Topics
13. Circles
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United States of America
Grades 9-12

13.04 Inscribed angles and polygons

Lesson
Practice

Introduction

We learned about several theorems related to chords and angles in the previous lesson. When two chords share an endpoint, they create an angle within the circle. This lesson will explore angles created by chords that share endpoints and the theorems specific to these cases.

Inscribed angles and polygons

Angles formed by chords of a circle are known as inscribed angles.

Inscribed angle in a circle

An angle which is formed in the interior of a circle when two chords share an endpoint

Two chords are on a circle which both have an endpoint at the same point on the circle. The angle between the chords is an inscribed angle in the circle.

Multiple inscribed angles which share segments can form an inscribed polygon.

Inscribed polygon in a circle

A polygon which has all of its vertices on a circle

Quadrilateral C E D B inscribed in circle A.
Inscribed right triangle theorem

If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle

Circle A with diameter B C. A right triangle is inscribed in circle A with B C as the hypotenuse of the triangle.
Inscribed semicircle theorem

An angle inscribed in a semicircle is a right angle

Circle A with diameter B D, chords B C and C D. Angle B C D is a right angle.

When a polygon is inscribed in a circle, we can equivalently say that the circle is circumscribed on the polygon.

A circle can also be inscribed into a polygon:

Inscribed circle in a polygon

The largest possible circle that can be drawn in the interior of a polygon. If it is a regular polygon, then each side of the polygon is tangent to the circle.

A circle inscribed in a regular hexagon.

Again, we can refer to this equivalently by saying that the polygon is circumscribed on the circle.

Opposite inscribed angle theorem

The opposite angles of an inscribed quadrilateral are supplementary

Quadrilateral B C D E inscribed in circle A. Angles B and E are marked.
Quadrilateral B C E D inscribed in circle A.

The opposite inscribed angle theorem says that two opposite angles in a quadrilateral inscribed in a circle add up to 180 \degree.

In this case, that results in the following equations:

m\angle BCE + m\angle EDB = 180

m\angle DBC + m\angle CED = 180

Examples

Example 1

Construct a square inscribed in a circle.

Worked Solution
Create a strategy

The diagonals of a square bisect each other and are perpendicular. We can use the properties to construct two segments with endpoints on the circle which are perpendicular bisectors of one another. We will use a compass and straightedge.

Apply the idea

We can follow these steps to construct a square inscribed in a circle:

  1. Draw a circle and mark its center, O.
  2. Mark a point A on the circle.
  3. Draw a diameter through points O and A, mark the other endpoint of the diameter as C.
  4. Construct the perpendicular bisector of \overline{AC}.
    • Set the width of the compass to between AO and AC
    • Put the compass at point C and draw an arc that crosses \overline{OA} and goes above and below O
    • Put the compass at point A and draw an arc that that crosses \overline{OC} and goes above and below O
    • Draw a line segment that goes through the points of intersection of the arcs and has endpoints on the circle, label these endpoints B and D, this is the perpendicular bisector of \overline{AC}
  5. Join points A, B, C, and D.
  6. This square ABCD is inscribed in a circle.

Example 2

Construct the circumscribed circle whose center is the point of intersection of the perpendicular bisectors of each side of the triangle (the circumcenter).

Worked Solution
Apply the idea

  1. Draw a triangle using the Polygon tool.

  2. Draw the perpendicular bisector of \overline{AB} using the Perpendicular Bisector tool. To use this tool, click on vertex A then vertex B.

  3. Draw the perpedicular bisector of \overline{BC} using the Perpendicular Bisector tool again.

  4. Draw the perpedicular bisector of \overline{CA} using the Perpendicular Bisector tool once more.

  5. The point of intersection of the perpendicular bisectors is called the circumcenter. Label the circumcenter using the Point tool.

  6. The circumcenter is the center of the circumscribed circle. The radius is the distance from the center to any of the vertices of the triangle. Use the Circle with Center tool to draw the circumscribed circle by clicking the center first, then a vertex of the triangle.

Example 3

In the diagram below, O is the center of the circle.

Solve for the values of p, q, and r.

Worked Solution
Create a strategy

Any chord that passes through the center of the circle is a diameter. By this, we know that \angle p is an angle inscribed in a semicircle.

Apply the idea

By the inscribed semicircle theorem, m\angle p=90\degree. The known angle, \angle p, and \angle q form a triangle, so these three angles will sum to 180\degree.

\displaystyle 27+90+m\angle q\displaystyle =\displaystyle 180
\displaystyle m\angle q\displaystyle =\displaystyle 63\degree

Finally, \angle r and \angle q are opposite angles of an inscribed quadrilateral, so they sum to 180\degree by the opposite inscribed angle theorem.

