Arc Proof
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help with proof for "angles inside the circle" theorem?
hi! how do you prove the theorem stating that if two chords intersect inside a circle, then the measure of each angle is one half the measures of the arcs intercepted by the angle and its vertical angle?
the given being that the intersecting chords are segments AC and BD, and proving that m<1 = 1/2 (m of arc DC + m of arc AB)
also segment BC is drawn in
thanks so much!!
Use the theorem that tells you that angle with its vertex on the circle and rays passing through two other points on the circle intercepts an arc that is twice the measure of the angle. In your setup, assuming that ABCD are listed cyclically, then that means that
m∠ACB = m∠ADB = (1/2) m arc AB.
m∠CAD = m∠CBD = (1/2) m arc CD
Let X be (appropriately) the intersection of segments AD and BC. Then ∠BXC and ∠CXD make a straight angle. So do ∠BXC, ∠XBC and ∠XCB as the three angles of a triangle. So
m∠CXD = m∠XBC + m∠XCB
But m∠XBC = m∠CBD = (1/2) m arc CD
and m∠XCB = m∠ACB = (1/2) m arc AB
So m∠CXD = (1/2)m arc CD + (1/2)m arc AB