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# Angle Measures in Given Quadrilaterals

## Find missing angle measures of quadrilaterals given a sum of 360 degrees.

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Understanding the Angle Measures of Quadrilaterals

Have you ever used a quadrilateral in a real-world object? Take a look at this dilemma.

Margie makes jewelry. She made this necklace to sell at a craft fair.

Can you identify the quadrilateral? In this Concept, you will learn how to accomplish this task.

### Guidance

A quadrilateral is any four-sided figure.

In the word “quadrilateral”, we find the word “quad” which means four. This means that any four-sided figure is considered a quadrilateral. Now, there are different types of quadrilaterals that we are going to learn about in this lesson.

We can say that a quadrilateral is any four-sided figure. We could consider this an umbrella category meaning that there are different types of quadrilaterals that we can identify in a specific way even though they are still quadrilaterals too.

Let’s look at identify the types of quadrilaterals.

The first type of quadrilateral to learn about is called a parallelogram. A parallelogram is a quadrilateral with opposite sides parallel and congruent.

Here is a picture of a parallelogram.

When you look at this picture, you can see that the opposite sides of the figure are parallel. They are also the same length-meaning congruent.

There are three main kinds of parallelograms.

Parallelograms can be plain old parallelograms like the one in the picture. They could also be a rectangle, square and rhombus.

A rectangle is a parallelogram with four right angles, where opposite sides are congruent and parallel. You have been looking at rectangles for a long time, but now you need to notice that there are specific properties that make a rectangle a rectangle.

A rhombus is a parallelogram with four congruent sides, but not necessarily four right angles. A rhombus can look like a square, but while a square is always a rhombus, a rhombus is not necessarily a square. A rhombus can only be a square if it has four right angles.

A square is a parallelogram too. The big difference between a square and a rectangle is that a square has four congruent sides. It also has four right angles though just like a rectangle.

There is one other type of quadrilateral. This quadrilateral is NOT a parallelogram. It is a special kind of quadrilateral. It is called a trapezoid. A trapezoid is a quadrilateral with one pair of opposite sides parallel.

One important thing to remember about quadrilaterals is that their four angles always have a sum of \begin{align*}360^\circ\end{align*}. This is true no matter what shape or size the quadrilateral is.

Notice how different the angles and the sides of the quadrilaterals are. Look closely, though. If you add up the measures of the four angles, they always equal \begin{align*}360^\circ\end{align*}. This is because every quadrilateral is actually two triangles put together. As we know, the three angles in all triangles always add up to \begin{align*}180^\circ\end{align*}.

This quadrilateral has been divided into two congruent triangles, each with angles of \begin{align*}120^\circ, 25^\circ\end{align*}, and \begin{align*}35^\circ\end{align*}. If we add these angles together, we get a sum of \begin{align*}180^\circ\end{align*}. If we step back and look at the whole quadrilateral, we see that it has two \begin{align*}120^\circ\end{align*} angles and two \begin{align*}60^\circ\end{align*} angles \begin{align*}(25^\circ + 35^\circ = 60^\circ)\end{align*}. When we add these together, we get a sum of \begin{align*}360^\circ\end{align*}: \begin{align*}60^\circ + 120^\circ + 60 + 120^\circ = 360^\circ\end{align*}. This will be true no matter what size each angle in the quadrilateral measures.

We can use what we know about quadrilaterals to analyze them. When we analyze quadrilaterals, we can find the measure of an unknown angle or side. Remember, one of the most important things to know about quadrilaterals is that their angles always add up to \begin{align*}360^\circ\end{align*}. That means that if we know the measure of any three angles, we can set up an equation to solve for the measure of the fourth. Let’s see how this works.

Find the measure of the unknown angle in the quadrilateral below.

We know that the four angles must have a sum of \begin{align*}360^\circ\end{align*}, so we can add the four angles, using \begin{align*}m\end{align*} to represent the unknown angle.

By solving for \begin{align*}m\end{align*}, we have found that the fourth angle has a measure of \begin{align*}110^\circ\end{align*}.

We can check our work by adding the four angles to see if they total \begin{align*}360^\circ\end{align*}.

\begin{align*}55^\circ + 90^\circ + 105^\circ + 110^\circ = 360^\circ\end{align*}

Our calculation was correct. We can always use this method when given three out of the four angles in a quadrilateral.

Often we can use what we know about the properties of quadrilaterals to find unknown measures without having to set up an equation. We can simply use reasoning to put the pieces together.

Identify each missing angle.

