In this explainer, we will learn how to find the measures of inscribed angles subtended by the same arc or by congruent arcs.

Letβs begin by defining the meaning of some of these key terms.

### Definition: Inscribed Angles

An inscribed angle is the angle that is formed by the intersection of two chords on the circumference of a circle. In the diagram below, is an inscribed angle.

The angle is also said to be *subtended* by arc .

There are a number of properties that apply to such angles. In this explainer, we will investigate one such property.

### Property: The Angles Subtended by the Same Arc Are Equal

In the following diagram, , since both angles are subtended by arc .

In a similar way, since and are both subtended by arc , they are equal.

This property is sometimes equivalently stated as βangles in the same segment are equal.β

This is sometimes informally referred to as the βbow tieβ property, since the pair of inscribed angles form the shape of a bow tie. It is important to note that this is an informal definition and should not be referred to in a mathematical proof or otherwise!

An incredibly powerful aspect of this property is that we can construct any number of angles subtended by arc , and they will all be equal. Similarly, any number of angles can be subtended by arc , and those will also all be equal.

Before we demonstrate an application of this inscribed angle property, we will investigate a short geometric proof. In this proof, we begin by defining the center of the circle to be and constructing radii and .

Next, we apply a known property. That is, the inscribed angle is half the central angle that subtends the same arc. In other words, the angle at the center is double the angle at the circumference. For ease, we will define , although we might have alternatively chosen . Hence,

Similarly, by the same property,

Thus, , as required.

We will now demonstrate a simple application of this property.

### Example 1: Finding the Measure of an Unknown Inscribed Angle That is Subtended by the Same Arc in a Circle with Another Given Inscribed Angle

Given that and , find and .

### Answer

While not entirely necessary, it can be sensible to begin by adding the given angles to the diagram. and , so the diagram is as shown.

Next, we observe that the first unknown angle, , is subtended by the same arc, , as . We know that the angles subtended by the same arc are equal, so

Similarly, is subtended by the same arc, , as . Hence,

In our first example, we demonstrated an application of the inscribed angles property using numeric expressions. We can also apply this property to solve problems involving algebraic expressions. In our second example, we will see what this process looks like.

### Example 2: Solving Equations Using Two Inscribed Angles Subtended by the Same Arc in a Circle

If and , determine the value of .

### Answer

We recall that angles subtended by the same arc are equal. In this diagram, and are both subtended by arc , so these two angles are equal. This allows us to form and solve an equation in . Since and ,

Hence, .

An especially interesting property that results from the properties of inscribed angles is that the angle inscribed in a semicircle is . This is demonstrated in the following diagram, where is the diameter of the circle and .

In our next example, we will combine this property with angle facts to find missing values.

### Example 3: Finding the Measure of an Unknown Inscribed Angle Using Another Inscribed Angle Subtended by Congruent Arcs in a Circle

Given that is a diameter of the circle and , find .

### Answer

With questions involving a lot of details, it can be difficult to work out exactly how to get started. In these cases, we can begin by finding any βeasy-to-calculateβ angles.

Recall that the angle inscribed in a semicircle is . Using this property, we can see that .

Since the interior angles in a triangle add to , we can work out the measure of :

This is useful since we also know that line segments and are parallel, so we can use the fact that alternate angles are equal to calculate .

Finally, we observe that and are subtended by the same arc, . Since angles inscribed by the same arc are equal,

So, is .

It comes as no surprise that we can extend the properties of inscribed angles to work with distinct circles or congruent arcs. In particular, if a pair of circles are congruent, then inscribed angles subtended by congruent arcs, or arcs of equal measure, will be equal.

In the following diagram showing a pair of congruent circles, if , then .

Similarly, all inscribed angles subtended by congruent arcs in a circle are equal in measure. It might be tempting to look for the typical βbow tieβ shape, but in our next example we will demonstrate why that is not always sensible.

### Example 4: Solving Equations Using Two Inscribed Angles Subtended by Two Congruent Arcs in a Circle

Given that and , calculate the value of .

### Answer

Recall that inscribed angles subtended by congruent arcs in a circle are equal in measure. In this diagram, we see that arc is congruent to arc . The angles inscribed by these arcs are and , respectively, so it follows that .

Hence, we can form and solve the following equation:

Now, consider a pair of concentric circles. Two circles will always be similar to one another, and these share the same center. This means that we can solve problems involving concentric circles by using the fact that inscribed angles subtended by two arcs of equal measures must be equal to one another. In our next example, we will see what that looks like.

### Example 5: Solving Equations Using Two Inscribed Angles Subtended by Two Arcs of Equal Measures in Two Circles

In the figure, and pass through the center of the circles. Given that and , find .

### Answer

The diagram shows a pair of concentric circles. Since arc and arc have the same central angle, they must have the same measure. Hence, the angle subtended by arc will be equal to the angle subtended by arc . These angles are and respectively.

We use this information to form and solve an equation for :

In our previous examples, we used the properties of inscribed angles in a circle to find missing values. It follows that we can apply the corollary to our property to prove statements about circles. That is, if there are two congruent angles subtended by the same line segment and are on the same side of it, then their vertices and the segmentβs endpoints lie on a circle in which that segment is a chord.

Letβs demonstrate this in our final example.

### Example 6: Determining If a Circle Can Pass through Four Given Points Using the Measures of Angles at the Side of a Line Segment

Given that and , can a circle pass through the points , , , and ?

### Answer

Remember, if there are two congruent angles subtended by the same line segment and are on the same side of it, then their vertices and the segmentβs endpoints lie on a circle in which that segment is a chord.

In order to establish whether the points , , , and lie on the circumference of a circle, we begin by identifying angles subtended by the same line segments. and are both subtended by and lie on the same side of this segment. So, if and are congruent, that is, , then the points , , , and lie on the circumference of a circle.

We are given and . We can use this information to calculate since is an isosceles triangle:

So, . They are not congruent, so we deduce that a circle cannot pass through the points , , , and .

We will now recap the key concepts from this explainer.

### Key Points

- An inscribed angle is the angle that is formed by the intersection of two chords on the circumference of a circle.
- Inscribed angles subtended by the same arc are equal.
- If a pair of arcs in the same circle are congruent, their inscribed angles are equal.
- If a pair of circles are congruent, then inscribed angles subtended by congruent arcs, or arcs of equal measure, will be equal.
- If there are two congruent angles subtended by the same line segment and are on the same side of it, then their vertices and the segmentβs endpoints lie on a circle in which that segment is a chord.