Parallelogram Proofs - 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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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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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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

Question

Prove the following parallelogram has opposite congruent sides.

Answer

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

Compare your answer with the correct one above

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

Compare your answer with the correct one above

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