Prove Parallelogram Theorems: CCSS.Math.Content.HSG-CO.C.11 - Common Core: High School - Geometry
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Which of the following is the definition of a parallelogram?
Which of the following is the definition of a parallelogram?
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.
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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Which of the following is the definition of a parallelogram?
Which of the following is the definition of a parallelogram?
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.
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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Which of the following is the definition of a parallelogram?
Which of the following is the definition of a parallelogram?
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.
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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Prove the following parallelogram has opposite congruent sides.
Prove the following parallelogram has opposite congruent sides.
Thus, we have proven this parallelogram has congruent opposite sides.
Thus, we have proven this parallelogram has congruent opposite sides.
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Prove the following parallelogram has two pairs of opposite congruent angles.
Prove the following parallelogram has two pairs of opposite congruent angles.
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.
is parallel to 
is parallel to 
We know this to be true according to the definition of a parallelogram.
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.
is a common side between 
and 
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, ,
Congruent triangles have congruent corresponding parts by definition
Because these angles are congruent they are also equal
Since 
we can add one of these angles to each side and still keep the equation balanced and equal
The Angle Addition Postulate says that two side by side angles create a new angle whose measure is equal to their sum
We are simply substituting these equalities into the equation 
Equal angles are congruent
Therefore 
and 
Thus, we have proven that this parallelogram has two pairs of opposite congruent angles.
Statement Reasoning
We can add the line 
Therefore 
Thus, we have proven that this parallelogram has two pairs of opposite congruent angles.
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Prove that the following parallelogram has four pairs of consecutive supplementary angles.
Prove that the following parallelogram has four pairs of consecutive supplementary angles.
Thus, we have shown that this parallelogram has four pairs of consecutive supplementary angles.
Thus, we have shown that this parallelogram has four pairs of consecutive supplementary angles.
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Prove the following parallelogram has diagonals that bisect each other.
Prove the following parallelogram has diagonals that bisect each other.
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Prove that the following parallelogram has diagonals that prove two congruent triangles.
Prove that the following parallelogram has diagonals that prove two congruent triangles.
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True or False: Is the following quadrilateral a parallelogram?
True or False: Is the following quadrilateral a parallelogram?
This is true because a theorem states that if a quadrilateral has two consecutive angles that are supplementary, then the quadrilateral is a parallelogram.
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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Prove that since this quadrilateral has two pairs of opposite congruent sides, it is a parallelogram.
Prove that since this quadrilateral has two pairs of opposite congruent sides, it is a parallelogram.
Thus, we have shown that if a quadrilateral has two pairs of congruent opposite sides, then it is a parallelogram.
Thus, we have shown that if a quadrilateral has two pairs of congruent opposite sides, then it is a parallelogram.
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Prove that since this quadrilateral has two pairs of opposite congruent angles it is a parallelogram.
Prove that since this quadrilateral has two pairs of opposite congruent angles it is a parallelogram.
Thus, we have shown that if a quadrilateral has two pairs of opposite congruent angles, it is a parallelogram.
Thus, we have shown that if a quadrilateral has two pairs of opposite congruent angles, it is a parallelogram.
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Prove the following parallelogram has two pairs of opposite congruent angles.
Prove the following parallelogram has two pairs of opposite congruent angles.
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Prove that if the following quadrilateral has a pair of opposite parallel, congruent sides, it is a parallelogram.
Prove that if the following quadrilateral has a pair of opposite parallel, congruent sides, it is a parallelogram.
Thus, we have shown that if a quadrilateral has one pair of parallel, congruent opposite sides, then it is a parallelogram.
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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What is known if 
is said to be a parallelogram?
What is known if
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,
and 
.
Therefore, all of the possible answer choices are correct.
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
Looking at the image it can be said that,
Therefore, all of the possible answer choices are correct.
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Which of the following helps prove an image is a parallelogram?
Which of the following helps prove an image is a parallelogram?
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 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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Which of the following helps prove an image is a parallelogram?
Which of the following helps prove an image is a parallelogram?
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 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.
Compare your answer with the correct one above
Which of the following helps prove an image is a parallelogram?
Which of the following helps prove an image is a parallelogram?
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 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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Determine whether the statement is true or false.
Bisecting a parallelogram along one of its diagonals creates two congruent triangles.
Determine whether the statement is true or false.
Bisecting a parallelogram along one of its diagonals creates two congruent triangles.
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.
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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Determine whether the statement is true or false:
A rhombus is an example of a parallelogram.
Determine whether the statement is true or false:
A rhombus is an example of a parallelogram.
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.
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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Determine whether the statement is true or false:
A trapezoid is a parallelogram.
Determine whether the statement is true or false:
A trapezoid is a parallelogram.
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.
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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