Given: Parallelogram ABCD with and

Quadrilaterals That Are Parallelograms Rectangle Kite The Parallelogram and Kite 4.2 KEY CONCEPTS The quadrilaterals discussed in this section have two pairs of congruent sides. THE PARALLELOGRAM Because the hypothesis of each theorem in Section 4.1 included a given parallelogram, our goal was to develop the properties of parallelograms. In this section, Theorems 4.2.1 to 4.2.3 take the form “If . . . , then this quadrilateral is a parallelogram.” In this section, we find that quadrilaterals having certain characteristics must be parallelograms. General Rule: This method answers the question, “Why would the last statement be true?” The answer often provides insight into the statements preceding the last statement. Illustration: In line 8 of Example 1, we state that RSTV is a parallelogram by definition. With in line 1, we need to show that as shown in line 7. RV ‘ ST RS ‘ VT STRATEGY FOR PROOF 왘 The “Bottom Up” Approach to Proof EXAMPLE 1 Give a formal proof of Theorem 4.2.1. If two sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram. THEOREM 4.2.1 GIVEN: In Figure 4.12a, and PROVE: RSTV is a ⵥ RS ⬵ VT RS ‘ VT PROOF Statements Reasons

1. and

2. Draw diagonal , as in Figure 4.12b

3. 4.

5. 6. ⬖ 7. 8. RSTV is a ⵥ RV ‘ ST ∠RVS ⬵ ∠VST 䉭RSV ⬵ 䉭TVS ∠RSV ⬵ ∠SVT VS ⬵ VS VS RS ⬵ VT RS ‘ VT 1. Given 2. Exactly one line passes through two points 3. Identity 4. If two lines are cut by a transversal, alternate interior are 5. SAS 6. CPCTC 7. If two lines are cut by a transversal so that alternate interior are , these lines are 8. If both pairs of opposite sides of a quadrilateral are , the quadrilateral is a parallelogram ‘ ‘ ⬵ ∠s ⬵ ∠s ‘ 쮿 V T R S a Figure 4.12 V T R S b Consider the Discover activity at the left. Through it, we discover another type of quadrilateral that must be a parallelogram. This activity also leads to the following theorem; proof of the theorem is left to the student. Discover Take two straws and cut each straw into two pieces so that the lengths of the pieces of one straw match those of the second. Now form a quadrilateral by placing the pieces end to end so that congruent sides lie in opposite positions. What type of quadrilateral is always formed? ANSWER A parallelogram Figure 4.13 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. THEOREM 4.2.2 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. THEOREM 4.2.3 In a kite, one pair of opposite angles are congruent. THEOREM 4.2.4 Another quality of quadrilaterals that determines a parallelogram is stated in Theorem 4.2.3. Its proof is also left to the student. To clarify the meaning of Theorem 4.2.3, see the drawing for Exercise 3 on page 193. When a figure is drawn to represent the hypothesis of a theorem, we should not in- clude more conditions than the hypothesis states. Relative to Theorem 4.2.3, if we drew two diagonals that not only bisected each other but also were equal in length, then the quadrilateral would be the special type of parallelogram known as a rectangle. We will deal with rectangles in the next section. THE KITE The next quadrilateral we consider is known as a kite. This quadrilateral gets its name from the child’s toy pictured in Figure 4.13. In the construction of the kite, there are two pairs of congruent adjacent sides. See Figure 4.14a on page 189. This leads to the for- mal definition of a kite. Exs. 1–4 A kite is a quadrilateral with two distinct pairs of congruent adjacent sides. DEFINITION The word distinct is used in this definition to clarify that the kite does not have four congruent sides. In Example 2, we verify Theorem 4.2.4 by proving that . With congruent sides as marked, . ∠A ⬵ ∠C ∠B ⬵ ∠D Discover Take two straws and cut them into pieces so the lengths match. Now form a quadrilateral by placing congruent pieces together. What type of quadrilateral is always formed? ANSWER Kite © Elemental ImagingShutterstock