To determine the properties of trapezoids and kites, we will draw and analyze each of the polygons and use the theorems we already know about parallelograms and triangle congruence.

Trapezoids

Let's consider a trapezoid whose base angles are congruent. Since a trapezoid has two bases, we need to study two different cases.

Case $1$

Let's draw a trapezoid where the base angles corresponding to the larger base are congruent.

Next, let's draw a segment that passes through $C$ such that it is parallel to $\overline{A D} .$

Since $\overline{C M} ∥ \overline{A D},$ by the Corresponding Angles Theorem we have that $∠ A ≅ ∠ B M C .$ From this and the Converse of the Base Angles Theorem, we get that $\overline{C M} ≅ \overline{C B} .$

Finally, since $A M C D$ is a parallelogram, by the Parallelogram Opposite Sides Theorem we have that $\overline{C M} ≅ \overline{A D} .$ Then, the Transitive Property of Congruence tells us that $\overline{A D} ≅ \overline{B C},$ which implies that the trapezoid is isosceles.

Case $2$

This time we will consider a trapezoid where the base angles corresponding to the smaller base are congruent.

Since $\overline{A B} ∥ \overline{C D}$ and since $\overline{A D}$ and $\overline{B C}$ are transversals, by the Consecutive Interior Angles Theorem we have that both $∠ A$ and $∠ D,$ as well as $∠ B$ and $∠ C,$ are supplementary. Thus, by applying the Congruent Supplements Theorem, we get that $∠ A ≅ ∠ B .$

Notice that we've arrived to the same situation we studied in case $1 .$ Then, we conclude that the trapezoid is isosceles.

Property of Trapezoids
If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.

Kites

Let's consider a kite $A B C D .$

If we draw the diagonal $\overline{A C},$ it will divide the kite into two triangles which have three corresponding congruent sides.

Then, by the Side-Side-Side (SSS) Congruence Theorem, we have that $△ A B C ≅ △ A D C .$ This implies that $∠ B ≅ ∠ D .$

Notice that $∠ A$ cannot be congruent to $∠ C$ because this would imply that $A B C D$ would have both pairs of opposite angles congruent, and this would cause $A B C D$ to be a parallelogram due to the Parallelogram Opposite Angles Converse. Since a kite is not a parallelogram, this is a contradiction.

Property of Kites
A kite has exactly one pair of opposite angles congruent (the ones between the noncongruent adjacent sides).

Additionally, since $△ A B C ≅ △ A D C,$ we have that $∠ A C B ≅ ∠ A C D$ and $∠ C A B ≅ ∠ C A D,$ which means that $\overline{A C}$ bisects $∠ A$ and $∠ C .$

Next, let's draw the second diagonal $\overline{B D} .$ This time, the kite will be divided into four triangles.

By the Side-Angle-Side (SAS) Congruence Theorem, we have that $△ P C B ≅ △ P C D,$ which implies that $∠ B P C ≅ ∠ D P C .$ Notice that these angles form a linear pair, so they both are right angles.

Thus, we can write another property of kites.

Property of Kites
The diagonals of a kite are perpendicular.

Trapezoids

Case 1

Case 2

Kites

Case $1$

Case $2$

Exercise