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.

Inscribed Angles Inscribed Angles

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Contents
Table of contents

    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.
    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.

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