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Problem 308

Given two intersecting chords of a circle, show that the measure of the angle formed by the intersection is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Expert verified
Given two intersecting chords of a circle, let their point of intersection be O and the intercepted arcs be Arc AC, Arc CE, Arc EA, and Arc EB. We use the Inscribed Angle Theorem that states the measure of an inscribed angle is half the measure of its intercepted arc. Also, the sum of measures of an angle and its vertical angle is 180 degrees. Thus, we can establish relationships between angles and arcs: The measure of angle AOE + angle EOC is half the measure of Arc AC + Arc EB, and this sum equals the sum of the measures of angle AOC and angle COE (vertical angles). Therefore, the measure of the angle formed by the intersection (angle AOC or angle EOC) is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle (Arc AC + Arc EB).
See the step by step solution

Step 1: Identify the problem

The problem states that given two intersecting chords of a circle, the measure of the angle formed by the intersection is equal to half the sum of the measures of the arcs intercepted by the angle and its vertical angle. The task here is to establish and demonstrate this relation using the properties of angles and arcs in a circle.

Step 2: Draw an illustrative diagram

Start by drawing a circle and two intersecting chords inside the circle. Label the point of intersection as O. These intersecting chords will divide the circle into four sections, each associated with an arc. Label the intercepted arcs as Arc AC, Arc CE, Arc EA and Arc EB.

Step 3: Identify relevant geometric properties

Do you remember the Inscribed Angle Theorem? It states that the measure of an inscribed angle is half the measure of its intercepted arc. Also, remember that vertical angles are congruent (or equal in measure), and the sum of measures of an angle and its vertical angle is 180 degrees.

Step 4: Establish relationships between angles and arcs

By the Inscribed Angle Theorem, the measure of angle AOE is half the measure of Arc AC and the measure of angle EOC is half the measure of Arc EB. Hence, the measure of angle AOE + angle EOC is half the measure of Arc AC + Arc EB.

Step 5: Use the properties of vertical angles

Angle AOC and angle EOB are vertical angles, as they occur at the intersection of the chords, hence they are of equal measure. Additionally, angle AOE and angle COE are also vertical angles. This means, the sum of the measures of angle AOC and angle COE equals the sum of the measures of angle AOE and angle EOC.

Step 6: Finalize the theorem

From step 4, we know that the sum of the measures of angle AOE and angle EOC, which are vertical angles, is half the sum of the measures of Arc AC and Arc EB. From step 5, we know that this sum also equals the sum of the measures of angle AOC and angle COE. Therefore, the measure of the angle formed by the intersection (angle AOC or angle EOC) is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle (Arc AC + Arc EB). This is the proof of the theorem in question.

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