In these lessons, we will learn

More Lessons on Circle Theorems and Geometry

### Inscribed Angles

### Bow Theorem

Now we will look at the Bow Theorem.

This video explains the inscribed angle property and how it can be used to find missing angles.

(1) Inscribed angles from equal arcs are equal.

(2) Arcs that contain equal angles are equal.
This video shows the two useful properties when solving problems that involve inscribed angles.

(1) The measure of the inscribed angle is half the measure of the central angle.

(2) Inscribed angles that intercept congruent arcs are congruent.

#### Inscribed Angles and Quadrilaterals and Arcs

What is the relationship between an inscribed angle and its intercepted arc? The angle is half the arc (or the arc is twice the angle). Also, if a quadrilateral is inscribed in a circle, opposite angle are supplementary.
This problem looks at the way theorems about inscribed angles can be used to solve a problem in high school geometry.

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- inscribed angles
- a circle theorem about inscribed angles which is sometimes called the Bow Theorem. It states that the inscribed angles subtended by the same arc or chord are equal.

More Lessons on Circle Theorems and Geometry

We will first look at what is meant by inscribed angle or angle at the circumference.

An inscribed angle has its vertex on the circle. ∠*ABC, *in the diagram below, is called an inscribed angle. The angle is also said to be **subtended by **(i.e. opposite to)** arc ADC** or chord

∠*ABC *is also called an **angle at the circumference.**

The theorem states

The inscribed angles subtended by the same arc or chord are equal.

or

Arcs that contain equal angles are equal.

or

Angles in the same segment are equal.

or

**The angles at the circumference subtended by the same arc are equal. **

∠

x= ∠ybecause they are subtended by the same arcAEC.

You can see from the above diagram that the inscribed angles form a "bow-tie". That is why it is sometimes called the bow theorem.

*Example: *

Find the value of ∠* x* in the figure below.

* Solution: *

∠* x* = 38˚ because they are both subtended by the same arc *PRQ. *

(1) Inscribed angles from equal arcs are equal.

(2) Arcs that contain equal angles are equal.

(1) The measure of the inscribed angle is half the measure of the central angle.

(2) Inscribed angles that intercept congruent arcs are congruent.

What is the relationship between an inscribed angle and its intercepted arc? The angle is half the arc (or the arc is twice the angle). Also, if a quadrilateral is inscribed in a circle, opposite angle are supplementary.

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You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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