**Learning Goal**

By the end of the lesson you will be able to . . . describe the properties and relationships of the interior angles of quadrilaterals.

Have you ever measured something and made a mistake that you couldn't figure out? Look at what happened to Tara.

At a skateboard park, one of the designs for the grind box shows that it has a rectangular face. Tara has measured the angles using a protractor, but something isn't right. Three of the four angles measured 95 degrees. Tara knows that her measurement is off because the face of the grind box is rectangular.

Why does Tara know this?

What should each angle measure if the face is rectangular?

**This Concept is about angle measures and quadrilateral. Pay attention and you will know the answers to these questions at the end of the Concept.**

### Guidance

In an earlier Concept, you learned that the sum of the interior angles of a triangle is equal to 180 degrees. You learned to draw specified triangles and to find missing angle measures.

**What about a quadrilateral?**

This Concept will teach you about the sum of the interior angles of a quadrilateral. We will use this information in problem solving.

**What is the sum of the interior angles of a quadrilateral?**

**To best understand this, let’s look at a square.**

**A square has four right angles. Each right angle is** \begin{align*}90^\circ\end{align*}. **We can add up the sum of the interior angles of a square and see how this is related to all quadrilaterals.**

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

**The sum of the interior angles of all quadrilaterals is**

\begin{align*}360^\circ\end{align*}.

**How can we use this information to find the measure of missing angles?**

**We can write an equation using the variable and given measurements and figure out the measure of the missing angle.**

\begin{align*}80 + 75 + 105 + x & = 360 \\ 260 + x & = 360 \\ 360 - 260 & = x \\ 100 & = x\end{align*}

**The missing angle is equal to** \begin{align*}100^\circ\end{align*}.

**You can use this information to help you when figuring out missing angle measures in different quadrilaterals.**

Example A

If one angle of a rectangle is 90 degrees, what are the measures of the other three angles?

**Solution: 90 degrees**

#### Example B

A quadrilateral has the following angle measures: 105, 90 and 88. What is the measure of the missing angle?

**Solution: 77 degrees**

#### Example C

A parallelogram has two congruent angles that are both 85 degrees. The other two angles are congruent. What is the measure of each missing angle?

**Solution: 95 degrees**

Now back to Tara and the grind box design.

At the skateboard park, one of the designs for the grind box shows that it has a rectangular face. Tara has measured the angles using a protractor, but something isn't right. Three of the four angles measured 95 degrees. Tara knows that her measurement is off because the face of the grind box is rectangular.

Why does Tara know this?

What should each angle measure if the face is rectangular?

Tara knows that she has made a mistake because a rectangle has four congruent angles. These angles are also all right angles. The measure of each angle should measure 90 degrees.

With this information, Tara began her design again and was able to correct all of her errors.

### Vocabulary

Here are the vocabulary words in this Concept.

- Quadrilateral
- closed figure with four sides and four vertices.

- Trapezoid
- Quadrilateral with one pair of opposite sides parallel.

- Rectangle
- Parallelogram with four right angles.

- Parallelogram
- Quadrilateral with opposite sides congruent and parallel.

- Square
- Four congruent sides and four congruent angles.

- Rhombus
- Parallelogram with four congruent sides.

- Parallel
- lines that are equidistant and will never intersect

- Congruent
- exactly the same, having the same measure

### Guided Practice

Here is one for you to try on your own.

If a quadrilateral has four congruent angles, which two figures could it be?

**Answer**

You have to know a couple of things to answer this question.

First, a quadrilateral has four sides, so it has four angles.

Next, congruent means that the measures of those angles are the same.

The sum of the interior angles of a quadrilateral equal 360 degrees.

The angles are all right angles or 90 degree angles.

**The figure is either a rectangle or a square.**

### Video Review

Here are videos for review.

Khan Academy Quadrilateral Properties - This video provides supporting information to this Concept.

### Practice

Directions: Answer each of the following questions about quadrilaterals.

1. True or false. A quadrilateral will always have only four sides.

2. The interior angles of a quadrilateral add up to be _________ degrees.

3. A square will have four ___________ degree angles.

4. A rectangle will have four ___________ degree angles.

5. True or false. A rhombus will also always have four right angles.

6. If the sum of three of the angles of a quadrilateral is equal to \begin{align*}300^\circ\end{align*}, it means that the measure of the missing angle is ____________.

7.

What is the value of \begin{align*}x\end{align*}?

8.

9.

10. What are all four angles of this rectangle equal to?

11. If the sum of the interior angles of a quadrilateral is equal to \begin{align*}360^\circ\end{align*}, how many triangles can you draw inside a quadrilateral?

12. How many degrees are in a triangle?

13. Write an equation to show how the angles of the two triangles are equal to 360 degrees.

Directions: Identify the following figures.

14.

15.