Lesson 3 Cyclic Polygons Solidify Understanding

Learning Focus

Examine the relationship between inscribed angles and the intercepted arc.

We have seen that circles can be inscribed in triangles so that the circle touches all three sides, and that triangles can be inscribed in circles, so that the circle passes through all three vertices. What other types of polygons can be inscribed in circles, or can contain inscribed circles that touch all of their sides?

What relationships exist between central angles and inscribed angles that intercept the same arc?

Open Up the Math: Launch, Explore, Discuss

By definition, a cyclic polygon is a polygon that can be inscribed in a circle. That is, all of the vertices of the polygon lie on the same circle.

In Centers of a Triangle, your work on Kara’s notes and diagram should have convinced you that it is possible to locate a point that is equidistant from all three vertices of any triangle, and therefore, all triangles are cyclic polygons.

1.

Based on Kara’s work, use a compass and straightedge to construct the circles that contain all three vertices of at least one of the following triangles.

Since each vertex of an inscribed triangle lies on the circle, each angle of the triangle is an inscribed angle. We know that the sum of the measures of the interior angles of the triangle is $180 °$ and that the sum of the measures of the three intercepted arcs is $360 °$ .

2.

Using one of the diagrams of an inscribed triangle you created, illustrate and explain why this last statement is true.

The degree measure of an arc is, by definition, the same as the measure of the central angle formed by the radii that contain the endpoints of the arc (see $∠ A B C$ in the diagram). But how is the measure of an inscribed angle that intercepts this same arc, such as $∠ A D C$ , related to the measure of the central angle and the intercepted arc? That is something useful to find out.

3.

Using a protractor, work with your group to find the measure of each arc represented on each circle diagram in problem 1. Then find the measure of each corresponding inscribed angle. Make a conjecture based on this data.

Pause and Reflect

4.

The three circle diagrams you created for problem 3 have been reproduced in the following problem parts. One inscribed angle has been bolded in each triangle. A diameter of the circle has also been added to each diagram as an auxiliary line segment, as well as some additional line segments that will assist in writing proofs about the inscribed angles. In each diagram, prove your conjecture about the measure of an inscribed angle for the inscribed angle shown in bold.

a.

The diameter is a side of the inscribed angle. Hint: Look for isosceles triangles and an external angle of a triangle.

b.

The diameter lies in the interior of the inscribed angle. Hint: Can you see 4a in 4b?

c.

The diameter lies in the exterior of the inscribed angle.

We have found that all triangles are cyclic polygons. Now let’s examine possible cyclic quadrilaterals. Obviously, some generic quadrilaterals are cyclic, since you can select any 4 points on a circle as the vertices of a quadrilateral.

5.

Experiment with cyclic quadrilaterals by selecting any $4$ points on the circle as vertices of the quadrilateral. Using a protractor, find the measurements of the angles. Make a conjecture about the measures of the angles of a cyclic quadrilateral. Then prove your conjecture using what you know about inscribed angles. (We already know that the sum of the angles in a quadrilateral is $360 °$ , so your conjecture should focus on something else.)

a.

Conjecture about the angles of a cyclic quadrilateral:

b.

Proof of my conjecture:

(How might you use the following diagram to assist you in your proof?)

Ready for More?

Reasoning together in your small group, decide which word best completes each of the following statements:

a.

[Some, All, No] squares are cyclic.

b.

[Some, All, No] rhombuses are cyclic.

c.

[Some, All, No] trapezoids are cyclic.

d.

[Some, All, No] rectangles are cyclic.

e.

[Some, All, No] parallelograms are cyclic.

Takeaways

Complete each of the following descriptions and illustrate with an example:

The degree measure of an arc is, by definition,

The degree measure of an inscribed angle is

The degree measure of an inscribed angle whose rays intersect the endpoints of a diameter of a circle is

Vocabulary

arc of a circle, intercepted arc
cyclic polygon
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned about cyclic polygons—polygons whose vertices all lie on the circumference of a circle. All triangles are cyclic, and some quadrilaterals are. We used cyclic polygons to make and prove a conjecture about the measure of an inscribed angle relative to its intercepted arc. Once we understood this relationship, we could use it to make a conjecture about opposite angles in a cyclic quadrilateral.

Retrieval

1.

Given triangle $A B C$ and circle $D$ , identify each line segment and line in the diagram with its mathematical name as it relates to the circle. What is the mathematical name for point $D$ ? What is the mathematical name for circle $D$ ?

2.

Write the trigonometric equation needed to find $x$ and the trigonometric equation needed to find $y$ in the right triangle. Then solve for $x$ and $y$ .