Suggested languages for you:

Americas

Europe

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).

What do you think about this solution?

We value your feedback to improve our textbook solutions.

- Access over 3 million high quality textbook solutions
- Access our popular flashcard, quiz, mock-exam and notes features
- Access our smart AI features to upgrade your learning

Chapter 16

Let \(\mathrm{m} \angle \mathrm{A}=90^{\circ}\) in \(\triangle \mathrm{ABC}\). Let \(\mathrm{D}, \mathrm{E}\), and \(\mathrm{F}\) be the midpoints of $\underline{A B}, \underline{A C}\(, and \)\underline{B C}\(, respectively. Prove that \)F$ is the center of a semicircle which contains \(\mathrm{B}, \mathrm{A}\) and \(\mathrm{C}\).

Chapter 16

Prove that inscribed angles which intercept the same arc are congruent.

Chapter 16

In circle \(\mathrm{O}, \underline{\mathrm{BD}}\) is a diameter, \(\underline{\mathrm{AB}}\) and \(\underline{\mathrm{BC}}\) are chords, and \(\mathrm{AB}>\mathrm{BC}\). Prove that $\mathrm{m} \angle \mathrm{ABD}<\mathrm{m} \angle \mathrm{CBD}$.

Chapter 16

Let \(\angle \mathrm{B}\) be an angle inscribed in a circle and let it have measure greater than \(90^{\circ}\). (See figure.) Prove that $$ \mathrm{m} \angle \mathrm{B}=180^{\circ}-(1 / 2) \mathrm{m} \angle \mathrm{P} $$

Chapter 16

Given: Points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\), and \(\mathrm{D}\) are in \(\mathrm{OP} ; \mathrm{AB} \cong \mathrm{AD}^{\prime}\); \(\mathrm{BC}^{-} \cong \mathrm{DC}^{-}\) Prove: $\angle \mathrm{B} \cong \angle \mathrm{D}$

The first learning app that truly has everything you need to ace your exams in one place.

- Flashcards & Quizzes
- AI Study Assistant
- Smart Note-Taking
- Mock-Exams
- Study Planner