6.2.1 Properties of Special Parallelograms (continued) Example 3 Verifying Properties of Squares
The diagonals of a square in the coordinate plane are shown to be congruent and to be perpendicular bisectors of each other. To show the diagonals are congruent, calculate their lengths using the Distance Formula. The lengths are both found to be 5 2. By the definition of congruent line segments, the diagonals are congruent. To show the diagonals bisect each other, calculate their midpoints using the Midpoint Formula. Both midpoints are the same point on the coordinate plane. This means the two diagonals intersect at their midpoints, or bisect each other. To show the diagonals are perpendicular, calculate their slopes. The slope of AC is − 1/7 and the slope of BD is 7. The product of the slopes is equal to − 1, indicating that the lines are perpendicular.
Example 4 Using Properties of Special Parallelograms in Proofs
A triangle is proven to be isosceles using properties of rectangles and parallelograms. ABCD is given to be a rectangle with E the midpoint of AD . By the definition of a rectangle, ∠ A and ∠ D are right angles and are congruent by the Right Angle Congruence Theorem. ABCD is a parallelogram since a rectangle is a parallelogram. AB ≅ DC according to the Properties of Parallelograms. AE ≅ ED by the definition of a midpoint. Therefore, △ ABE ≅ △ DCE by the SAS Congruence Postulate. BE ≅ CE by the Corresponding Parts of Congruent Triangles are Congruent Theorem. Thus by the definition of an isosceles triangle, △ BCE is isosceles.
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