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Problem 330

# If two chords of a circle subtend different acute inscribed angles, the smaller angle belongs to the shorter chord. Prove the statement and its converse.

Expert verified
The proof has two parts: 1. For the given statement: If two chords of a circle subtend different acute inscribed angles, the smaller angle belongs to the shorter chord, use the inscribed angle theorem to find measures of arcs AB and CD. Since ∠ACB < ∠ADB, we get m(arc AB) < m(arc CD). The length of a chord is proportional to the length of its intercepted arc, so AB < CD. 2. For the converse: If AB < CD, then using the properties of chords and arcs, we have m(arc AB) < m(arc CD). Using the inscribed angle theorem, we get ∠ACB < ∠ADB. This completes the proof of the statement and its converse.
See the step by step solution

## Step 1: Diagram

Draw a circle with center O. Let AB and CD be two distinct chords of the circle. Let ∠ACB and ∠ADB be different acute inscribed angles such that ∠ACB < ∠ADB.

## Step 1: Prove Smaller Angle Belongs to Shorter Chord

We need to show that if ∠ACB < ∠ADB, then AB < CD. To prove this, we will first find the measures of the arcs AB and CD using the inscribed angle theorem. The inscribed angle theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. Therefore, we have: m(arc AB) = 2∠ACB and m(arc CD) = 2∠ADB Since ∠ACB < ∠ADB, we know: m(arc AB) < m(arc CD) Now, we'll use the fact that the length of a chord is proportional to the length of its intercepted arc. Thus, if m(arc AB) < m(arc CD), then AB < CD. This proves the first part of the statement.

## Step 2: Prove the Converse

Now, we need to show that if AB < CD, then ∠ACB < ∠ADB. If AB < CD, then by the same properties of chords and arcs that we used in Step 1, we know that: m(arc AB) < m(arc CD) Using the inscribed angle theorem again, we have: 2∠ACB = m(arc AB) and 2∠ADB = m(arc CD) Since m(arc AB) < m(arc CD), we know: 2∠ACB < 2∠ADB Divide both sides by 2 to get: ∠ACB < ∠ADB This proves the converse of the statement, completing the solution.

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