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 RSTV 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 \overline {RT}. Remember that the slope is the change in the y-coordinate over the change in the x-coordinate.

\text{slope of }\overline{RT} &= \frac{(y_2 - y_1)} {(x_2 - x_1)}\\ &= \frac{(2 - 3)} {(3 - 2)}\\ &= \frac{-1}{1}\\ &=-1

The slope of \overline {RT} is -1. You can also find the slope of \overline {VS} using the same method.

\text{slope of }\overline{VS} &= \frac{(y_2 - y_1)} {(x_2 - x_1)}\\ &= \frac{(4 - (-1))} {(4 - (-1))}\\ &= \frac{5}{5}\\ &= 1

The slope of \overline {VS} is 1. If you think of both of these numbers as fractions, -\frac{1} {1} and \frac{1} {1}, 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 PART with \overline {PA} \cong \overline {PT} and \overline {AR} \cong \overline {RT}
  • Prove: \overline {PR} bisects \angle {APT} and \angle {ART}

Statement

Reason

1. \overline {PA} \cong \overline {PT} and \overline {AR} \cong \overline {RT}

 1. Given

2. \overline {PR} \cong \overline {PR}

 2. Reflexive Property

3. \triangle {PAR} \cong \triangle {PTR}

 3. SSS Congruence Postulate

4. \angle {APR} \cong \angle {TPR}

 4. Corresponding parts of congruent triangles are congruent

5. \angle {ARP} \cong \angle {TRP}

 5. Corresponding parts of congruent triangles are congruent

6. \overline {PR} bisects \angle {APT} and \angle {ART}

 6. Definition of angle bisector \blacklozenge

Now we will prove that the diagonals are perpendicular.

Part 2:

  • Given: Kite PART with \overline {PA} \cong \overline {PT} and \overline {AR} \cong \overline {RT}
  • Prove: \overline {PR} \perp \overline {AT}

Statement

Reason

1. Kite PART with \overline {PA} \cong \overline {PT} and \overline {AR} \cong \overline {RT}

1. Given

2. \overline {PY} \cong \overline {PY}

2. Reflexive Property of \cong

3. \angle {APR} \cong \angle {TPR}

3. By part 1 above: The diagonal between vertex angles bisects the angles

4. \triangle {PAY} \cong \triangle {PTY}

4. SAS Congruence Postulate

5. \angle {AYP} \cong \angle {TYP}

5. Corresponding parts of congruent triangles are congruent

6. \angle {AYP} and \angle {TYP} are supplementary

6. Linear Pair Postulate

7. \angle {AYP} and \angle {TYP} are right angles

7. Congruent supplementary angles are right angles

8. \overline {PR} \perp \overline {AT}

8. Definition of perpendicular \blacklozenge

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, \angle {PAR} \cong \angle {PTR}.

Example 2

Complete the two-column proof below.

  • Given: \overline {PA} \cong \overline {PT} and \overline {AR} \cong \overline {RT}
  • Prove: \angle {PAR} \cong \angle {PTR}

Statement

Reason

1. \overline {PA} \cong \overline {PT} 1. Given

2. \overline{AR} \cong \overline {RT}

2. Given

3. _____________

3. Reflexive Property

4. ______________

4. SSS \cong SSS

If two triangles have three pairs of congruent sides, the triangles are congruent.

5. \angle {PAR} \cong \angle {PTR}

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.


Last modified: Monday, June 28, 2010, 5:09 PM