Congruence: Formal Geometric Constructions (CCSS.G-CO.11)
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Common Core High School Geometry › Congruence: Formal Geometric Constructions (CCSS.G-CO.11)
Parallelogram ABCD has vertices A(0,0), B(6,2), C(7,7), and D(1,5).
Which property holds in all parallelograms?
The diagonals are perpendicular.
All four sides are congruent.
Opposite sides are congruent.
All angles are right angles.
Explanation
In every parallelogram, opposite sides are congruent. Here AB and CD match in length, and AD and BC match in length. Choice B is rhombus-specific, A is a rhombus/kite property, and D is rectangle-specific.
Parallelogram ABCD has vertices A(1,1), B(6,2), C(7,7), and D(2,6).
Which property holds in all parallelograms?
The diagonals bisect each other at (4,4).
The diagonals are congruent.
The diagonals are perpendicular.
The diagonals intersect at (3,3).
Explanation
In a parallelogram, diagonals bisect each other. Midpoint of AC is ((1+7)/2,(1+7)/2)=(4,4) and midpoint of BD is ((6+2)/2,(2+6)/2)=(4,4). Choice D is an arithmetic slip; B is rectangle-specific; C is not true in general.
In parallelogram WXYZ, the diagonals have lengths WY = 13 and XZ = 13.
Which conclusion follows if diagonals are congruent?
The figure must be a rhombus.
The diagonals are perpendicular.
Opposite angles are acute.
The parallelogram is a rectangle.
Explanation
In a parallelogram, congruent diagonals imply the parallelogram is a rectangle. Choice B confuses with a rhombus/kite property; A and C are not guaranteed by congruent diagonals.
Parallelogram PQRS has vertices P(1,2), Q(7,3), R(9,8), and S(3,7).
Which property holds in all parallelograms?
Consecutive angles are congruent.
Opposite angles are congruent.
Diagonals are perpendicular.
All angles are right angles.
Explanation
Opposite angles in a parallelogram are congruent. Consecutive angles are supplementary, not congruent (unless it is a rectangle). Perpendicular diagonals and all right angles are special cases, not true for all parallelograms.
In parallelogram ABCD, diagonals AC and BD intersect at E. Suppose AE = 7, CE = 7, BE = 5, and DE = 5.
Which property holds in all parallelograms?
The diagonals are congruent.
Opposite sides are all perpendicular.
The diagonals bisect each other.
BD = 14.
Explanation
Diagonals of a parallelogram bisect each other, so AE = CE and BE = DE, as given. The diagonals need not be congruent (here AC = 14 and BD = 10), and BD = 14 is an arithmetic slip confusing AC with BD. Perpendicular sides are not required.