Review Answers
The perimeter is .
Perpendicular bisector
A quadrilateral is a square if and only if it is a rhombus. This is FALSE because some rhombi are not squares. Quadrilateral below is a counterexample—it is a rhombus, but not a square
A quadrilateral has four right angles if and only if it is a rectangle. This is TRUE by the definition of rectangle.
Answers will vary, but any geometric definition can be written as a biconditional.
Trapezoids
Learning Objectives
Understand and prove that the base angles of isosceles trapezoids are congruent.
Understand and prove that if base angles in a trapezoid are congruent, it is an isosceles trapezoid.
Understand and prove that the diagonals in an isosceles trapezoid are congruent.
Understand and prove that if the diagonals in a trapezoid are congruent, the trapezoid is isosceles.
Identify the median of a trapezoid and use its properties.
Introduction
Trapezoids are particularly unique figures among quadrilaterals. They have exactly one pair of parallel sides so unlike rhombi, squares, and rectangles, they are not parallelograms. There are special relationships in trapezoids, particularly in isosceles trapezoids. Remember that isosceles trapezoids have non-parallel sides that are of the same lengths. They also have symmetry along a line that passes perpendicularly through both bases.
Isosceles Trapezoid
Non-isosceles Trapezoid
Base Angles in Isosceles Trapezoids
Previously, you learned about the Base Angles Theorem. The theorem states that in an isosceles triangle, the two base angles (opposite the congruent sides) are congruent. The same property holds true for isosceles trapezoids. The two angles along the same base in an isosceles triangle will also be congruent. Thus, this creates two pairs of congruent angles—one pair along each base.
Theorem: The base angles of an isosceles trapezoid are congruent
Example 1
Examine trapezoid below.
What is the measure of angle ?
This problem requires two steps to solve. You already know that base angles in an isosceles triangle will be congruent, but you need to find the relationship between adjacent angles as well. Imagine extending the parallel segments and on the trapezoid and the transversal . You’ll notice that the angle labeled is a consecutive interior angle with .
Consecutive interior angles along two parallel lines will be supplementary. You can find by subtracting from .
So, measures . Since is adjacent to the same base as in an isosceles trapezoid, the two angles must be congruent. Therefore,
Here is a proof of this property.
Given: Isosceles trapezoid with and
Prove:
Statement
Reason
1. is an isosceles trapezoid with
1. Given
2. Extend
2. Line Postulate
3. Construct as shown in the figure below such that
3. Parallel Postulate
with added auxiliary lines and markings
4. is a parallelogram
4. Definition of a parallelogram
5.
5. Opposite angles in a parallelogram are
6.
6. Opposite sides of a parallelogram are congruent
7. is isosceles
7. Definition of isosceles triangle
8.
8. Base angles in an isosceles triangle are
9.
9. Alternate Interior Angles Theorem
10.
10. Transitive Property of
11.
11. Transitive Property of
Identify Isosceles Trapezoids with Base Angles
In the last lesson, you learned about biconditional statements and converse statements. You just learned that if a trapezoid is an isosceles trapezoid then base angles are congruent. The converse of this statement is also true. If a trapezoid has two congruent angles along the same base, then it is an isosceles trapezoid. You can use this fact to identify lengths in different trapezoids.
First, we prove that this converse is true.
Theorem: If two angles along one base of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid
Given: Trapezoid with and
Prove:
This proof is very similar to the previous proof, and it also relies on isosceles triangle properties.
Statement
Reason
1. Trapezoid has and
1. Given
2. Construct
2. Parallel Postulate
3.
3. Corresponding Angles Postulate
4. is a parallelogram
4. Definition of a parallelogram
5.
5. Opposite sides of a parallelogram are
Trapezoid with auxiliary lines
6.
6. Transitive Property
7. is isosceles
7. Definition of isosceles triangle
8.
8. Converse of the Base Angles Theorem
9.
9. Transitive Property
Example 2
What is the length of in the trapezoid below?
Notice that in trapezoid , two base angles are marked as congruent. So, the trapezoid is isosceles. That means that the two non-parallel sides have the same length. Since you are looking for the length of , it will be congruent to . So, .
