Geometry: An Interactive Journey to Mastery

Lesson 14: Exploring Special Quadrilaterals

Example 1

Opposite angles in a quadrilateral are congruent. Prove that the quadrilateral must

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Solution

Following the labeling given in the diagram, because the angles in a quadrilateral

sum to 360°, we have 2x + 2y = 360°, giving x + y = 180°. This means that each pair of angles x and y in the

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Example 2

Suppose that A ͼ௘í௘ͽB ͼ௘௘ͽC ͼ௘௘ͽDQGD ͼ௘í௘ͽ:KDWW\SHRITXDGULODWHUDOLVABCD?

Solution

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Midpoint AC §· ̈ ̧©¹^32 ,9.

Midpoint BD §· ̈ ̧©¹^32 ,9.

The diagonals bisect each other, so we have a parallelogram, at the very least.

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AC  34 5.^22

BD  50 5.^22

These are the same. The parallelogram is a rectangle, at the very least.

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Slope AC 43.

Slope BD (^) ^05 0.
These are not negative reciprocals, so the diagonals are not perpendicular. The rectangle is not a square.
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y
y
x
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Figure 14.1