More geometry proof help?

Question

The measure of an inscribed arc is half the measure of the central angle that intercepts the same endpoints

Answer 1

Start by assuming that one of the line segments of the inscribed angle is a diameter. In that case, you can draw a radius from the center of the circle (also the midpoint of one of the segments) to the endpoint of the other segment, constructing an isoceles triangle. This also creates an exterior angle to the triangle that is a central angle of the circle, subtending the arc defined by the inscribed angle. Label the congruent angles of the triangle as a and the exterior angle as b. Triangle geometry shows you that b is equal to the sum of the other two angles of the triangle, both equal to a, so b is twice a, QED.

When neither of the line segments is a diameter, construct a diameter and treat the subtended arc as either a sum of two arcs subtended by a diameter and another segment, or as the difference of two such arcs.

Answer 2

I have slightly reworded your problem statement as follows:

"The measure of an inscribed angle is half the measure of the central angle that intercepts the same arc."

The proof of this theorem requires you to look at three different cases.
Case 1: One side of the inscribed angle lies on the circle center
Case 2 The circle lies between the two sides.
Case 3 The circle center lies outside the inscribed angle.

Draw circle with center O. Label two points on circle O A and B.
Draw a third point A on the circle. Then the angle BAC is an inscribed angle and we are to prove it is 1/2 the measure of
arc BC. This is the same as proving that the inscribed anle is 1/2 the measure of the central angle because the measur of the central angle = measur of arc.

Case 1 proof:
Draw a line from A through O to C. Draw AB and OC.
1. measure of angle BOC measure of arc BC {definition}
2. measure angle BOC = measure angle BAC + m angle OBA
because exterior Angle = sum of non adjacent interior angles.
3. But angle BAC = angle OBA because OA = OB
4 Therefore angle BOC = twice the measure angle BAC {Substiution Axiom}
5. Therefore angle BAC = 1/2 measure of Angle BOC or arc BC
{This is just another way of stating Step 4.}

Case 2

This time O lies between the sides of the inscribed angle. So draw a line from A through O to D. Now the arc AC = arc BD + arc CD. So use case 1 twice and add the angles and arcs.

Case 3

Same as Case 2 except Use Case1 twice and subtract the angles and arcs.

Answer 3

education.yahoo.com/homework_help/math_help/problem_list?

mathforum.org/dr.math/faq/faq.proof.html

shs.issaquah.wednet.edu/teachers/.../Geometry/Unit 10 Answer Key.doc