By now you might expect that if you add up various angles in polygons, there will be some sort of pattern or rule. For example, you know that the sum of the interior angles of a triangle will always be . From that fact, you have learned that you can find the sums of the interior angles of any polygons with sides using the expression . There is also a rule for exterior angles in a polygon. Let’s begin by looking at a triangle.
To find the exterior angles at each vertex, extend the segments and find angles supplementary to the interior angles.
The sum of these three exterior angles is:
So, the exterior angles in this triangle will sum to .
To compare, examine the exterior angles of a rectangle.
In a rectangle, each interior angle measures . Since exterior angles are supplementary to interior angles, all exterior angles in a rectangle will also measure .
Find the sum of the four exterior angles in a rectangle.
So, the sum of the exterior angles in a rectangle is also .
In fact, the sum of the exterior angles in any convex polygon will always be . It doesn’t matter how many sides the polygon has, the sum will always be .
We can prove this using algebra as well as the facts that at any vertex the sum of the interior and one of the exterior angles is always , and the sum of all interior angles in a polygon is .
Exterior Angle Sum: The sum of the exterior angles of any convex polygon is
Proof. At any vertex of a polygon the exterior angle and the interior angle sum to . So summing all of the exterior angles and interior angles gives a total of 180 degrees times the number of vertices:
On the other hand, we already saw that the sum of the interior angles was:
Putting these together we have
Example 2
What is in the diagram below?
in the diagram is marked as an exterior angle. So, we need to find the measure of one exterior angle on a polygon given the measures of all of the others. We know that the sum of the exterior angles on a polygon must be equal to , regardless of how many sides the shape has. So, we can set up an equation where we set all of the exterior angles shown (including ) summed and equal to . Using subtraction, we can find the value of .
The measure of the missing exterior angle is .
We can verify that our answer is reasonable by inspecting the diagram and checking whether the angle in question is acute, right, or obtuse. Since the angle should be obtuse, is a reasonable answer (assuming the diagram is accurate).
Lesson Summary
In this lesson, we explored exterior angles in polygons. Specifically, we have learned:
How to identify the exterior angles of convex polygons.
How to find the sums of exterior angles in convex polygons.
We have also shown one example of how knowing the sum of the exterior angles can help you find the measure of particular exterior angles.
Review Questions
For exercises 1-3, find the measure of each of the labeled angles in the diagram.
Draw an equilateral triangle with one set of exterior angles highlighted. What is the measure of each exterior angle? What is the sum of the measures of the three exterior angles in an equilateral triangle?
Recall that a regular polygon is a polygon with congruent sides and congruent angles. What is the measure of each interior angle in a regular octagon?
How can you use your answer to 5 to find the measure of each exterior angle in a regular octagon? Draw a sketch to justify your answer.
Use your answer to 6 to find the sum of the measures of the exterior angles of an octagon.
Complete the following table assuming each polygon is a regular polygon. Note: This is similar to a previous exercise with more columns—you can use your answer to that question to help you with this one.
Regular Polygon name Number of sides Sum of measures of interior angles Measure of each interior angle Measure of each exterior angle Sum of measures of exterior angles
triangle
octagon
decagon
Each exterior angle forms a linear pair with its adjacent internal angle. In a regular polygon, you can use two different formulas to find the measure of each exterior angle. One way is to compute (measure of each interior angle) in symbols
Alternatively, you can use the fact that all exterior angles in an gon sum to and find the measure of each exterior angle with by dividing the sum by . Again, in symbols this is
Use algebra to show these two expressions are equivalent.
Review Answers
,
Below is a sample sketch.
Each exterior angle measures , the sum of the three exterior angles is
Sum of the angles is . So, each angle measures
Since each exterior angle forms a linear pair with its adjacent interior angle, we can find the measure of each exterior angle with
Regular Polygon name Number of sides Sum of measures of interior angles Measure of each interior angle Measure of each exterior angle Sum of measures of exterior angles
triangle
square
pentagon
hexagon
heptagon
octagon
decagon
dodecagon
gon
One possible answer.
Classifying Quadrilaterals
Learning Objectives
Identify and classify a parallelogram.
Identify and classify a rhombus.
Identify and classify a rectangle.
Identify and classify a square.
Identify and classify a kite.
Identify and classify a trapezoid.
Identify and classify an isosceles trapezoid.
Collect the classifications in a Venn diagram.
Identify how to classify shapes on a coordinate grid.
