Circles, Angle Measures, Arcs, Central & Inscribed Angles, Tangents, Secants & Chords - Geometry

A central angle has its vertex at the center of the circle. The measure of the intercepted arc is equal to the measure of the central angle. For example, if angle ACB is 50 degrees, arc AB is also 50 degrees.

This angle is formed by a secant segment and a tangent segment intersecting outside the circle. Similar to the secant-secant angle, its measure is one-half the difference of the measures of the two intercepted arcs. If arc AC is 130 degrees and angle B is 30 degrees, arc DC is 70 degrees.

A tangent-tangent angle is formed by two tangent segments that intersect outside the circle. Its measure is one-half the difference of the measures of the major and minor intercepted arcs. If the major arc AXC is 220 degrees, the minor arc AC is 140 degrees, and angle B is 40 degrees.

Given a central angle BDC of 40 degrees, the intercepted arc BC is also 40 degrees. An inscribed angle BAC that intercepts the same arc BC will be half its measure, so angle BAC is 20 degrees.

If arc BC is 60 degrees, the inscribed angle A (intercepting BC) is 30 degrees. If AC is a diameter, the arc it creates is 180 degrees, making angle ABC (intercepting AC) 90 degrees. Using the sum of angles in a triangle, angle C is 60 degrees.

If angle D is 60 degrees, its intercepted arc BE is 120 degrees. If angle A intercepts the same arc BE, then angle A must also be 60 degrees, as angles intercepting the same arc are congruent.

Using the chord-chord angle formula, set up an algebraic equation with the given arc and angle expressions. Solve for x, then substitute x back into the expressions to find the measures of the arcs and angles. For x=8, arc AC is 90 degrees.

Calculate the chord-chord angle (AFE) using the sum of intercepted arcs (130+70)/2 = 100 degrees. Calculate the secant-secant angle (C) using the difference of intercepted arcs (130-70)/2 = 30 degrees.

An inscribed angle has its vertex on the circle. The measure of the inscribed angle is half the measure of its intercepted arc. Conversely, the arc is twice the value of the inscribed angle. For instance, if angle ABC is 30 degrees, arc AC is 60 degrees.

This angle is formed by a tangent segment and a chord that meet at a point on the circle. The intercepted arc is twice the value of the tangent-chord angle. If angle ABC is 25 degrees, then arc AB is 50 degrees.

A chord-chord angle is formed by the intersection of two chords inside a circle. The measure of this angle is half the sum of the measures of the two intercepted arcs. For example, if arc AC is 100 degrees and arc DE is 60 degrees, angle ABC (or DBE) is 80 degrees.

A secant-secant angle is formed by two secant segments that intersect outside the circle. The angle's measure is one-half the difference of the measures of the two intercepted arcs. For instance, if arc AC is 110 degrees and arc DE is 60 degrees, angle B is 25 degrees.