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Proving Properties of Inscribed QuadrilateralsAdapted from Walch Education

Leonardo da Vincis Vitruvian Man3.2.3: Proving Properties of Inscribed Quadrilaterals2

2inscribed quadrilateral An inscribed quadrilateral is a quadrilateral whose vertices are on a circle. The opposite angles of an inscribed quadrilateral are supplementary. mA + mC = 180mB + mD = 180

Rectangles and squares can always be inscribed within a circle.

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PracticeConsider the inscribed quadrilateral in the diagram at right. What are the relationships between the measures of the angles of an inscribed quadrilateral? 3.2.3: Proving Properties of Inscribed Quadrilaterals4

Find the measure of B. B is an inscribed angle. Therefore, its measure will be equal to half the measure of the intercepted arc.The intercepted arc has a measure of 122 + 52, or 174.The measure of B is of 174, or 87.3.2.3: Proving Properties of Inscribed Quadrilaterals5

Find the measure of D. The intercepted arc has a measure of 82 + 104, or 186.The measure of D is of 186, or 93.3.2.3: Proving Properties of Inscribed Quadrilaterals6

What is the relationship between B and D? Since the sum of the measures of B and D equals 180, B and D are supplementary angles.3.2.3: Proving Properties of Inscribed Quadrilaterals7Does this same relationship exist between A and C ? The intercepted arc has a measure of 104 + 52=156.

The measure of A is of 156, or 78.The intercepted arc has a measure of 82 + 122=204.

The measure of C is of 204, or 102.The sum of the measures of A and C also equals 180; therefore, A and C are supplementary.

The opposite angles of an inscribed quadrilateral are supplementary. 3.2.3: Proving Properties of Inscribed Quadrilaterals8

Your TurnConsider the inscribed quadrilateral to the right. Do the relationships discovered between the angles in the practiceproblem still hold for the angles in this quadrilateral?3.2.3: Proving Properties of Inscribed Quadrilaterals9

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