Introduction
There are many different classifications of quadrilaterals. In this lesson, you will explore what defines each type of quadrilateral and also what properties each type of quadrilateral has. You have probably heard of many of these shapes before, but here we will focus on things we’ve learned about other polygons—the relationships among interior angles, and the relationships among the sides and diagonals. These issues will be explored in later lessons to further your understanding.
Parallelograms
A parallelogram is a quadrilateral with two pairs of parallel sides. Each of the shapes shown below is a parallelogram.
As you can see, parallelograms come in a variety of shapes. The only defining feature is that opposite sides are parallel. But, once we know that a figure is a parallelogram, we have two very useful theorems we can use to solve problems involving parallelograms: the Opposite Sides Theorem and the Opposite Angles Theorem.
We prove both of these theorems by adding an auxiliary line and showing that a parallelogram can be divided into two congruent triangles. Then we apply the definition of congruent triangles—the fact that if two triangles are congruent, all their corresponding parts are congruent (CPCTC).
An auxiliary line is a line that is added to a figure without changing the given information. You can always add an auxiliary line to a figure by connecting two points because of the Line Postulate. In many of the proofs in this chapter we use auxiliary lines.
Opposite Sides of Parallelogram Theorem: The opposite sides of a parallelogram are congruent.
Proof.
Given Parallelogram
Prove and
Statement
Reason
1. is a parallelogram.
1. Given
2. Draw Auxiliary segment and label the angles as follows.
2. Line Postulate
3.
3. Definition of parallelogram
4.
4. Alternate Interior Angles Theorem
5.
5. Definition of parallelogram
6.
6. Alternate Interior Angles Theorem
&
nbsp; 7.
7. Reflexive Property
8.
8. ASA Triangle Congruence Postulate
9. and
9. Definition of congruent triangles (all corresponding sides and angles are congruent)
Opposite Angles in Parallelogram Theorem: The opposite angles of a parallelogram are congruent.
Proof. This proof is nearly the same as the one above and you will do it as an exercise.
Rhombi
A rhombus (plural is rhombi or rhombuses) is a quadrilateral that has four congruent sides. While it is possible for a rhombus to have four congruent angles, it’s only one example. Many rhombi do NOT have four congruent angles.
Theorem: A rhombus is a parallelogram
Proof.
Given: Rhombus
Prove: and
Statement
Reason
1. is a Rhombus.
1. Given
2.
2. Definition of a rhombus
3. Add auxiliary segment .
3. Line Postulate
4.
4. Reflexive Property
5.
5. SSS
6.
6. Definition of Congruent Triangles
7.
7. Converse of AIA Theorem
8.
8. Definition of Congruent Triangles
9.
9. Converse of AIA Theorem
That may seem like a lot of work just to prove that a rhombus is a parallelogram. But, now that you know that a rhombus is a type of parallelogram, then you also know that the rhombus inherits all of the properties of a parallelogram. This means if you know something is true about parallelograms, it must also be true about a rhombus.
Rectangle
A rectangle is a quadrilateral with four congruent angles. Since you know that any quadrilateral will have interior angles that sum to (using the expression ), you can find the measure of each interior angle.
Rectangles will have four right angles, or four angles that are each equal to .
Square
A square is both a rhombus and a rectangle. A square has four congruent sides as well as four congruent angles. Each of the shapes shown below is a square.
Kite
A kite is a different type of quadrilateral. It does not have parallel sides or right angles. Instead, a kite is defined as a quadrilateral that has two distinct pairs of adjacent congruent sides. Unlike parallelograms or other quadrilaterals, the congruent sides are adjacent (next to each other), not opposite.
Trapezoid
A trapezoid is a quadrilateral that has exactly one pair of parallel sides. Unlike the parallelogram that has two pairs, the trapezoid only has one. It may or may not contain right angles, so the angles are not a distinguishing characteristic. Remember that parallelograms cannot be classified as trapezoids. A trapezoid is classified as having exactly one pair of parallel sides.
Isosceles Trapezoid
An isosceles trapezoid is a special type of trapezoid. Like an isosceles triangle, it has two sides that are congruent. As a trapezoid can only have one pair of parallel sides, the parallel sides cannot be congruent (because this would create two sets of parallel sides). Instead, the non-parallel sides of a trapezoid must be congruent.
Example 1
Which is the most specific classification for the figure shown below?
