Consider any kite. TEST 534

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. The following theorem is true for rectangles but not for parallelograms in general. All angles of a rectangle are right angles. COROLLARY 4.3.1 The diagonals of a rectangle are congruent. THEOREM 4.3.2 All sides of a square are congruent. COROLLARY 4.3.3 Reminder A rectangle is a parallelogram. Thus, it has all the properties of a parallelogram plus some properties of its own. NOTE: To follow the flow of the proof in Example 1, it may be best to draw triangles NMQ and PQM of Figure 4.21 separately. EXAMPLE 1 Complete a proof of Theorem 4.3.2. GIVEN: Rectangle MNPQ with diagonals and See Figure 4.21. PROVE: MP ⬵ NQ NQ MP M Q N P Figure 4.21 PROOF Statements Reasons 1. Rectangle MNPQ with diagonals and 2. MNPQ is a 3. 4. 5. and are right 6. 7. 8. MP ⬵ NQ 䉭NMQ ⬵ 䉭PQM ∠NMQ ⬵ ∠PQM ∠s ∠PQM ∠NMQ MQ ⬵ MQ MN ⬵ QP ⵥ NQ MP 1. Given 2. By definition, a rectangle is a with a right angle 3. Opposite sides of a are 4. Identity 5. By Corollary 4.3.1, the four of a rectangle are right 6. All right are 7. SAS 8. CPCTC ⬵ ∠s ∠s ∠s ⬵ ⵥ ⵥ 쮿 Discover Given a rectangle MNPQ like a sheet of paper, draw diagonals and . From a second sheet, cut out formed by two sides and a diagonal of MNPQ. Can you position so that it coincides with ? ANSWER 䉭 NQP 䉭 MPQ 䉭 MPQ NQ MP Yes Exs. 1–4 THE SQUARE All rectangles are parallelograms; some parallelograms are rectangles; and some rectangles are squares. A B D C Square ABCD Figure 4.22 A square is a rectangle that has two congruent adjacent sides. See Figure 4.22. DEFINITION