a As shown, must

34. RSTV is a kite, with and

. If , how large is the angle formed: a by the bisectors of and ? b by the bisectors of and ?

35. In concave kite ABCD, there is an

interior angle at vertex B that is a reflex angle. Given that , find the measure of the indicated reflex angle.

36. If the length of side for kite

ABCD is 6 in., find the length of not shown. Recall that

37. Prove that the segment that joins

the midpoints of two sides of a triangle has a length equal to one- half the length of the third side. HINT: In the drawing, is extended to D, a point on . Also, is parallel to . AB CD CD MN m ∠A = m∠C = m∠D = 30° AC AB m ∠A = m∠C = m∠D = 30° ∠RST ∠SRV ∠STV ∠RST m ∠STV = 40° RV ⬜ VT RS ⬜ ST

26. If the diagonals of a quadrilateral bisect each other, then

the quadrilateral is a parallelogram.

27. In a kite, one diagonal is the perpendicular bisector of the

other diagonal.

28. One diagonal of a kite bisects two of the angles of the

kite. In Exercises 29 to 31, has M and N for midpoints of sides and , respectively.

29. Given:

Find: y , MN, and ST

30. Given:

Find: x , MN, and ST

31. Given:

Find: x , RM, and ST

32. In kite ABCD not shown, and

. If and find x .

33. In kite ABCD of Exercise 32, ,

and . Find the perimeter sum of lengths of all sides of kite ABCD. BC = x - 2 AD = x 3 + 3, AB = x 6 + 5 m ∠D = 9x 4 - 3, m ∠B = 3x 2 + 2 BC ⬵ DC AB ⬵ AD m ∠R = 60° ST = 5x - 3 RM = RN = 2x + 1 ST = x 2x + 5 MN = x 2 + 5 ST = 3y MN = 2y - 3 RT RS 䉭RST R M S N T Exercises 29–31 ? ? T S V R 40º A C B D Exercises 35, 36 A M B C D N

38. Prove that when the midpoints of consecutive sides of a

quadrilateral are joined in order, the resulting quadrilateral is a parallelogram. Rectangle Square Rhombus Pythagorean Theorem The Rectangle, Square, and Rhombus 4.3 KEY CONCEPTS THE RECTANGLE In this section, we investigate special parallelograms. The first of these is the rectangle abbreviated “rect.”, which is defined as follows: A rectangle is a parallelogram that has a right angle. See Figure 4.20. DEFINITION A D B C Figure 4.20 Any reader who is familiar with the rectangle may be confused by the fact that the pre- ceding definition calls for only one right angle. Because a rectangle is a parallelogram by definition, the fact that a rectangle has four right angles is easily proved by applying Corollaries 4.1.3 and 4.1.5. The proof of Corollary 4.3.1 is left to the student.