Parallelogram Proofs - Common Core: High School - Geometry

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Question

Which of the following is the definition of a parallelogram?

Answer

A parallelogram is a quadrilateral with two pairs of opposite parallel sides such as the following.

Opposite sides of a parallelogram are congruent as well as its opposite angles.

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Question

Which of the following is the definition of a parallelogram?

Answer

A parallelogram is a quadrilateral with two pairs of opposite parallel sides such as the following.

Opposite sides of a parallelogram are congruent as well as its opposite angles.

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Question

Which of the following is the definition of a parallelogram?

Answer

A parallelogram is a quadrilateral with two pairs of opposite parallel sides such as the following.

Opposite sides of a parallelogram are congruent as well as its opposite angles.

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Question

Prove the following parallelogram has opposite congruent sides.

Answer

Thus, we have proven this parallelogram has congruent opposite sides.

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Question

Prove the following parallelogram has two pairs of opposite congruent angles.

Answer

Statement Reasoning

is a parallelogram. This is given in the problem.

We can add the line across the diagonal Connecting any two points make a line, so this is a valid line we can add.

$\overline{AB}$ is parallel to $\overline{CD}$

$\overline{AC}$ is parallel to $\overline{BD}$ We know this to be true according to the definition of a parallelogram.

$\angle 1\cong \angle 4$

$\angle 2\cong \angle 3$ Line is a transversal line intersection two parallel lines. We could extend lines and or lines and to make this relationship more clear. Alternate interior angles are formed by this transversal line.

$\overline{CB}$ is a common side between $\Delta ABC$ and $\Delta DCB$

$\Delta ABC\cong \Delta DCB$ We are able to use the Angle-Side-Angle Theorem because we have one congruent side between these two triangles, , and two pairs of congruent angles, $\angle 1\cong \angle 4$ , $\angle 2\cong \angle 3$

$\angle A\cong \angle D$ Congruent triangles have congruent corresponding parts by definition

$\angle 1= \angle 4, \angle 2= \angle 3$ Because these angles are congruent they are also equal

$\angle 1+\angle 2= \angle 4+\angle 3$ Since $\angle 2= \angle 3$ we can add one of these angles to each side and still keep the equation balanced and equal

$\angle 1+\angle 2= \angle C$

$\angle 4+\angle 3= \angle B$ The Angle Addition Postulate says that two side by side angles create a new angle whose measure is equal to their sum

$\angle C= \angle B$ We are simply substituting these equalities into the equation $\angle 1+\angle 2= \angle 4+\angle 3$

$\angle C\cong \angle B$ Equal angles are congruent

Therefore $\angle A\cong \angle D$ and $\angle C\cong \angle B$

Thus, we have proven that this parallelogram has two pairs of opposite congruent angles.

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Question

Prove that the following parallelogram has four pairs of consecutive supplementary angles.

Answer

Thus, we have shown that this parallelogram has four pairs of consecutive supplementary angles.

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Question

Prove the following parallelogram has diagonals that bisect each other.

Answer

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Question

Prove that the following parallelogram has diagonals that prove two congruent triangles.

Answer

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Question

True or False: Is the following quadrilateral a parallelogram?

Answer

This is true because a theorem states that if a quadrilateral has two consecutive angles that are supplementary, then the quadrilateral is a parallelogram.

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Question

Prove that since this quadrilateral has two pairs of opposite congruent sides, it is a parallelogram.

Answer

Thus, we have shown that if a quadrilateral has two pairs of congruent opposite sides, then it is a parallelogram.

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Question

Prove that since this quadrilateral has two pairs of opposite congruent angles it is a parallelogram.

Answer

Thus, we have shown that if a quadrilateral has two pairs of opposite congruent angles, it is a parallelogram.

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Question

Prove the following parallelogram has two pairs of opposite congruent angles.

Answer

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Question

Prove that if the following quadrilateral has a pair of opposite parallel, congruent sides, it is a parallelogram.

Answer

Thus, we have shown that if a quadrilateral has one pair of parallel, congruent opposite sides, then it is a parallelogram.

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