The inscribed angle theorem mentions that the angle inscribed inside a circle is always half the measure of the central angle or the intercepted arc that shares the endpoints of the inscribed angle's sides. In a circle, the angle formed by two chords with the common endpoints of a circle is called an inscribed angle and the common endpoint is considered as the vertex of the angle. In this section, we will learn about the inscribed angle theorem, the proof of the theorem, and solve a few examples.
| 1. | What is Inscribed Angle Theorem? |
| 2. | Properties of Inscribed Angle Theorem |
| 3. | Proof of Inscribed Angle Theorem |
| 4. | FAQs on Inscribed Angle Theorem |
What is Inscribed Angle Theorem?
The inscribed angle theorem is also called the angle at the center theorem as the inscribed angle is half of the central angle. Since the endpoints are fixed, the central angle is always the same no matter where it is on the same arc between the endpoints. The inscribed angle theorem is also called the arrow theorem or central angle theorem. This theorem states that: The measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc. OR. An inscribed angle is half of a central angle that subtends the same arc. OR. The angle at the center of a circle is twice any angle at the circumference subtended by the same arc. We need to keep in mind these three terms for the theorem:
- An inscribed angle is an angle whose vertex lies on the circle with its two sides as the chords of the same circle.
- A central angle is an angle whose vertex lies at the center of the circle with two radii as the sides of the angle.
- The intercepted arc is an angle formed by the ends of two chords on a circle's circumference.
In the above image, AB = the intercepted arc, θ = the inscribed angle, and 2θ = the central angle.
Properties of Inscribed Angle Theorem
An inscribed angle theorem has three basic properties that are connected with the central angle, they are:
- The inscribed angle subtended by the same arc is equal. (see below image for reference)
- The inscribed angle in a semicircle is 90°.
- Central angles subtended by arcs are of the same length.
In the image above, we see that....
Proof of Inscribed Angle Theorem
To prove the inscribed angle theorem we need to consider three cases:
- Inscribed angle is between a chord and the diameter of a circle.
- Diameter is between the rays of the inscribed angle.
- Diameter is outside the rays of the inscribed angle.
Case 1. Inscribed angle is between a chord and the diameter of a circle.
Here we need to prove that ∠AOB = 2θ
In the above image, let us consider that ∆OBD is an isosceles triangle where OD = OB = radius of the circle. Therefore, ∠ODB = ∠DBO = inscribed angle = θ. The diameter AD is a straight line hence ∠BOD = 180 - ∠AOB(call it x). According to the angle sum property, ∠ODB + ∠DBO + ∠BOD = 180°.
θ + θ + (180 - x) = 180
2θ + 180 - x = 180
2θ - x = 180 - 180
2θ - x = 0
x = 2θ.
Therefore, ∠AOB = 2θ. Hence proved.
Case 2: Diameter is between the rays of the inscribed angle.
Here we need to prove that ∠ACB = 2θ
In the above image, we draw a diameter in dotted lines that bisect both the angles as seen i.e. θ = θ1 + θ2 and a = a1 + a2. From case 1, we already that a1 = 2θ1 and a2 = 2θ2. When we add the angles, we get:
a1 + a2 = 2θ1 + 2θ2
a1 + a2 = 2 (θ1 + θ2)
a1 + a2 = 2θ
a = 2θ
Hence proved that ∠ACB = 2θ.
Case 3: Diameter is outside the rays of the inscribed angle.
Need to prove a = 2θ in the below circle.
From the above circle, we already know,
a1 = 2θ1
2 (θ1 + θ) = a1 + a
But, a1 = 2θ1 and a2 = 2θ2. By substituting we get,
2θ1 + 2θ = 2θ1 + a
a = 2θ.
Hence proved.
Related Topics
Listed below are a few topics related to the inscribed angle theorem, take a look.
- Consecutive Interior Angle
- Exterior Angle Theorem
- Central Angle Calculator
- Bisect
FAQs on Inscribed Angle Theorem
What is Meant by Inscribed Angle Theorem?
Inscribed angle theorem is also called as central angle theorem where it states that the angle subtended by an arc at the center of the circle is double the angle subtended by it at any other point on the circumference of the circle.
What Does the Inscribed Angle Theorem State?
The inscribed angle theorem states that an angle inscribed in a circle is half of the central angle that is subtends the same arc on the circle.
What is an Inscribed Angle?
The angle subtended by an arc at any point on the circle is called an inscribed angle.
What is the Difference Between Central Angle and Inscribed Angle?
