Geometry - Parallelogram Proofs

Which of the following is the definition of a parallelogram?

A quadrilateral with four right angles

A quadrilateral with two pairs of opposite parallel sides

A quadrilateral with four congruent sides and four congruent angles

A quadrilateral with one pair of parallel sides

Explanation

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.

Which of the following is the definition of a parallelogram?

A quadrilateral with four right angles

A quadrilateral with two pairs of opposite parallel sides

A quadrilateral with four congruent sides and four congruent angles

A quadrilateral with one pair of parallel sides

Explanation

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.

Which of the following is the definition of a parallelogram?

A quadrilateral with four right angles

A quadrilateral with two pairs of opposite parallel sides

A quadrilateral with four congruent sides and four congruent angles

A quadrilateral with one pair of parallel sides

Explanation

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.

Prove the following parallelogram has opposite congruent sides.

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Proof:

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Proof:

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Proof:

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Explanation

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

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

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Proof:

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Proof:

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Proof:

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Explanation

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

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

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Proof:

is a parallelogram (given)

We can add the line across the diagonal

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

$\overline{AC}$ is parallel to $\overline{BD}$ (definition of a parallelogram)

$\angle 1\cong \angle 4$

$\angle 2\cong \angle 3$ (alternate interior angles)

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

$\Delta ABC\cong \Delta DCB$ (By Angle-Side-Angle Theorem)

Because the triangles are congruent we can assume:

$\angle C\cong \angle B$ , $\angle A\cong \angle D$

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

Proof:

is a parallelogram (given)

We can add the line across the diagonal

Screen shot 2020 08 20 at 10.04.42 am

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

$\overline{AC}$ is parallel to $\overline{BD}$ (definition of a parallelogram)

$\angle 1\cong \angle 4$

$\angle 2\cong \angle 3$ (corresponding angles)

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

$\Delta ABC\cong \Delta DCB$ (By Angle-Side-Angle Theorem)

$\angle A\cong \angle D$ (corresponding parts of congruent triangles)

$\angle 1= \angle 4, \angle 2= \angle 3$

$\angle 1+\angle 2= \angle 4+\angle 3$ (Addition of equalities)

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

$\angle 4+\angle 3= \angle B$ (Angle Addition Postulate)

$\angle C= \angle B$

$\angle C\cong \angle B$

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

Proof:

is a parallelogram (given)

We can add the line across the diagonal

Screen shot 2020 08 20 at 10.04.42 am

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

$\overline{AC}$ is parallel to $\overline{BD}$ (definition of a parallelogram)

$\angle 1\cong \angle 4$

$\angle 2\cong \angle 3$ (alternate interior angles)

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

$\Delta ABC\cong \Delta DCB$ (By Angle-Side-Angle Theorem)

$\angle A\cong \angle D$ (corresponding parts of congruent triangles)

$\angle 1= \angle 4, \angle 2= \angle 3$

$\angle 1+\angle 2= \angle 4+\angle 3$ (Addition of equalities)

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

$\angle 4+\angle 3= \angle B$ (Angle Addition Postulate)

$\angle C= \angle B$

$\angle C\cong \angle B$

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

Explanation

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.

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$\overline{AB}$ 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.

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$\overline{CB}$ 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.

Prove the following parallelogram has diagonals that bisect each other.

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Proof:

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Proof:

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Proof:

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Explanation

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

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Proof:

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Proof:

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Proof:

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Explanation

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True or False: Is the following quadrilateral a parallelogram?

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False

True

Explanation

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

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

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