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  • Euclidean Geometry: Triangles and Circles

Why do opposite angles of a cyclic quadrilateral add up to 180 degrees?

Quick answer

The opposite angles of a cyclic quadrilateral add up to 180 degrees because each opposite angle intercepts one of two arcs that together make a full 360-degree circle. By the inscribed angle theorem, an angle inscribed in a circle measures half of its intercepted arc. Adding the two opposite angles yields half of 360 degrees, which equals 180 degrees.

The topic explained

The rule and why it works

A cyclic quadrilateral is any four-sided polygon whose four vertices sit directly on the circumference of a single circle. A key property in geometry is that opposite angles of a cyclic quadrilateral add up to 180 degrees, making them supplementary.

This property comes from the inscribed angle theorem, which states that an angle on the circle equals half the measure of the arc it intercepts. Imagine quadrilateral ABCD on a circle. Angle B intercepts arc ADC, and angle D intercepts arc ABC. Together, arc ADC and arc ABC form the entire circle of 360 degrees. Therefore, angle B plus angle D equals half of 360 degrees, which is exactly 180 degrees.

How to use the theorem and its converse

When solving circle geometry problems, look for four points joined on a circle. If angle A measures 70 degrees, its opposite angle C must equal 180 minus 70, which is 110 degrees.

You can also run this logic in reverse using the converse. If you test any quadrilateral and find that one pair of opposite angles adds up to 180 degrees, all four vertices must lie on a single circle. This converse helps you prove that points are concyclic in multi-step proofs.

Which special quadrilaterals are always cyclic

Not every quadrilateral can fit inside a circle. A rectangle is always cyclic because all four angles are 90 degrees, so opposite angles sum to 180 degrees. Squares and isosceles trapezoids are also always cyclic.

Parallelograms and rhombuses are not cyclic unless their angles happen to be right angles. Watch out for non-adjacent angles: the rule applies only to angles across from each other, never to consecutive angles along the same edge.

Free printable worksheet

Cyclic Quadrilaterals: Angle Properties Worksheet

For 10th grade geometry students. Use the properties of cyclic quadrilaterals and inscribed angles to find each missing value or determine whether four points form a circle.

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Cyclic Quadrilaterals: Angle Properties WorksheetName:

  1. Quadrilateral ABCD is inscribed in a circle. If angle A = 82 degrees, find the measure of angle C.

  2. Quadrilateral JKLM is cyclic. Angle K = 105 degrees and angle L = 64 degrees. Find angles M and J.

  3. In cyclic quadrilateral PQRS, angle P = 2x + 10 and angle R = 3x - 5. Solve for x and find the measure of angle P.

  4. Quadrilateral WXYZ has angles W = 78 degrees, X = 104 degrees, Y = 102 degrees, and Z = 76 degrees. Can a circle pass through all four vertices? Explain.

  5. Can a general rhombus with an acute angle of 60 degrees be inscribed in a circle? Explain.

  6. Quadrilateral ABCD is cyclic. Side AB is extended past B to point E. If angle ADC = 115 degrees, find the measure of exterior angle CBE.

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Common questions

Can any four-sided shape be inscribed in a circle?

No. A quadrilateral can only be inscribed in a circle if its opposite angles add up to 180 degrees. Most random four-sided shapes do not satisfy this condition.

What is the exterior angle property of a cyclic quadrilateral?

An exterior angle formed by extending one side of a cyclic quadrilateral is always equal to the interior opposite angle. This happens because both angles are supplementary to the same interior angle.

Is an isosceles trapezoid always cyclic?

Yes. In an isosceles trapezoid, base angles are equal and consecutive interior angles between the parallel bases are supplementary. This forces opposite angles to add up to 180 degrees.

How do I prove four points lie on a circle?

Connect the four points into a quadrilateral and show that one pair of opposite angles sums to 180 degrees. Alternatively, show that two angles subtended by the same segment are equal.

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