Central angle is the angle subtended by an arc at the center of a circle. Inscribed angle is an angle subtended by an arc at any point on the circumference of a circle.
FAQs
Inscribed angle is between a chord and the diameter of a circle. Here we need to prove that ∠AOB = 2θ In the above image, let us consider that ∆OBD is an isosceles triangle where OD = OB = radius of the circle. Therefore, ∠ODB = ∠DBO = inscribed angle = θ.
How do you prove the inscribed angle theorem? ›
Inscribed Angle Theorem:
The measure of an inscribed angle is half the measure of the intercepted arc. That is, m ∠ A B C = 1 2 m ∠ A O C . This leads to the corollary that in a circle any two inscribed angles with the same intercepted arcs are congruent. Here, ∠ A D C ≅ ∠ A B C ≅ ∠ A F C .
What is an inscribed angle and examples? ›
An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of a circle. This is different than the central angle, whose vertex is at the center of a circle. If you recall, the measure of the central angle is congruent to the measure of the minor arc.
What is the proof of the angle angle theorem? ›
Thus, it is true that angle C has the same measure as angle D. This means that there are two equal angles with an equal side between them. The angle C is equal to the angle D. Therefore, by the ASA congruency theorem, the triangle ABC is congruent to the triangle DEF, because A = E, C = D, and |AC| = |DE|.
How do you prove the angle theorem? ›
To prove this theorem, let's assume a pair of intersecting straight lines that form an angle A between them. Now, we know that any two points on a straight line form an angle of 180 degrees between them. So, for the given pair of lines, the remaining angles on both the straight lines would be 180 - A.
What are the four theorems on inscribed angles? ›
Inscribed Angles Intercepting Arcs Theorem
Inscribed angles that intercept the same arc are congruent. Angles Inscribed in a Semicircle Theorem Angles inscribed in a semicircle are right angles. Cyclic Quadrilateral Theorem The opposite angles of a cyclic quadrilateral are supplementary.
How do you prove the angle angle side theorem? ›
In angle-angle side(AAS) if two angles and the one non-included side of one triangle are congruent to two angles and the non-included side of another triangle, then these two triangles are congruent.
Is the central angle twice the inscribed angle proof? ›
A central angle is twice the measure of an inscribed angle subtended by the same arc. COB since both are subtended by arc(CB). CAB since both are subtended by arc(CB). Note that a consequence of this property is that any inscribed angle subtended by a semicircle is a right angle, as shown in the example above right.
What is the theorem of inscribed angle and intercepted arc? ›
The vertex of an inscribed angle can be anywhere on the circle as long as its sides intersect the circle to form an intercepted arc. The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of its intercepted arc.
What is the proof of the theorem? ›
The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. The Pythagorean theorem has at least 370 known proofs.
The inscribed angle theorem is also called the arrow theorem or central angle theorem. This theorem states that: The measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc. OR. An inscribed angle is half of a central angle that subtends the same arc.
What is a theorem proof in geometry? ›
A geometric proof is a deduction reached using known facts such as axioms, postulates, lemmas, etc. with a series of logical statements. While proving any geometric proof statements are listed with the supporting reasons.
Why is the inscribed angle theorem true? ›
The measure of ∠AOB, where O is the center of the circle, is 2α. The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle. Therefore, the angle does not change as its vertex is moved to different positions on the circle.
Is an inscribed angle half the arc? ›
Theorem 70: The measure of an inscribed angle in a circle equals half the measure of its intercepted arc.
Is an inscribed angle an angle with its vertex? ›
An inscribed angle is an angle with its vertex on the circle and whose sides are chords. The intercepted arc is the arc that is inside the inscribed angle and whose endpoints are on the angle.
How do you prove the corresponding angle theorem? ›
How do you prove the Corresponding Angles Theorem? Let there be two parallel lines crossed by a transversal forming an angle, a, and an adjacent angle, b, that is below it. These two angles must be supplementary since they form a straight angle.
How do you prove the central angle theorem? ›
The Central Angle Theorem states that the central angle from two chosen points A and B on the circle is always twice the inscribed angle from those two points. The inscribed angle can be defined by any point along the outer arc AB and the two points A and B.
How do you prove the angle sum theorem? ›
Consider a ∆ABC, as shown in the figure below. To prove the above property of triangles, draw a line PQ parallel to the side BC of the given triangle. Thus, the sum of the interior angles of a triangle is 180°.
How do you prove angle congruence? ›
The complement theorem states that if angle A and angle B are both complementary to the same angle, then A and B are congruent. The supplement theorem states that if angle A and angle B are both supplementary to the same angle, then A and B are congruent.
