Inscribed Angles

A circle is unique because it does not have any corners or angles, which makes it different from other figures such as triangles, rectangles, and triangles. But specific properties can be explored in detail by introducing angles inside a circle. For instance, the simplest way to create an angle inside a circle is by drawing two chords such that they start at the same point. This might seem unnecessary at first, but by doing so, we can employ many rules of trigonometry and geometry, thus exploring circle properties in more detail.

Get started

Millions of flashcards designed to help you ace your studies

Sign up for free

Review generated flashcards

Sign up for free
You have reached the daily AI limit

Start learning or create your own AI flashcards

StudySmarter Editorial Team

Team Inscribed Angles Teachers

  • 7 minutes reading time
  • Checked by StudySmarter Editorial Team
Save Article Save Article
Contents
Contents
Table of contents

    Jump to a key chapter

      What is an Inscribed Angle of a Circle?

      Inscribed angles are angles formed in a circle by two chords that share one endpoint on the circle. The common endpoint is also known as the vertex of the angle. This is shown in figure 1, where two chords AB¯ and BC¯ form an inscribed angle m<ABC, where the symbol ‘m<' is used to describe an inscribed angle.

      Inscribed Angles, Inscribed Angles, StudySmarterInscribed Angles, StudySmarter Originals

      The other endpoints of the two chords form an arc on the circle, which is the arc AC shown below. There are two kinds of arcs that are formed by an inscribed angle.

      • When the measure of the arc is less than a semicircle or 180°, then the arc is defined as a minor arc which is shown in figure 2a.

      • When the measure of the arc is greater than a semicircle or 180°, then the arc is defined as a major arc which is shown in figure 2b.

      But how do we create such an arc? By drawing two cords, as we discussed above. But what exactly is a chord? Take any two points on a circle and join them to make a line segment:

      A chord is a line segment that joins two points on a circle.

      Inscribed Angles, Major arc and minor arc of a circle, StudySmarterMajor arc and Minor arc of a circle, StudySmarter Originals

      Now that a chord has been defined, what can one build around a chord? Let‘s start with an arc, and as obvious as it sounds, it is a simple part of the circle defined below:

      An arc of a circle is a curve formed by two points in a circle. The length of the arc is the distance between those two points.

      • An arc of a circle that has two endpoints on the diameter, then the arc is equal to a semicircle.
      • The measure of the arc in degrees is the same as the central angle that intercepts that arc.

      The length of an arc can be measured using the central angle in both degrees or radians and the radius as shown in the formula below, where θ is the central angle, and π is the mathematical constant. At the same time, r is the radius of the circle.

      Arc length (degrees)= θ 360 · 2π·r Arc length ( radians) = θ·r

      Inscribed Angles Formula

      Several types of inscribed angles are modeled by various formulas based on the number of angles and their shape. Thus a generic formula cannot be created, but such angles can be classified into certain groups.

      Inscribed Angle Theorems

      Let's look at the various Inscribed Angle Theorems.

      Inscribed angle

      The inscribed angle theorem relates the measure of the inscribed angle and its intercepted arc.

      It states that the measure of the inscribed angle in degrees is equal to half the measure of the intercepted arc, where the measure of the arc is also the measure of the central angle.

      m<ABC = 12·m<AOC

      Inscribed Angles, Inscribed Angle Theorem,  StudySmarterInscribed Angle Theorem, StudySmarter Originals

      Inscribed angles in the same arc

      When two inscribed angles intercept the same arc, then the angles are congruent. Congruent angles have the same degree measure. An example is shown in figure 4, where m<ADC and m<ABC and m<ABC are equal as they intercept the same arc AC:

      m<ABC=m<ADC

      Congruent Inscribed Angles, Inscribed Angles, StudySmarterCongruent Inscribed Angles, StudySmarter Originals

      Inscribed angle in a Semicircle

      When an inscribed angle intercepts an arc that is a semicircle, the inscribed angle is a right angle equal to 90°. This is shown below in the figure, where arc AB is a semicircle with a measure of 180° and its inscribed angle m<ACB is a right angle with a measure of 90°.

      Inscribed Angles, Inscribed Angle in a semicircle, StudySmarterInscribed Angle in a Semicircle, StudySmarter Originals

      Inscribed Quadrilateral

      If a quadrilateral is inscribed in a circle, which means that the quadrilateral is formed in a circle by chords, then its opposite angles are supplementary. For example, the following diagram shows an inscribed quadrilateral, where m<A is supplementary to m<C and m<B is supplementary to m<D:

      m<B+m<D=180°

      m<A+m<C=180°

      Inscribed Angles, Inscribed Quadrilateral, StudySmarterInscribed Quadrilateral, StudySmarter Originals

      Inscribed Angles Examples

      Find angles m<ABC and m<ACD if the central angle m<AQD shown below is 75°.

