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Euclidean Geometry: Triangles and Circles · Ages 15–16

Cyclic Quadrilaterals: Angle Properties Worksheet

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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.

  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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Cyclic Quadrilaterals: Angle Properties Worksheet

Answer key

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

    98 degrees. Opposite angles sum to 180 degrees: 180 - 82 = 98.

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

    Angle M = 116 degrees (180 - 64) and angle J = 75 degrees (180 - 105).

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

    x = 35, angle P = 80 degrees. Equation: (2x + 10) + (3x - 5) = 180; 5x + 5 = 180; 5x = 175; x = 35; 2(35) + 10 = 80.

  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.

    Yes. Opposite pairs sum to 180 degrees: 78 + 102 = 180 and 104 + 76 = 180. By the converse theorem, WXYZ is cyclic.

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

    No. The opposite angle of a rhombus equals the given angle, so opposite angles would sum to 60 + 60 = 120 degrees instead of 180 degrees.

  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.

    115 degrees. Angle ABC and angle ADC are supplementary (angle ABC = 65 degrees). Angles ABC and CBE form a straight line, so angle CBE = 180 - 65 = 115 degrees. An exterior angle of a cyclic quadrilateral equals the interior opposite angle.