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Inscribed Quadrilaterals in Circles

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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? After completing this Concept, you'll be able to apply the Inscribed Quadrilateral Theorem to solve problems like this one.

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Inscribed Quadrilaterals in Circles CK-12

Guidance

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

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.

If ABCD is inscribed in \bigodot E , then m\angle A+m\angle C=180^\circ and m\angle B+m\angle D=180^\circ . Conversely, If m\angle A+m\angle C=180^\circ and m\angle B+m\angle D=180^\circ , then ABCD is inscribed in \bigodot E .

Example A

Find the values of the missing variables.

a)

b)

Answers:

a) x+80^\circ &= 180^\circ && y+71^\circ = 180^\circ\\x &= 100^\circ && y=109^\circ

b) z+93^\circ &=180^\circ && x=\frac{1}{2} (58^\circ+106^\circ) && y+82^\circ=180^\circ\!\\z &= 87^\circ && x=82^\circ && y = 98^\circ

Example B

Find x and y in the picture below.

(7x+1)^\circ+105^\circ& =180^\circ && (4y+14)^\circ+(7y+1)^\circ=180^\circ\\7x+106^\circ&=180^\circ && \qquad \qquad \quad \ 11y+15^\circ=180^\circ\\7x&=74 && \qquad \qquad \qquad \qquad 11y=165\\x&=10.57 && \qquad \qquad \qquad \qquad \quad y=15

Example C

Find the values of x and y in \bigodot A .

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

Inscribed Quadrilaterals in Circles CK-12

Guided Practice

Quadrilateral ABCD is inscribed in \bigodot E . Find:

  1. m\angle A
  2. m\angle B
  3. m\angle C
  4. m\angle D

Answers:

First, note that m\widehat{AD}=105^\circ because the complete circle must add up to 360^\circ .

1. m\angle A=\frac{1}{2}m\widehat{BD}=\frac{1}{2}(115+86)=100.5^\circ

2. m\angle B=\frac{1}{2}m\widehat{AC}=\frac{1}{2}(86+105)=95.5^\circ

3. m\angle C=180^\circ-m\angle A=180^\circ-100.5^\circ=79.5^\circ

4. m\angle D=180^\circ-m\angle B=180^\circ-95.5^\circ=84.5^\circ

Practice

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 \bigodot E . Find:

  1. m\angle DBC
  2. m \widehat{BC}
  3. m \widehat{AB}
  4. m\angle ACD
  5. m\angle ADC
  6. m\angle ACB

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

Solve for x .

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