Processing math: 100%
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
K12 LibreTexts

6.15: Inscribed Quadrilaterals in Circles

( \newcommand{\kernel}{\mathrm{null}\,}\)

Quadrilaterals with every vertex on a circle and opposite angles that are supplementary.

An inscribed polygon is a polygon where every vertex is on the circle, as shown below.

f-d_3b423cece4d07f83fc74f35d7ecac4b7b0f212fe4f6e6236319fde1d+IMAGE_TINY+IMAGE_TINY.png
Figure 6.15.1

For inscribed quadrilaterals in particular, the opposite angles will always be supplementary.

Inscribed Quadrilateral Theorem: A quadrilateral can be inscribed in a circle if and only if the opposite angles are supplementary.

f-d_234aed0c25ab3088ee1539f4f8cbda8d779cd55821268cf106c165f1+IMAGE_TINY+IMAGE_TINY.png
Figure 6.15.2

If ABCD is inscribed in E, then mA+mC=180 and mB+mD=180. Conversely, If mA+mC=180 and mB+mD=180, then ABCD is inscribed in E.

What if you were given a circle with a quadrilateral inscribed in it? How could you use information about the arcs formed by the quadrilateral and/or the quadrilateral's angle measures to find the measure of the unknown quadrilateral angles?

Example 6.15.1

  1. f-d_f24b37126d5836fd55db753eeb521107c3754f643667625dacaf1479+IMAGE_TINY+IMAGE_TINY.png
    Figure 6.15.3
  2. f-d_4e9fa23995200ff2bce5845a55ac0b9a3feedb20776807da7788843c+IMAGE_TINY+IMAGE_TINY.png
    Figure 6.15.4

Solution

  1. x+80=180y+71=180x=100y=109
  1. z+93=180x=12(58+106)y+82=180z=87x=82y=98

Example 6.15.2

Find x and y in the picture below.

f-d_64f8d45d7bded8e76a6d0c885eca321e6148062fea389c6837ff75dd+IMAGE_TINY+IMAGE_TINY.png
Figure 6.15.5

Solution

(7x+1)+105=180(4y+14)+(7y+1)=1807x+106=18011y+15=1807x=7411y=165x=10.57y=15

Example 6.15.3

Find the values of x and y in A.

f-d_f2ea4fb83ec882d7580bd9e11efb85d1defafd3b1188c5c7fa7f5ec3+IMAGE_TINY+IMAGE_TINY.png
Figure 6.15.6

Solution

Use the Inscribed Quadrilateral Theorem. x+108=180 so x=72. Similarly, y+88=180 so y=92.

Example 6.15.4

Quadrilateral ABCD is inscribed in E. Find mA, mB, mC, and mD.


f-d_28916d1f401e34988707414adb44c915487a4f08050767bfff944c90+IMAGE_TINY+IMAGE_TINY.png
Figure 6.15.7

Solution

First, note that m^AD=105 because the complete circle must add up to 360.

mA=12m^BD=12(115+86)=100.5mB=12m^AC=12(86+105)=95.5mC=180mA=180100.5=79.5mD=180mB=18095.5=84.5

Review

Fill in the blanks.

  1. A(n) _______________ polygon has all its vertices on a circle.
  2. The _____________ angles of an inscribed quadrilateral are ________________.

Quadrilateral ABCD is inscribed in E. Find:

f-d_3bb0b500960d9fbd9502dbf49ab2b3b335d590165bbcc4228e01f369+IMAGE_TINY+IMAGE_TINY.png
Figure 6.15.8
  1. mDBC
  2. m^BC
  3. m^AB
  4. mACD
  5. mADC
  6. mACB

Find the value of x and/or y in A.

  1. f-d_e5c6113c5f81b51a1b335af83993babf54f2020eeaedd687d985c21a+IMAGE_TINY+IMAGE_TINY.png
    Figure 6.15.9
  2. f-d_6aeedd2059d1d9ebfc81e7df42693c5f93332a5b3dc846355d840c85+IMAGE_TINY+IMAGE_TINY.png
    Figure 6.15.10
  3. f-d_0652ac7ccf2739e340734a85a42de79c2ce279e0970e3f4844de4115+IMAGE_TINY+IMAGE_TINY.png
    Figure 6.15.11

Solve for x.

  1. f-d_4c1df032bf17250331001e38694b77100784e74570d46982b9061952+IMAGE_TINY+IMAGE_TINY.png
    Figure 6.15.12
  2. f-d_2365c18f60fd2ba0361e3eef433fbd5a400423169540c3c1d6abc3e0+IMAGE_TINY+IMAGE_TINY.png
    Figure 6.15.13

Vocabulary

Term Definition
central angle An angle formed by two radii and whose vertex is at the center of the circle.
chord A line segment whose endpoints are on a circle.
circle The set of all points that are the same distance away from a specific point, called the center.
diameter A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.
inscribed angle An angle with its vertex on the circle and whose sides are chords.
intercepted arc The arc that is inside an inscribed angle and whose endpoints are on the angle.
radius The distance from the center to the outer rim of a circle.
Inscribed Polygon An inscribed polygon is a polygon with every vertex on a given circle.
Inscribed Quadrilateral Theorem The Inscribed Quadrilateral Theorem states that a quadrilateral can be inscribed in a circle if and only if the opposite angles of the quadrilateral are supplementary.
Cyclic Quadrilaterals A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle.

Additional Resources

Interactive Element

Video: Inscribed Quadrilaterals in Circles Principles - Basic

Activities: Inscribed Quadrilaterals in Circles Discussion Questions

Study Aids: Inscribed in Circles Study Guide

Practice: Inscribed Quadrilaterals in Circles

Real World: Sunrise at Stonehenge


This page titled 6.15: Inscribed Quadrilaterals in Circles is shared under a CK-12 license and was authored, remixed, and/or curated by CK-12 Foundation via source content that was edited to the style and standards of the LibreTexts platform.

CK-12 Foundation
LICENSED UNDER
CK-12 Foundation is licensed under CK-12 Curriculum Materials License

Support Center

How can we help?