∠ A | |
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∠ B | |
∠ C | |
∠ D |
A parallelogram is a quadrilateral made from two pairs of intersecting parallel lines. There are several rules involving:
Interactive Parallelogram
Two Pairs of Parallel Lines
![Parallelograms Shape and Properties](images/parallelogram-from-lines.png)
To create a parallelogram just think of 2 different pairs of parallel lines intersecting. ABCD is a parallelogram.
![Parallelograms Shape and Properties](images/parallelogram-from-lines.png)
Angles of Parallelogram
Opposite Angles are Congruent
![Parallelogram Angles Picture](images/parallelogram-angles-color-coded.png)
$$ \angle D \cong \angle B \\ \angle A \cong \angle C $$
Triangles can be used to prove this rule about the opposite angle.
Consecutive angles are supplementary
![consecutive angles of a parallelogram](images/consecutive-angles-of-parallelogram.png)
The following pairs of angles are supplementary
$$ \angle C $$ and $$ \angle D $$
$$ \angle C $$ and $$ \angle B $$
$$ \angle A $$ and $$ \angle B $$
$$ \angle A $$ and $$ \angle D $$
To explore these rules governing the angles of a parallelogram use Math Warehouse's interactive parallelogram.
Problem 1
![Picture Angles of Parallelogram](images/angles/practice-problem-1.png)
There are many different ways to solve this question. You know that the opposite angles are congruent and the adjacent angles are supplementary.
$$ \angle \red W = 40^{\circ} $$ since it is opposite $$ \angle Y $$ and opposite angles are congruent.
Since consecutive angles are supplementary
$$
m \angle Y + m \angle Z = 180 ^{\circ}
\\
40^{\circ} + m \angle Z = 180 ^{\circ}
\\
m \angle Z = 180 ^{\circ} - 40^{\circ}
\\
m \angle \red Z = 140 ^{\circ}
$$
Problem 2
![Paralelogram Angles Diagram](images/angles/parallelogram-angles-diagram.jpg)
Sides of A Parallelogram
![Sides of Parallelogram](images/sides/opposite-sides-parallelogram-congruent.gif)
The opposite sides of a parallelogram are congruent.
Triangles can be used to prove this rule about the opposite sides.
To explore these rules governing the sides of a parallelogram use Math Warehouse's interactive parallelogram.
Problem 3
Problem 4
![Parallelogram Sides Picture](images/sides/parallelogram-sides4.jpg)
![Sides Practice Problem](images/sides/answer.jpg)
Problem 5
Since opposite sides are congruent you can set up the following equations and solve for $$x $$: $ \text{ Equation 1} \\ 2x − 10 = x + 80 \\ x - 10 = 80 \\ x = 90 $
Since opposite sides are congruent you can set up the following equations and solve for $$y $$: $ \text{ Equation 2} \\ 3y − 4 = y + 20 \\ 2y − 4 = 24 \\ 2y = 24 \\ y = 12 $
Diagonals
![Parallelogram Diagonals Bisect Each Other](images/sides/parallelogram-sides-diags-v2.png)
The diagonals of a parallelogram bisect each other.
AO = OD
CO = OB
To explore these rules governing the diagonals of a parallelogram use Math Warehouse's interactive parallelogram.
Problem 6
Since the diagonals bisect each other, y = 16 and x = 22
Problem 7
$$ x + 40 = 2x + 18 \\ 40 = x +18 \\ 40 = x + 18 \\ 22 = x $$
A classic activity: have the students construct a quadrilateral and its midpoints, then create an inscribed quadrilateral. Then ask the students to measure the angles, sides etc.. of inscribed shape and use the measurements to classify the shape (parallelogram). Designed with Geometer's Sketchpad in mind .