\displaystyle m\angle r +m\angle q\displaystyle =\displaystyle 180Opposite angles of an inscribed quadrilateral
\displaystyle m\angle r +63\displaystyle =\displaystyle 180Substitute m\angle q=63
\displaystyle m\angle r\displaystyle =\displaystyle 117Subtract both sides by 63

Therefore, m\angle p=90\degree, m\angle q=63\degree, and m\angle r=117\degree.

Idea summary

A polygon whose vertices lie on the edge of a circle is called an inscribed polygon. We can also say the circle is circumscribed on the polygon. To construct an inscribed polygon, we use perpendicular bisectors.

When a quadrilateral is inscribed in a polygon, the opposite angles are supplementary.

Properties of inscribed angles

Exploration

Rotate the points around the circle.

  1. What relationship do you notice between the central angle and the inscribed angle?
  2. How does this relationship translate to the inscribed angle and the arc it intercepts?
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The following theorems relate to angles inscribed in circles:

Inscribed angle theorem

If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc

An angle with a measure of theta degrees inscribed in a circle. The angle intercepts the arc A B with a measure of 2 theta degrees.
Congruent inscribed angle theorem

If two inscribed angles of a circle intercept the same arc, then the angles are congruent

A circle with points A, C, D, and B placed clockwise on a circle. Chords A C, A D, C B, and B D. Chords A D and C B intersect at a point. Angles A C B and A D B are congruent.

Examples

Example 4

Prove opposite angles of an inscribed quadrilateral are supplementary.

Worked Solution
Create a strategy

Let us begin by drawing an inscribed quadrilateral and labeling the opposite angles. This will help us visualize the problem and decide on a plan.

Our goal is to prove that m\angle x +m\angle y=180.

Apply the idea

By the inscribed angle theorem, we know that the measure of the arc intercepted by \angle x is twice m\angle x.

We also know the the measure of the arc intercepted by \angle y is twice m\angle y.

Together, both of the intercepted arcs make up the entire circle, so their measures will sum to 360\degree.

This allows us to set up an equation that can be simplified:\begin{aligned} 2\cdot m\angle x + 2\cdot m\angle y&=360\\2\left(m\angle x+m\angle y\right)&=360\\m\angle x+m\angle y&=180\end{aligned}

The other two angles can be shown to be supplementary using the same reasoning. Therefore, opposite angles of an inscribed quadrilateral are supplementary.

Example 5

Given m\angle CEB = 4x + 11 and m\angle CDB = 12x - 5. Find m\angle CDB.

Circle A with points C, E, D, and B placed clockwise on the circle. Chords C E, C D, C B, E B, and D B are drawn.
Worked Solution
Create a strategy

By the congruent inscribed angle theorem, we know m\angle CEB = m\angle CDB. We want to write an equation relating the two angles and then solve for x.

Apply the idea
\displaystyle 4x + 11\displaystyle =\displaystyle 12x - 5Congruent inscribed angle theorem
\displaystyle 4x + 16\displaystyle =\displaystyle 12xAdd 5 to both sides
\displaystyle 16\displaystyle =\displaystyle 8xSubtract 4x from both sides
\displaystyle 2\displaystyle =\displaystyle xDivide both sides by 8
\displaystyle x\displaystyle =\displaystyle 2Symmetric property of equality

We have established x = 2, so we can substitute that into the equation for \angle CDB.

Substituting x = 2 into 12x - 5 we get 12\left(2\right) - 5 = 19. Therefore, m\angle CDB = 19 \degree.

Reflect and check

Note that since m\angle CDB = m\angle CEB, we could have used the expression for m\angle CEB to calculate the size of the angle instead:

\displaystyle m\angle CEB\displaystyle =\displaystyle 4x + 11Given
\displaystyle =\displaystyle 4\left(2\right) + 11Substitute x = 2
\displaystyle =\displaystyle 19Simplify the expression

which is the same result.

Example 6

Solve for x.

Angle A T B with a measure of negative 10 plus x degrees inscribed in a circle. Angle A T B intercepts an arc A B which has a measure of 4 plus x degrees.
Worked Solution
Create a strategy

We can use the inscribed angle theorem to write an equation and then solve for x.

Apply the idea
\displaystyle 2(m\angle{ATB}) \displaystyle =\displaystyle m \overset{\large\frown}{AB}
\displaystyle 2\left(-10+x\right)\displaystyle =\displaystyle 4+xSubstitution
\displaystyle -20+2x\displaystyle =\displaystyle 4+xDistribute the 2
\displaystyle 2x\displaystyle =\displaystyle 24+xAdd 20 to both sides
\displaystyle x\displaystyle =\displaystyle 24Subtract x from both sides

x=24

Reflect and check

The value of x is 24, which means the inscribed angle m \angle ATB = 14 \degree and the intercepted arc m\overset{\large\frown}{AB} = 28 \degree.

Idea summary

The measure of an inscribed angle is half the measure of the intercepted arc. If two inscribed angles intercept the same arc, the angles are congruent.

Outcomes

G.CO.D.13

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

G.C.A.2

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

G.C.A.3

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

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