What if you were told that is a kite and you are given information about some of its angles or its diagonals? How would you find the measure of its other angles or its sides? After completing this Concept, you'll be able to find the value of a kite's unknown angles and sides.
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Guidance
A kite is a quadrilateral with two distinct sets of adjacent congruent sides. It looks like a kite that flies in the air.
From the definition, a kite could be concave. If a kite is concave, it is called a dart. The word distinct in the definition means that the two pairs of congruent sides have to be different. This means that a square or a rhombus is not a kite.
The angles between the congruent sides are called vertex angles. The other angles are called non-vertex angles. If we draw the diagonal through the vertex angles, we would have two congruent triangles.
Facts about Kites
1) The non-vertex angles of a kite are congruent.
If is a kite, then .
2) The diagonal through the vertex angles is the angle bisector for both angles.
If is a kite, then and .
3) Kite Diagonals Theorem: The diagonals of a kite are perpendicular.
and are isosceles triangles, so is the perpendicular bisector of (Isosceles Triangle Theorem).
Example A
Find the missing measures in the kites below.
a)
b)
Answer:
a) The two angles left are the non-vertex angles, which are congruent.
b) The other non-vertex angle is also . To find the fourth angle, subtract the other three angles from .
Example B
Use the Pythagorean Theorem to find the lengths of the sides of the kite.
Recall that the Pythagorean Theorem says , where is the hypotenuse. In this kite, the sides are the hypotenuses.
Example C
Prove that the non-vertex angles of a kite are congruent.
Given : with and
Prove :
Statement | Reason |
---|---|
1. and | 1. Given |
2. | 2. Reflexive PoC |
3. | 3. SSS |
4. | 4. CPCTC |
Guided Practice
is a kite.
Find:
Answers:
- by the Triangle Sum Theorem (remember that is a right angle because the diagonals are perpendicular.)
- because the diagonals are perpendicular.
- because it is congruent to .
Practice
For questions 1-6, find the value of the missing variable(s). All figures are kites.
For questions 7-11, find the value of the missing variable(s).
- Fill in the blanks to the proof below.
Given : and
Prove : is the angle bisector of and
Statement | Reason |
---|---|
1. and | 1. |
2. | 2. |
3. | 3. |
4. | 4. CPCTC |
5. is the angle bisector of and | 5. |
- Fill in the blanks to the proof below.
Given :
Prove :
Statement | Reason |
---|---|
1. and | 1. |
2. | 2. Definition of isosceles triangles |
3. is the angle bisector of and | 3. |
4. | 4. Isosceles Triangle Theorem |
5. | 5. |