#### Example A

\begin{align*}110^\circ, 110^\circ, 70^\circ,?\end{align*}

Solution:  \begin{align*}70^\circ\end{align*}

#### Example B

\begin{align*}90^\circ, 90^\circ, 90^\circ,?\end{align*}

Solution:  \begin{align*}90^\circ\end{align*}

#### Example C

\begin{align*}100^\circ, 100^\circ, 80^\circ,?\end{align*}

Solution:  \begin{align*}80^\circ\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

Look at the necklace that Margie made once again.

Now let’s examine this picture. We can look for the qualities that identify this quadrilateral. Notice that it has two parallel sides. The other two sides aren’t parallel or congruent. With one pair of parallel sides, this figure must be a trapezoid.

### Vocabulary

any four-sided figure.
Trapezoid
a quadrilateral with one pair of parallel sides.
Parallelogram
a quadrilateral with two pairs of opposite sides that are congruent and parallel.
Rhombus
a parallelogram with four congruent sides.
Rectangle
a parallelogram with opposites congruent and four right angles.
Square
a parallelogram with four congruent sides and four right angles.
Congruent
means exactly the same.

### Guided Practice

Here is one for you to try on your own.

Find the measures of the unknown angles in the quadrilateral below.

Solution

This time we have only been given the measures of two angles and we need to solve for the other two. First let’s determine what we know about the figure. What kind of quadrilateral is it? It has two pairs of parallel sides, so it must be a parallelogram. It doesn’t have \begin{align*}90^\circ\end{align*} angles, so it’s not a rectangle or square. Finally, the side lengths are not all congruent so it cannot be a rhombus. It is a regular parallelogram.

Now, what do we know about the angles of parallelograms? Not only do they add up to \begin{align*}360^\circ\end{align*}, they fall into two congruent pairs. The congruent angles are opposite each other. Take a look back at the figure.

Angle \begin{align*}x\end{align*} is opposite the \begin{align*}56^\circ\end{align*} angle. Therefore it must also be \begin{align*}56^\circ\end{align*}. Angle \begin{align*}y\end{align*} is opposite the \begin{align*}124^\circ\end{align*} angle, so it must also be \begin{align*}124^\circ\end{align*}. This gives us two pairs of congruent angles.

Let’s check to make sure these are the correct measurements by adding them to see if they total \begin{align*}360^\circ\end{align*}.

\begin{align*}124^\circ + 124^\circ + 56^\circ + 56^\circ = 360^\circ\end{align*}

They do, so our answers are correct.

### Practice

Directions: Use what you have learned about quadrilaterals to figure out the missing angle measure of each quadrilateral based on three given angles.

1. \begin{align*}120^\circ, 120^\circ, 60^\circ,?\end{align*}
2. \begin{align*}50^\circ, 70^\circ, 130^\circ,?\end{align*}
3. \begin{align*}52^\circ, 128^\circ, 52^\circ,?\end{align*}
4. \begin{align*}47^\circ, 55^\circ, 120^\circ,?\end{align*}
5. \begin{align*}80^\circ, 80^\circ, 100^\circ,?\end{align*}
6. \begin{align*}105^\circ, 105^\circ, 85^\circ,?\end{align*}
7. \begin{align*}97^\circ, 97^\circ, 35^\circ,?\end{align*}
8. \begin{align*}120^\circ, 120^\circ, 40^\circ,?\end{align*}
9. \begin{align*}88^\circ, 90^\circ, 60^\circ,?\end{align*}
10. \begin{align*}25^\circ, 85^\circ, 85^\circ,?\end{align*}
11. \begin{align*}90^\circ, 90^\circ, 90^\circ,?\end{align*}
12. \begin{align*}140^\circ, 150^\circ, 45^\circ,?\end{align*}
13. \begin{align*}80^\circ, 80^\circ, 120^\circ,?\end{align*}
14. \begin{align*}75^\circ, 95^\circ, 110^\circ,?\end{align*}
15. \begin{align*}80^\circ, 50^\circ, 95^\circ,?\end{align*}

### Vocabulary Language: English

Congruent

Congruent

Congruent figures are identical in size, shape and measure.
Parallelogram

Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides.

A quadrilateral is a closed figure with four sides and four vertices.
Rectangle

Rectangle

A rectangle is a quadrilateral with four right angles.
Rhombus

Rhombus

A rhombus is a quadrilateral with four congruent sides.
Square

Square

A square is a polygon with four congruent sides and four right angles.
Trapezoid

Trapezoid

A trapezoid is a quadrilateral with exactly one pair of parallel opposite sides.