Measures Of Parallelograms, Including Angles, Sides, And Diagonals

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Defining all the measures of a parallelogram

A parallelogram is a quadrilateral that has opposite sides that are parallel.

The parallel sides let you know a lot about a parallelogram.

Here are the special properties of parallelograms:

Parallelogram

Two pairs of opposite parallel sides

Opposite sides are equal lengths

Opposite angles are congruent

???m\angle 1=m\angle 3???

???m\angle 2=m\angle 4???

Consecutive angles are supplementary

???m\angle 1+m\angle 2=180^\circ???

???m\angle 2+m\angle 3=180^\circ???

???m\angle 3+m\angle 4=180^\circ???

???m\angle 4+m\angle 1=180^\circ???

Diagonals bisect each other (cut each other in half)

How to solve for every measure of a parallelogram, including angles, side lengths, and the lengths of diagonals

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Finding the measure of a interior angle of a parallelogram

Example

Find the measure of angle ???y???, given ???JKLM??? is a parallelogram.

Opposite angles of parallelograms are congruent, so

???m\angle JML=m\angle JKL=57^\circ???

Now we can use the fact that opposite sides of a parallelogram are parallel to state that ???JK\parallel ML???. This means that the diagonal ???JL??? of the parallelogram is also a transversal of these two parallel lines. This means that ???\angle KLJ??? and ???\angle MJL??? are alternate interior angles. Alternate interior angle pairs are congruent, so ???m\angle KLJ=m\angle MJL=y???.

The measures of the three interior angles of a triangle add up to ???180^\circ???, so we can set up an equation for the sum of the interior angles of ???\triangle JML??? and solve for ???y???.

???y+57^\circ+64^\circ=180^\circ???

???y=59^\circ???

Example

If ???STUV??? is a parallelogram, and if ???VT=4n+34??? and ???VE=7n-3???, what is the length of ???ET????

We know that the diagonals of a parallelogram bisect each other. Let’s add this information into the diagram.

Now we can see the relationships we need. Because the diagonals bisect, ???VE=ET??? and ???VE=(1/2)VT???. We can use what we know to find the length of ???VE??? and then we’ll know the length of ???ET??? as well.

???VE=\frac{1}{2}VT???

???7n-3=\frac{1}{2}(4n+34)???

???7n-3=2n+17???

???5n=20???

???n=4???

Now we can substitute back in to find the length of ???VE???, which is equal to the length of ???ET???.

???VE=ET=7n-3???

???VE=ET=7(4)-3???

???VE=ET=25???

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Get started Learn mathKrista KingNovember 16, 2020math, learn online, online course, online math, geometry, parallelograms, measures of parallelograms, angles of a parallelogram, sides of a parallelogram, side lengths of a parallelogram, diagonals of a parallelogram, parallel sides, equivalent angles, equal angles, bisecting diagonals Facebook0 Twitter LinkedIn0 Reddit Tumblr Pinterest0 0 Likes Previous

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