      Inscribed Angles, Inscribed Angles Example , StudySmarterInscribed angles example, StudySmarter Originals

      Solution:

      Since angles m<ACD and m<ABD intercept the same arc AD, then they are congruent.

      m<ABD= m <ACD

      Using the inscribed angle theorem, we know that the central angle is twice the inscribed angle that intercepts the same arc.

      m<AQD = 2·m<ACD 75° = 2·m<ACD M<ACD = 37.5°

      Hence the angle is 37.5°.

      What is the measure of angle m<ABD in the circle shown below if m<ACD is 30°?

      Inscribed Angles, Congruent Inscribed Angles , StudySmarterCongruent Inscribed Angles, StudySmarter Originals

      Solution:

      As angles m<ABD and m<ACD intercept the same arc , then they are equal . Hence, if m<ACD is 30° then m<ABD must also be 30°.

      Method for Solving Inscribed Angle Problems

      To solve any example of inscribed angles, write down all the angles given. Recognize the angles given by drawing a diagram if not given. Let’s look at some examples.

      Find m<ABC if its intercepted arc has a measure of 80°.

      Solution:

      Using the inscribed angle theorem, we derive that the inscribed angle equals half of the central angle.

      m<ABC = 12·m<AOC m<ABC = 802=40 °

      Find m<C and m<D in the inscribed quadrilateral shown below.

      Inscribed Angles, Inscribed Quadrilateral Example, StudySmarterInscribed quadrilateral Example, StudySmarter Originals

      Solution:

      As the quadrilateral shown is inscribed in a circle, its opposite angles are complementary.

      <A + <C = 180° <B + <D = 180 °

      Then we substitute the given angles into the equations, and we re-arrange the equations to make the unknown angle the subject.

      98°+<C = 180° <C= 180°-98° = 82° 85° +<D = 180° <D = 180°- 85°=95°

      Find m<b, m<d, and m<c in the diagram below.

      Inscribed Angles, An Inscribed quadrilateral, StudySmarterAn inscribed quadrilateral, StudySmarter Originals

      Solution:

      Inscribed angles m<BAC and m<BDC intercept the same arc BC. Hence they are equal, therefore

      <d = 50°

      Angle m<BCD is inscribed in a semicircle. Hence <c must be a right angle.

      <c = 90°

      As quadrilateral ABCD is inscribed in a circle, its opposite angles must be supplementary.

      <B + <D = 180 ° B + (d+35) = 180° B= 180-50-35 <b= 95 °

      Inscribed Angles - Key takeaways

      • An inscribed angle is an angle formed in a circle by two chords with a common end point that lies on the circle.
      • Inscribed angle theorem states that the inscribed angle is half the measure of the central angle.
      • Inscribed angles that intercept the same arc are congruent.
      • Inscribed angles in a semicircle are right angles.
      • If a quadrilateral is inscribed in a circle, its opposite angles are supplementary.
      Inscribed Angles Inscribed Angles
      Learn with 0 Inscribed Angles flashcards in the free StudySmarter app

      We have 14,000 flashcards about Dynamic Landscapes.

      Sign up with Email

      Already have an account? Log in

      Frequently Asked Questions about Inscribed Angles

      What is an inscribed angle?

      An inscribed angle is an angle that is  formed in a circle by two chords that have a common end point that lies on the circle.

      What is the difference between inscribed and central angles?

      A central angle is formed by two line segments that are equal to the radius of the circle and inscribed angles are formed by two chords, which are line segments that intersect the circle in two points.

      How to solve inscribed angles?

      Inscribed angles can be solved using the various inscribed angles theorem, depending on the angle, number of angles and the polygons formed in the circle.

      What is the formula for calculating inscribed angles?

      There is not a general formula for calculating inscribed angles. Inscribed angles can be solved using the various inscribed angles theorem, depending on the angle, number of angles and the polygons formed in the circle.

      What is an example of an inscribed angle?

      A typical example would be a quadrilateral inscribed in a circle where the angles formed at the corners are inscribed angles.

      Save Article

      Discover learning materials with the free StudySmarter app

      Sign up for free
      1
      About StudySmarter

      StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

      Learn more
      StudySmarter Editorial Team

      Team Math Teachers

      • 7 minutes reading time
      • Checked by StudySmarter Editorial Team
      Save Explanation Save Explanation

      Study anywhere. Anytime.Across all devices.

      Sign-up for free

      Sign up to highlight and take notes. It’s 100% free.

      Join over 22 million students in learning with our StudySmarter App

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

      • Flashcards & Quizzes
      • AI Study Assistant
      • Study Planner
      • Mock-Exams
      • Smart Note-Taking
      Join over 22 million students in learning with our StudySmarter App
      Sign up with Email