Diagonals in Isosceles Trapezoids
The angles in isosceles trapezoids are important to study. The diagonals, however, are also important. The diagonals in an isosceles trapezoid will not necessarily be perpendicular as in rhombi and squares. They are, however, congruent. Any time you find a trapezoid that is isosceles, the two diagonals will be congruent.
Theorem: The diagonals of an isosceles trapezoid are congruent
Example 3
Review the two-column proof below.
Given: is a trapezoid and
Prove:
Statement
Reason
1.
1. Given
2.
2. Base angles in an isosceles trapezoid are congruent
3.
3. Reflexive Property.
4.
4.
5.
5. Corresponding parts of congruent triangles are congruent
So, the two diagonals in the isosceles trapezoid are congruent. This will be true in any isosceles trapezoids.
Identifying Isosceles Trapezoids with Diagonals
The converse statement of the theorem stating that diagonals in an isosceles triangle are congruent is also true. If a trapezoid has congruent diagonals, it is an isosceles trapezoid. You can either use measurements shown on a diagram or use the distance formula to find the lengths. If you can prove that the diagonals are congruent, then you can identify the trapezoid as isosceles.
Theorem: If a trapezoid has congruent diagonals, then it is an isosceles trapezoid
Example 4
Is the trapezoid on the following grid isosceles?
It is true that you could find the lengths of the two sides to identify whether or not this trapezoid is isosceles. However, for the sake of this lesson, compare the lengths of the diagonals.
Begin by finding the length of . The coordinates of are and the coordinates of are .
Now find the length of . The coordinates of are and the coordinates of are .
Thus, we have shown that the diagonals are congruent. . Therefore, trapezoid is isosceles.
Trapezoid Medians
Trapezoids can also have segments drawn in called medians. The median of a trapezoid is a segment that connects the midpoints of the non-parallel sides in a trapezoid. The median is located half way between the bases of a trapezoid.
Example 5
In trapezoid below, segment is a median. What is the length of
The media
n of a trapezoid is a segment that is equidistant between both bases. So, the length of will be equal to half the length of . Since you know that , you can divide that value by . Therefore, is .
Theorem: The length of the median of a trapezoid is equal to half of the sum of the lengths of the bases
This theorem can be illustrated in the example above,
Therefore, the measure of segment is . We leave the proof of this theorem as an exercise, but it is similar to the proof that the length of the triangle midsegment is half the length of the base of the triangle.
Lesson Summary
In this lesson, we explored trapezoids. Specifically, we have learned to:
Understand and prove that the base angles of isosceles trapezoids are congruent.
Understand that if base angles in a trapezoid are congruent, it is an isosceles trapezoid.
Understand that the diagonals in an isosceles trapezoid are congruent.
Understand that if the diagonals in a trapezoid are congruent, the trapezoid is isosceles.
Identify the properties of the median of a trapezoid.
It is helpful to be able to identify specific properties in trapezoids. You will be able to use this information in many different ways.
Review Questions
Use the following figure for exercises 1-2.
Use the following figure for exercises 3-5.
________
Use the following diagram for exercises 6-7.
Can the parallel sides of a trapezoid be congruent? Why or why not? Use a sketch to illustrate your answer.
Can the diagonals of a trapezoid bisect each other? Why or why not? Use a sketch to illustrate your answer.
Prove that the length of the median of a trapezoid is equal to half of the sum of the lengths of the bases.
Review Answers
No, if the parallel (and by definition opposite) sides of a quadrilateral are congruent then the quadrilateral MUST be a parallelogram. When you sketch it, the two other sides must also be parallel and congruent to each other (proven in a previous section).
No, if the diagonals of a trapezoid bisect each other, then you have a parallelogram. We also proved this in a previous section.
We will use a paragraph proof. Start with trapezoid and midsegment .
Now, using the parallel postulate, construct a line through point that is parallel to . Label the new intersections as follows:
Now quadrilateral is a parallelogram by construction. Thus, the theorem about opposite sides of a parallelogram tells us The triangle midsegment theorem tells us that
or
So,
Kites
Learning Objectives
Identify the relationship between diagonals in kites.
