Geometry: Let's All Fly a Kite!

Let's All Fly a Kite!

Although there are many designs for kites, my kites all look like the kite in Figure 15.5. These kites are constructed by attaching two sticks of different lengths together so that the sticks are perpendicular and one of the sticks bisects the other. In this construction, there are two pairs of congruent adjacent sides. And that is what makes a kite a kite. A kite is defined as a quadrilateral with two distinct pairs of congruent adjacent sides.

Figure 15.5The kite ABCD.

In a kite, one pair of opposite angles are congruent. Also, in a kite, the diagonals are perpendicular and one of them bisects the other. I'll walk you through the proofs of these statements and give you the opportunity to write the proof itself.

• Theorem 15.3: In a kite, one pair of opposite angles are congruent.
• Theorem 15.4: The diagonals of a kite are perpendicular, and the diagonal opposite the congruent angles bisects the other diagonal.

You can prove Theorem 15.3 by using the SSS Postulate. The kite ABCD has ¯AB ~= ¯AD and ¯BC ~= ¯CD, and the reflexive property of ~= enables you to write ¯AC ~= ¯AC. Then by CPOCTAC, you have ∠B ~= ∠D. As an added bonus, you also have ∠BAC ~= ∠DAC. This will come in handy when proving Theorem 15.4.

To prove Theorem 15.4, suppose that the two diagonals intersect at M, as shown in Figure 15.6. Consider the two small triangles formed: ΔAMD and ΔAMB. Because ¯AB ~= ¯AD , ¯AM ~= ¯AM , and ∠BAM ~= ∠DAM , you can use the SAS Postulate to show ΔAMD ~= ΔAMB. Thus ¯BM ~= ¯MD (and you've got bisection ofone of the diagonals) and ∠BMA ~= ∠DMA (and you have right angles).

Figure 15.6Kite ABCD.