Do all angles have to be congruent for quadrilaterals to be parallelograms?

Answer 1

The broadest term we've used to describe any kind of shape is "polygon." When we discussed quadrilaterals in the last section, we essentially just specified that they were polygons with four vertices and four sides. Still, we will get more specific in this section and discuss a special type of quadrilateral: the parallelogram. Before we do this, however, let's go over some definitions that will help us describe different parts of quadrilaterals.

Quadrilateral TerminologySince this entire section is dedicated to the study of quadrilaterals, we will use some terminology that will help us describe specific pairs of lines, angles, and vertices of quadrilaterals. Let's study these terms now. Consecutive AnglesTwo angles whose vertices are the endpoints of the same side are called consecutive angles.

∠Q and ∠R are consecutive angles because Q and R are the endpoints of the same side.

Opposite AnglesTwo angles that are not consecutive are called opposite angles.

∠Q and ∠S are opposite angles because they are not endpoints of a common side.

Consecutive SidesTwo sides of a quadrilateral that meet are called consecutive sides.

QR and RS are consecutive sides because they meet at point R.

Opposite SidesTwo sides that are not consecutive are called opposite sides.

QR and TS are opposite sides of the quadrilateral because they do not meet.

Now, that we understand what these terms refer to, we are ready to begin our lesson on parallelograms.

Properties of Parallelograms: Sides and AnglesA parallelogram is a type of quadrilateral whose pairs of opposite sides are parallel.

Quadrilateral ABCD is a parallelogram because AB∥DC and AD∥BC.

Although the defining characteristics of parallelograms are their pairs of parallel opposite sides, there are other ways we can determine whether a quadrilateral is a parallelogram. We will use these properties in our two-column geometric proofs to help us deduce helpful information.

If a quadrilateral is a parallelogram, then…

(1) its opposite sides are congruent,

(2) its opposite angles are congruent, and

(2) its consecutive angles are supplementary.

Another important property worth noticing about parallelograms is that if one angle of the parallelogram is a right angle, then they all are right angles. Why is this property true? Let's examine this situation closely. Consider the figure below.

Given that ∠J is a right angle, we can also determine that ∠L is a right angle since the opposite sides of parallelograms are congruent. Together, the sum of the measure of those angles is 180 because

We also know that the remaining angles must be congruent because they are also opposite angles. By the Polygon Interior Angles Sum Theorem, we know that all quadrilaterals have angle measures that add up to 360. Since ∠Jand ∠L sum up to 180, we know that the sum of ∠K and ∠M will also be 180:

Since ∠K and ∠M are congruent, we can define their measures with the same variable, x. So we have

Therefore, we know that ∠K and ∠Mare both right angles. Our final illustration is shown below.

Let's work on a couple of exercises to practice using the side and angle properties of parallelograms.

Exercise 1Given that QRST is a parallelogram, find the values of x and y in the diagram below.

Solution:

After examining the diagram, we realize that it will be easier to solve for x first because y is used in the same expression as x (in ∠R), but x is by itself at segment QR. Since opposite sides of parallelograms are congruent, we have can set the quantities equal to each other and solve for x:

Now that we've determined that the value of x is 7, we can use this to plug into the expression given in ∠R. We know that ∠R and ∠T are congruent, so we have

Substitute x for 7 and we get

So, we've determined that x=7 and y=8.

Exercise 2Given that EDYF is a parallelogram, determine the values of x and y.

Solution:

In order to solve this problem, we will need to use the fact that consecutive angles of parallelograms are supplementary. The only angle we can figure out initially is the one at vertex Y because all it requires is the addition of angles. We have

Knowing that ∠Y has a measure of 115will allow us to solve for x and ysince they are both found in angles consecutive to ∠Y. Let's solve for y first. We have

All that is left for solve for is x now. We will use the same method we used when solving for y:

So, we have x=10 and y=13.

The sides and angles of parallelograms aren't their only unique characteristics. Let's learn some more defining properties of parallelograms.

Properties of Parallelograms: DiagonalsWhen we refer to the diagonals of a parallelogram, we are talking about lines that can be drawn from vertices that are not connected by line segments. Every parallelogram will have only two diagonals. An illustration of a parallelogram's diagonals is shown below.

We have two important properties that involve the diagonals of parallelograms.

If a quadrilateral is a parallelogram, then…

(1) its diagonals bisect each other, and

(2) each diagonal splits the parallelogram into two congruent triangles.

Segments AE and CE are congruent to each other because the diagonals meet at point E, which bisects them. Segments BE and DE are also congruent.

The two diagonals split the parallelogram up into congruent triangles.

Let's use these properties for solve the following exercises.

Exercise 3Given that ABCD is a parallelogram, find the value of x.

Solution:

We know that the diagonals of parallelograms bisect each other. This means that the point E splits up each bisector into two equivalent segments. Thus, we know that DEand BE are congruent, so we have

So, the value of x is 3.

Exercise 4Given that FGHI is a parallelogram, find the values of x and y.

Let's try to solve for x first. We are given that ∠FHI is a right angle, so it has a measure of 90°. We can deduce that ∠HFG is also a right angle by the Alternate Interior Angles Theorem.

If we look at ∆HIJ, we notice that two of its angles are congruent, so it is an isosceles triangle. This means that ∠HIJ has a measure of 9x since ∠IJH has that measure.

We can use the fact that the triangle has a right angle and that there are two congruent angles in it, in order to solve for x. We will use the Triangle Angle Sum Theoremto show that the angles must add up to 180°.

Now, let's solve for y. We do not know if segments IJ and FJ are congruent because they are not part of the same diagonal. Therefore, we cannot set them equal to each other, yet.

Since the sides opposite of congruent angles are congruent in isosceles triangles, we know that JH is congruent to IH.

Next, we know that IH is congruent to FG because opposite sides of parallelograms are congruent.

By the Alternate Interior Angles Theorem, we also know that ∠FGI is congruent to ∠GIH. This means that ∆FGJ is isosceles.

Since FG is on the side opposite of one of the congruent angles in ∆FGJ, then segments FG and FJ are congruent.

Finally, by transitivity, we can say that IJ and FJ are congruent, so we have

So, our answers are x=5 and y=4.