Lesson Explainer: Inscribed Angles Subtended by the Same Arc Mathematics

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

Let’s 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, ∠𝐴𝐡𝐢 is an inscribed angle.

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

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, ∠𝐴𝐡𝐢=∠𝐴𝐷𝐢, since both angles are subtended by arc 𝐴𝐢.

In a similar way, since ∠𝐡𝐴𝐷 and ∠𝐡𝐢𝐷 are both subtended by arc 𝐡𝐷, they are equal.

This property is sometimes equivalently stated as β€œangles in the same segment are equal.”

This is sometimes informally referred to as the β€œbow tie” 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 𝐴𝐢, and they will all be equal. Similarly, any number of angles can be subtended by arc 𝐡𝐷, 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 𝑂 and constructing radii 𝑂𝐴 and 𝑂𝐢.

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 βˆ π΄π‘‚πΆ=2π‘₯∘, although we might have alternatively chosen π‘₯∘. Hence, ∠𝐴𝐡𝐢=12Γ—2π‘₯=π‘₯.∘∘

Similarly, by the same property, ∠𝐴𝐷𝐢=12Γ—2π‘₯=π‘₯.∘∘

Thus, ∠𝐴𝐡𝐢=∠𝐴𝐷𝐢, 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 π‘šβˆ π΅π΄π·=36∘ and π‘šβˆ πΆπ΅π΄=37∘, find π‘šβˆ π΅πΆπ· and π‘šβˆ πΆπ·π΄.

Answer

While not entirely necessary, it can be sensible to begin by adding the given angles to the diagram. π‘šβˆ π΅π΄π·=36∘ and π‘šβˆ πΆπ΅π΄=37∘, so the diagram is as shown.

Next, we observe that the first unknown angle, ∠𝐡𝐢𝐷, is subtended by the same arc, 𝐡𝐷, as ∠𝐡𝐴𝐷. We know that the angles subtended by the same arc are equal, so π‘šβˆ π΅πΆπ·=π‘šβˆ π΅π΄π·=36.∘

Similarly, ∠𝐢𝐷𝐴 is subtended by the same arc, 𝐴𝐢, as ∠𝐢𝐡𝐴. Hence, π‘šβˆ πΆπ·π΄=π‘šβˆ πΆπ΅π΄=37.∘

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 π‘šβˆ π΅π΄π·=(2π‘₯+2)∘ and π‘šβˆ π΅πΆπ·=(π‘₯+18)∘, determine the value of π‘₯.

Answer

We recall that angles subtended by the same arc are equal. In this diagram, ∠𝐡𝐴𝐷 and ∠𝐡𝐢𝐷 are both subtended by arc 𝐡𝐷, so these two angles are equal. This allows us to form and solve an equation in π‘₯. Since π‘šβˆ π΅π΄π·=(2π‘₯+2)∘ and π‘šβˆ π΅πΆπ·=(π‘₯+18)∘, 2π‘₯+2=π‘₯+18βˆ’π‘₯βˆ’π‘₯π‘₯+2=18βˆ’2βˆ’2π‘₯=16.

Hence, π‘₯=16.

An especially interesting property that results from the properties of inscribed angles is that the angle inscribed in a semicircle is 90∘. This is demonstrated in the following diagram, where 𝐴𝐢 is the diameter of the circle and ∠𝐴𝐡𝐢=90∘.

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 𝐴𝐡 is a diameter of the circle and 𝐷𝐢⫽𝐴𝐡, find π‘šβˆ π΄πΈπ·.

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 β€œeasy-to-calculate” angles.

Recall that the angle inscribed in a semicircle is 90∘. Using this property, we can see that π‘šβˆ π΅πΆπ΄=90∘.

Since the interior angles in a triangle add to 180∘, we can work out the measure of ∠𝐡𝐴𝐢: π‘šβˆ π΅π΄πΆ=180βˆ’(90+68.5)=21.5.∘

This is useful since we also know that line segments 𝐷𝐢 and 𝐴𝐡 are parallel, so we can use the fact that alternate angles are equal to calculate π‘šβˆ π΄πΆπ·=21.5∘.

Finally, we observe that ∠𝐴𝐢𝐷 and ∠𝐴𝐸𝐷 are subtended by the same arc, 𝐴𝐷. Since angles inscribed by the same arc are equal, π‘šβˆ π΄πΆπ·=π‘šβˆ π΄πΈπ·=21.5.∘

So, π‘šβˆ π΄πΈπ· is 21.5∘.

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 𝐴𝐢=𝐷𝐹, then π‘šβˆ π΄π΅πΆ=π‘šβˆ π·πΈπΉ.

Similarly, all inscribed angles subtended by congruent arcs in a circle are equal in measure. It might be tempting to look for the typical β€œbow tie” 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 π‘šβˆ πΉπΈπ·=14∘ and π‘šβˆ πΆπ΅π΄=2π‘₯βˆ’96∘, calculate the value of π‘₯.

Answer

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

Hence, we can form and solve the following equation: 2π‘₯βˆ’96=142π‘₯=110π‘₯=55.

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, 𝐴𝐸 and 𝐡𝐢 pass through the center of the circles. Given that π‘šβˆ πΉπΈπ·=50∘ and π‘šβˆ πΆπ΅π΄=(2π‘₯βˆ’10)∘, find π‘₯.

Answer

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

We use this information to form and solve an equation for π‘₯: 2π‘₯βˆ’10=502π‘₯=60π‘₯=30.

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 segment’s endpoints lie on a circle in which that segment is a chord.

Let’s 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 π‘šβˆ π΅πΆπ΄=61∘ and π‘šβˆ π·π΄π΅=98∘, can a circle pass through the points 𝐴, 𝐡, 𝐢, and 𝐷?

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 segment’s endpoints lie on a circle in which that segment is a chord.

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

We are given π‘šβˆ π΅πΆπ΄=61∘ and π‘šβˆ π·π΄π΅=98∘. We can use this information to calculate π‘šβˆ π΅π·π΄ since 𝐡𝐴𝐷 is an isosceles triangle: π‘šβˆ π΅π·π΄=180βˆ’982=41.∘

So, π‘šβˆ π΅πΆπ΄β‰ π‘šβˆ π·π΄π΅. They are not congruent, so we deduce that a circle cannot pass through the points 𝐴, 𝐡, 𝐢, and 𝐷.

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 segment’s endpoints lie on a circle in which that segment is a chord.

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