Identify the relationship between opposite angles in kites.
Introduction
Among all of the quadrilaterals you have studied thus far, kites are probably the most unusual. Kites have no parallel sides, but they do have congruent sides. Kites are defined by two pairs of congruent sides that are adjacent to each other, instead of opposite each other.
A vertex angle is between two congruent sides and a non-vertex angle is between sides of different lengths.
Kites have a few special properties that can be proven and analyzed just as the other quadrilaterals you have studied. This lesson explores those properties.
Diagonals in Kites
The relationship of diagonals in kites is important to understand. The diagonals are not congruent, but they are always perpendicular. In other words, the diagonals of a kite will always intersect at right angles.
Theorem: The diagonals of a kite are perpendicular
This can be examined on a coordinate grid by finding the slope of the diagonals. Perpendicular lines and segments will have slopes that are opposite reciprocals of each other.
Example 1
Examine the kite on the following coordinate grid. Show that the diagonals are perpendicular.
To find out whether the diagonals in this diagram are perpendicular, find the slope of each segment and compare them. The slopes should be opposite reciprocals of each other.
Begin by finding the slope of . Remember that the slope is the change in the coordinate over the change in the coordinate.
The slope of is . You can also find the slope of using the same method.
The slope of is . If you think of both of these numbers as fractions, and , you can tell that they are opposite reciprocals of each other. Therefore, the two line segments are perpendicular.
Proving this property in general requires using congruent triangles (surprise!). We will do this proof in two parts. First, we will prove that one diagonal (connecting the vertex angles) bisects the vertex angles in the kite.
Part 1:
Given: Kite with and
Prove: bisects and
Statement
Reason
1. and
1. Given
2.
2. Reflexive Property
3.
3. Congruence Postulate
4.
4. Corresponding parts of congruent triangles are congruent
5.
5. Corresponding parts of congruent triangles are congruent
6. bisects and
6. Definition of angle bisector
Now we will prove that the diagonals are perpendicular.
Part 2:
Given: Kite with and
Prove:
Statement
Reason
1. Kite with and
1. Given
2.
2. Reflexive Property of
3.
3. By part 1 above: The diagonal between vertex angles bisects the angles
4.
4. Congruence Postulate
5.
5. Corresponding parts of congruent triangles are congruent
6. and are supplementary
6. Linear Pair Postulate
7. and are right angles
7. Congruent supplementary angles are right angles
8.
8. Definition of perpendicular
Opposite Angles in Kites
In addition to the bisecting property, one other property of kites is that the non-vertex angles are congruent.
So, in the kite PART above, .
Example 2
Complete the two-column proof below.
Given: and
Prove:
Statement
Reason
1. 1. Given
2.
2. Given
3. _____________
3. Reflexive Property
4. ______________
4.
If two triangles have three pairs of congruent sides, the triangles are congruent.
5.
5. ____________________________
We will let you fill in the blanks on your own, but a hint is that this proof is nearly identical to the first proof in this section.
So, you have successfully proved that the angles between the congruent sides in a kite are congruent.
Lesson Summary
In this lesson, we explored kites. Specifically, we have learned to:
Identify the relationship between diagonals in kites.
Identify the relationship between opposite angles in kites.
It is helpful to be able to identify specific properties in kites. You will be able to use this information in many different ways.
Points to Consider
Now that you have learned about different types of quadrilaterals, it is important to learn more about the relationships between shapes. The next chapter deals with similarity between shapes.
Review Questions
For exercises 1-5, use kite below with the given measurements.
For exercises 6-10, fill in the blanks in each sentence about Kite below:
The vertex angles of kite are _________ and __________.
___________ is the perpendicular bisector of
_______________.
Diagonal ___________ bisects ________ and _______.
_______, and .
The line of symmetry in the kite is along segment __________.
Can the diagonals of a kite be congruent to each other? Why or why not?
Review Answers
The vertex angles of kite are and
is the perpendicular bisector of
Diagonal bisects and
There are many possible answers:
is a line of reflection. Below is kite fully annotated with geometric markings.
CK-12 Geometry Page 25