Prove Parallelogram Theorems: CCSS.Math.Content.HSG-CO.C.11 - Geometry

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Question

What is known if is said to be a parallelogram?

Answer

A parallelogram is a special type of quadrilateral meaning, it is a shape that has four sides with opposite sides being parallel.

Drawing the parallelogram can be done as follows.

Looking at the image it can be said that,

$\overline{AB}||\overline{CD}$ and $\overline{AD}||\overline{BC}$ .

Therefore, all of the possible answer choices are correct.

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Question

Which of the following helps prove an image is a parallelogram?

Answer

A parallelogram is a special type of quadrilateral meaning, it is a shape that has four sides with opposite sides being parallel. Along with opposite sides being congruent, a parallelogram has two pairs of opposite angles that are congruent. Lastly, the diagonals of a parallelogram must bisect each other.

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Question

Which of the following helps prove an image is a parallelogram?

Answer

A parallelogram is a special type of quadrilateral meaning, it is a shape that has four sides with opposite sides being parallel. Along with opposite sides being congruent, a parallelogram has two pairs of opposite angles that are congruent. Lastly, the diagonals of a parallelogram must bisect each other.

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Question

Which of the following helps prove an image is a parallelogram?

Answer

A parallelogram is a special type of quadrilateral meaning, it is a shape that has four sides with opposite sides being parallel. Along with opposite sides being congruent, a parallelogram has two pairs of opposite angles that are congruent. Lastly, the diagonals of a parallelogram must bisect each other.

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Question

Determine whether the statement is true or false.

Bisecting a parallelogram along one of its diagonals creates two congruent triangles.

Answer

Given the statement:

"Bisecting a parallelogram along one of its diagonal creates two congruent triangles."

Recall that a parallelogram is a special type of quadrilateral meaning, it is a shape that has four sides with opposite sides being parallel. Along with opposite sides being congruent, a parallelogram has two pairs of opposite angles that are congruent. Lastly, the diagonals of a parallelogram must bisect each other. When a parallelogram is bisected along one of its diagonals it in fact creates two congruent triangles.

Therefore, the statement is true.

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Question

Determine whether the statement is true or false:

A rhombus is an example of a parallelogram.

Answer

Recall that a parallelogram is a special type of quadrilateral meaning, it is a shape that has four sides with opposite sides being parallel. Along with opposite sides being congruent, a parallelogram has two pairs of opposite angles that are congruent. Lastly, the diagonals of a parallelogram must bisect each other. Since a rhombus has all of these characteristics it too is considered a parallelogram.

Therefore, the statement is true.

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Question

Determine whether the statement is true or false:

A trapezoid is a parallelogram.

Answer

Recall that a parallelogram is a special type of quadrilateral meaning, it is a shape that has four sides with opposite sides being parallel. Along with opposite sides being congruent, a parallelogram has two pairs of opposite angles that are congruent. Lastly, the diagonals of a parallelogram must bisect each other.

A trapezoid has one set of opposite parallel sides however, they are not congruent. The opposite angles of a trapezoid are also not congruent therefore, the statement is false.

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Question

Determine whether the statement is true or false:

A quadrilateral has $\overline{AB}||\overline{CD}$ therefore, is a parallelogram.

Answer

Recall that a parallelogram is a special type of quadrilateral meaning, it is a shape that has four sides with opposite sides being parallel. Along with opposite sides being congruent, a parallelogram has two pairs of opposite angles that are congruent. Lastly, the diagonals of a parallelogram must bisect each other.

Since the statement only says that is an quadrilateral that has $\overline{AB}||\overline{CD}$ . This does not give enough information. A trapezoid has one pair of parallel sides but is not a parallelogram, therefore the statement, "A quadrilateral has $\overline{AB}||\overline{CD}$ therefore, is a parallelogram." is false.

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Question

Determine whether the statement is true or false:

To determine whether a figure is a parallelogram, you only need information on one pair of sides or one pair of angles.

Answer

Recall that a parallelogram is a special type of quadrilateral meaning, it is a shape that has four sides with opposite sides being parallel. Along with opposite sides being congruent, a parallelogram has two pairs of opposite angles that are congruent. Lastly, the diagonals of a parallelogram must bisect each other.

Since the statement gives one pair of sides and one pair of angles but does not specify whether the pairs are opposite it cannot be determined whether the figure is a parallelogram or not.

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Question

How can a parallelogram be constructed given an isosceles triangle?

Answer

Recall that a parallelogram is a special type of quadrilateral meaning, it is a shape that has four sides with opposite sides being parallel and congruent. Along with opposite sides being congruent, a parallelogram has two pairs of opposite angles that are congruent. Lastly, the diagonals of a parallelogram must bisect each other.

Since an isosceles triangle has two side lengths that are equal, if it is reflected over the base a rhombus is created which is a type of parallelogram.

If the isosceles triangle is rotated so that one of the sides is the diagonal of the parallelogram then the opposite sides of the parallelogram are congruent thus making the figure a parallelogram.

If side lengths are added the resulting image will have three equal sides and one side that isn't therefore, the new figure is not a parallelogram.

Thus, looking at the given answer options, "Draw a reflected isosceles triangle where the base is shared and becomes the diagonal of the parallelogram." is the best choice.

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Question

Determine whether the statement is true or false:

In the figure , $\measuredangle A\cong \measuredangle C$ and $\measuredangle A=60^\circ$ and $\measuredangle B=120^\circ$ , therefore is a parallelogram.

Answer

Recall that a parallelogram is a special type of quadrilateral meaning, it is a shape that has four sides with opposite sides being parallel and congruent. Along with opposite sides being congruent, a parallelogram has two pairs of opposite angles that are congruent. Lastly, the diagonals of a parallelogram must bisect each other.

Therefore, the statement is true.

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Question

Determine whether the statement is true or false:

In the figure , $\measuredangle A\cong \measuredangle C$ , $\measuredangle A=60^\circ$ , $\measuredangle B=120^\circ$ , and opposite side lengths are congruent, therefore is a parallelogram.

Answer

Recall that a parallelogram is a special type of quadrilateral meaning, it is a shape that has four sides with opposite sides being parallel and congruent. Along with opposite sides being congruent, a parallelogram has two pairs of opposite angles that are congruent. Lastly, the diagonals of a parallelogram must bisect each other.

Since the angles given represent those in a parallelogram, and the opposite sides are congruent, that makes the figure a parallelogram.

Therefore, the statement is true.

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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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