning they sum to 180 degrees. These properties make the Inscribed Angle Theorem one of the most useful tools in circle geometry, with applications in mathematics, engineering, and computer graphics.