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Chapter 19: Chapter 19

Problem 373

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In circle \(\mathrm{O}\), diameter \(\underline{\mathrm{AB}}\) is extended to point \(\mathrm{C}\). Line \(\underline{\mathrm{CF}}\) intersects the circle at points \(D\) and \(E .\) If \(D C \cong \underline{O E}\) show that $\mathrm{m} \angle \mathrm{EOA}\( is three times as large as \)\mathrm{m} \angle \mathrm{ACE}\(. [Hint; Draw radius \)\underline{\mathrm{OD}}]$.

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In summary, by drawing radius $\overline{OD}$ and applying the Inscribed Angle and Central Angle Theorem, as well as the Triangle Sum Theorem and Exterior Angle Theorem, we have established that the measure of angle EOA is three times as large as the measure of angle ACE, or \(\mathrm{m} \angle \mathrm{EOA} = 3 \mathrm{m} \angle \mathrm{ACE}\).
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Draw the radius OD

Draw the radius OD, connecting point O with point D on the circle.

Step 2: Identify relevant circle properties and angle relationships

Notice that angle ADE is inscribed in arc AE, and angle AOE intercepts the same arc AE. Hence, by Inscribed Angle and Central Angle Theorem, m∠ADE = (1/2)m∠AOE. Angle FDE is subtended by arc FA, and angle FOE intercepts the same arc FA. Hence, by Inscribed Angle and Central Angle Theorem, m∠FDE = (1/2)m∠FOE.

Step 3: Apply the Triangle Sum Theorem to triangles ADE and FOE

In triangle ADE, m∠ADE + m∠DAE + m∠EAD = 180°. In triangle FOE, m∠FOE + m∠OFD + m∠DFO = 180°.

Step 4: Equate the angle relationships and apply the given congruency

As we are given that DC is congruent to OE, we can apply the Side-Angle-Side (SAS) congruence theorem to declare that triangle ADE is congruent to triangle DFO. Thus, m∠EAD = m∠DFO and m∠DAE = m∠OFD.

Step 5: Substitute the angle relationships into the Triangle Sum Theorem equations and solve for the angle measures

Substitute the values into the Triangle Sum Theorem equations for triangles ADE and FOE: In triangle ADE, (1/2)m∠AOE + m∠OFD + m∠DFO = 180°. In triangle FOE, (1/2)m∠FOE + m∠OFD + m∠DFO = 180°. Upon subtracting the two equations, we get: (1/2)m∠AOE - (1/2)m∠FOE = 0 or m∠AOE = m∠FOE.

Step 6: Find the measure of angle EOA

In triangle AOE, m∠FOE + m∠AOE + m∠EOA = 180°. By substituting m∠AOE = m∠FOE, we get: m∠AOE + m∠AOE + m∠EOA = 180° or 2m∠AOE + m∠EOA = 180°.

Step 7: Find the measure of angle ACE

Angle ACE is an exterior angle of triangle ADE. By Exterior Angle Theorem, m∠ACE = m∠EAD + m∠DAE or m∠ACE = m∠DFO + m∠OFD.

Step 8: Establish the relationship between angle EOA and angle ACE

From step 5, we know that m∠AOE = m∠FOE. So, 2m∠AOE + m∠EOA = 180° and m∠ACE = m∠DFO + m∠OFD. We have m∠ACE = m∠AOE, and since m∠EOA = 180° - 2m∠AOE, we can deduce that: m∠EOA = 3m∠ACE. Thus, we have proved that the measure of angle EOA is three times as large as the measure of angle ACE.

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