Lesson | Loading...

Sign in to access protected content and invite other educators

Sign in using your existing Kendall Hunt account. If you don’t have one, create an educator account.

Curriculum

IM K-5 Math
IM 6-8 Math
IM 9-12 Math

Professional Learning

Illustrative Mathematics® has operated as an independent 501(c)3 non-profit organization since 2013.

©2024 Illustrative Mathematics®. Licensed under CC BY-NC 4.0.

The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written permission of Illustrative Mathematics.

This site includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses. See the image attribution section for more information.

Navigation

9-12 Integrated Math 2 Unit 7 Lesson 8

Gradeband

K-5 Math 6-8 Math •9-12 Math

Course

Integrated Math 1 •Integrated Math 2 Integrated Math 3

Unit

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 •Unit 7 Unit 8

Modeling Prompts

Lesson

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5 Lesson 6 Lesson 7 •Lesson 8 Lesson 9 Lesson 10 Lesson 11 Lesson 12 Lesson 13 Lesson 14 Lesson 15 Lesson 16 Lesson 17 Lesson 18

K-5
6-8
9-12

Integrated Math 1
Integrated Math 2
Integrated Math 3

Algebra 1
Geometry
Algebra 2
Extra Support Materials for Algebra 1

Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8

Modeling Prompts

Lesson 1
Lesson 2
Lesson 3
Lesson 4
Lesson 5
Lesson 6
Lesson 7
Lesson 8
Lesson 9
Lesson 10
Lesson 11
Lesson 12
Lesson 13
Lesson 14
Lesson 15
Lesson 16
Lesson 17
Lesson 18

Settings

Audience

family

Not all roles available for this page.

Assessment Access

Sign in to view assessments and invite other educators

Sign in using your existing Kendall Hunt account. If you don’t have one, create an educator account.

teacher
student
family

Lesson 8

8.1 Warm-up 8.2 Activity 8.3 Activity Student Lesson Summary Glossary

Unit 7, Lesson 8

Quadrilaterals in Circles

$Course Icon$

Integrated Math 2

8.1

Warm-up

Connecting the Dots

For each quadrilateral, use a compass to see if you can draw a circle that passes through all 4 of the quadrilateral’s vertices.

<p>A square with right angles marked.</p> — A

<p>Trapezoid with points at vertices.</p> — B

<p>Parallelogram with points at vertices.<br>
</p> — C

Are You Ready for More?

8.2

Activity

Inscribed Angles and Circumscribed Circles

The images show 3 quadrilaterals with circumscribed circles.

A

B

C

For each one, highlight the arc from to passing through . Then find the measures of:
1. the arc you highlighted
2. the other arc from to
3. angle
Here is another quadrilateral with a circumscribed circle. What is the value of ? Explain or show your reasoning.
$<p>Circle with center A. Quadrilateral DCFE inside circle. Angle C = <span class="math">\(\alpha\)</span>. Angle E = <span class="math">\(\beta\)</span>.</p>$

Are You Ready for More?

8.3

Activity

Construction Ahead

Quadrilateral is a cyclic quadrilateral.

<p>Quadrilateral A B C D. Angle B C D is right angle.<br>
</p>

Draw diagonal . How will this diagonal relate to the circumscribed circle? Explain your reasoning.
Construct the center of the circumscribed circle for quadrilateral . Label this point . Explain why your method worked.
Construct the circumscribed circle for quadrilateral .
Could we follow this procedure to construct a circumscribed circle for any cyclic quadrilateral? Explain your reasoning.

Student Lesson Summary

A circle is said to be circumscribed about a polygon if all the vertices of the polygon lie on the circle. If it is possible to draw a circumscribed circle for a quadrilateral, the figure is called a cyclic quadrilateral. Not all quadrilaterals have this property.

<p>Quadrilateral A B C D, with all points lying on a circle.</p>

<p>Quadrilateral E F G H inscribed in a circle. Vertices E, F and H lie on the circle.<br>
</p>

We can prove that the opposite angles of a cyclic quadrilateral are supplementary. Consider opposite angles and , labeled and .

<p>Circle with quadrilateral ABCD inside. Angle C = Alpha. Angle A = beta.</p>

Angle is inscribed in the arc from to through . Angle is inscribed in the arc from to through . Together, the two arcs trace out the entire circumference of the circle, so their measures add to 360 degrees. By the Inscribed Angle Theorem, the sum of and must be half of 360 degrees, or 180 degrees. So angles and are supplementary. The same argument can be applied to the other pair of opposite angles.

Glossary

circumscribed

When a figure is circumscribed, it is completely surrounded by another figure, so their sides, edges, vertices, or curves touch.

A polygon is circumscribed by a circle if it fits inside the circle and every vertex of the polygon is on the circle.
A circle is circumscribed by a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.

cyclic quadrilateral

If a circle can be drawn around a quadrilateral so that it passes through every vertex, the quadrilateral is called a cyclic quadrilateral.

That is, a cyclic quadrilateral can be circumscribed by a circle.

Have feedback on the curriculum?

Help us improve by sharing suggestions or reporting issues.

Submit Feedback

Are You Ready for More?