prove that in a parallelogram both pairs of opposite sides are congruent (theorem)?
310745
1. Opposite Sides Theorem Converse: If both pairs of opposite sides of a quadrilateral are congruent, then the figure is a parallelogram. 2. Opposite Angles Theorem Converse: If both pairs of opposite angles of a quadrilateral are congruent, then the figure is a parallelogram.
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All the properties of a parallelogram apply (the ones that matter here are parallel sides, opposite angles are congruent, and consecutive angles are supplementary). All sides are congruent by definition. The diagonals bisect the angles. The diagonals are perpendicular bisectors of each other.
60111
All the properties of a parallelogram apply (the ones that matter here are parallel sides, opposite sides are congruent, and diagonals bisect each other). All angles are right angles by definition. The diagonals are congruent.
603283
Which property is not true for all parallelograms? (1) Opposite angles are congruent. (2) Opposite sides are congruent. (3) Consecutive angles are supplementary.
81234
Opposite sides are parallel to each other. The diagonals are congruent. The diagonals are perpendicular to and bisect each other. ... Also, a parallelogram becomes a square when the diagonals are equal and right bisectors of each other.
63993
The opposite angles of a parallelogram are congruent. ... The diagonals of a parallelogram are sometimes congruent. The diagonals of a rhombus are always perpendicular. The consecutive angles of a parallelogram are never complementary.
569289
The opposite angles of a parallelogram are equal (and conversely: if the opposite angles of a quadrilateral are equal, it is a parallelogram). ... If one pair of opposite sides of a quadrilateral is equal and parallel, then the quadrilateral is a parallelogram.
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If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If — AB ≅ — CD and — BC ≅ — DA , then ABCD is a parallelogram. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
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THEOREM: If a quadrilateral is a parallelogram, it has 2 sets of opposite angles congruent. THEOREM: If a quadrilateral is a parallelogram, it has consecutive angles which are supplementary. THEOREM: If a quadrilateral is a parallelogram, it has diagonals which bisect each other.
815592
DEFINITION: A parallelogram is a quadrilateral with both pairs of opposite sides parallel. ... THEOREM: If a quadrilateral has consecutive angles which are supplementary, then it is a parallelogram. THEOREM: If a quadrilateral has diagonals which bisect each other, then it is a parallelogram.
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We have proved that the opposite sides of a parallelogram are equal, so if two adjacent sides are equal, then all four sides are equal and it is a rhombus. If two adjacent sides of a parallelogram are equal, then it is a rhombus. ... A quadrilateral whose diagonals bisect each other at right angles is a rhombus.
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Proof: Given ABCD, let the diagonals AC and BD intersect at E, we must prove that AE ∼ = CE and BE ∼ = DE. The converse is also true: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorem: Opposite angles of a parallelogram are congruent to each other.
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Theorem 2. In a parallelogram, opposite angles are equal. Conversely, if the opposite angles in a quadrilateral are equal, then it is a parallelogram.
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['All the properties of a parallelogram apply (the ones that matter here are parallel sides, opposite angles are congruent, and consecutive angles are supplementary).', 'All sides are congruent by definition.', 'The diagonals bisect the angles.']
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All sides and angles are congruent. Opposite sides are parallel to each other. The diagonals are congruent. The diagonals are perpendicular to and bisect each other.
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THEOREM: If a quadrilateral has 2 sets of opposite sides congruent, then it is a parallelogram. THEOREM: If a quadrilateral has 2 sets of opposite angles congruent, then it is a parallelogram. THEOREM: If a quadrilateral has consecutive angles which are supplementary, then it is a parallelogram.
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ANSWER: Show that a pair of corresponding angles is congruent, show that a pair of alternate interior angles is congruent, show that a pair of alternate exterior angles is congruent, or show that a pair of consecutive interior angles is supplementary.