A. parallelogram
B. rhombus
C. rectangle
D. square
The shape above has two sets of parallel sides, so it is a parallelogram. It also has four congruent sides, making it a rhombus. The angles are not right angles (and we can’t assume we know the angle measures since they are unmarked), so it cannot be a rectangle or a square. While the shape is a parallelogram, the most specific classification is rhombus. The answer is choice B.
Example 2
Which is the most specific classification for the figure shown below? You may assume the diagram is drawn to scale.
A. parallelogram
B. kite
C. trapezoid
D. isosceles trapezoid
The shape above has exactly one pair of parallel sides, so you can rule out parallelogram and kite as possible classifications. The shape is definitely a trapezoid because of the one pair of parallel sides. For a shape to be an isosceles trapezoid, the other sides must be congruent. That is not the case in this diagram, so the most specific classification is trapezoid. The answer is choice C.
Using a Venn Diagram for Classification
You have just explored many different rules and classifications for quadrilaterals. There are different ways to collect and understand this information, but one of the best methods is to use a Venn Diagram. Venn Diagrams are a way to classify objects according to their properties. Think of a rectangle. A rectangle is a type of parallelogram (you can prove this using the Converse of the Interior Angles on the Same Side of the Transversal Theorem), but not all parallelograms are rectangles. Here’s a simple Venn Diagram of that relationship:
Notice that all rectangles are parallelograms, but not all parallelograms are rectangles. If an item falls into more than one category, it is placed in the overlapping section between the appropriate classifications. If it does not meet any criteria for the category, it is placed outside of the circles.
To begin organizing the information for a Venn diagram, you can analyze the quadrilaterals we have discussed thus far by three characteristics: parallel sides, congruent sides, and congruent angles. Below is a table that shows how each quadrilateral fits these characteristics.
Shape Number of pairs of parallel sides Number of pairs of congruent sides Four congruent angles
Parallelogram No
Rhombus No
Rectangle Yes
Square Yes
Kite No
Trapezoid No
Isosceles trapezoid No
Example 3
Organize the classification information in the table above in a Venn Diagram.
To begin a Venn Diagram, you must first draw a large ellipse representing the biggest category. In this case, that will be quadrilaterals.
Now, one class of quadrilaterals are parallelograms—all quadrilaterals with opposite sides parallel. But, not all quadrilaterals are parallelograms: kites have no pairs of parallel sides, and trapezoids only have one pair of parallel sides. In the diagram we can show this as follows:
Okay, we are almost there, but there are several types of parallelograms. Squares, rectangles, and rhombi are all types of parallelograms. Also, under the category of trapezoids we need to add isosceles trapezoids. The completed Venn diagram is like this:
You can use this Venn Diagram to quickly answer questions. For instance, is every square a rectangle? (Yes.) Is every rhombus a square? (No, but some are.)
Strategies for Shapes on a Coordinate Grid
You have already practiced some of the tricks for analyzing shapes on a coordinate grid. You actually have all of the tools you need to classify any quadrilateral placed on a grid. To find out whether sides are congruent, you can use the distance formula.
Distance Formula: Distance between points and
To find out whether lines are parallel, you can find the slope by computing . If the slopes are the same, the lines are parallel. Similarly, if you want to find out if angles are right angles, you can test the slopes of their lines. Perpendicular lines will have slopes that are opposite reciprocals of each other.
Example 4
Classify the shape on the coordinate grid below.
First identify whether the sides are congruent. You can use the distance formula four times to find the distance between the vertices.
For segment , find the distance between and .
For segment , find the distance between and .
For segment , find the distance between ) and .
For segment , find the distance between and .
So, the length of each segment is equal to , and the sides are al
l equal. At this point, you know that the figure is either a rhombus or a square. To distinguish, you’ll have to identify whether the angles are right angles. If one of the angles is a right angle, they all must be, so the shape will be a square. If it isn’t a right angle, then none of them are, and it is a rhombus.
You can check whether two segments form a right angle by finding the slopes of two intersecting segments. If the slopes are opposite reciprocals, then the lines are perpendicular and form right angles.
The slope of segment can be calculated by finding the “rise over the run”.
Now find the slope of an adjoining segment, like
The two slopes are and . These are opposite numbers, but they are not reciprocals. Remember that the opposite reciprocal of would be , so segments and are not perpendicular. Since the sides of do not intersect a right angle, you can rule out square. Therefore is a rhombus.
Lesson Summary
In this lesson, we explored quadrilateral classifications. Specifically, we have learned:
How to identify and classify a parallelogram.
How to identify and classify a rhombus.
How to identify and classify a rectangle.
CK-12 Geometry Page 22
