Given that and TEST 534

31. The bisectors of two consecutive angles of are

shown. What can you conclude about ? ∠P ⵥHJKL L P K H J A E B D C 50 100 150 N S W E 50 100 150 200 250 300 50 100 150 200 250 300 Exercises 35, 36

32. When the bisectors of two consecutive angles of a

parallelogram meet at a point on the remaining side, what type of triangle is: a ? b ? c ? 䉭BCE 䉭ADE 䉭DEC

33. Draw parallelogram RSTV with and

. Which diagonal of has the greater length?

34. Draw parallelogram RSTV so that the diagonals have

the lengths and . Which two angles of have the greater measure?

35. The following problem is based on the Parallelogram

Law. In the scaled drawing, each unit corresponds to 50 mph. A small airplane travels due east at 250 mph. The wind is blowing at 50 mph in the direction due north. Using the scale provided, determine the approximate length of the indicated diagonal and use it to determine the speed of the airplane in miles per hour. ⵥRSTV SV = 4 RT = 5 ⵥRSTV m ∠S = 110° m ∠R = 70°

36. In the drawing for Exercise 35, the bearing direction in

which the airplane travels is described as north east, where x is the measure of the angle from the north axis toward the east axis. Using a protractor, find the approximate bearing of the airplane.

37. Two streets meet to form an obtuse angle at point B.

On that corner, the newly poured foundation for a building takes the shape of a parallelogram. Which diagonal, or , is longer? BD AC x ° A B r o a d w a y G r a n d A v e . A v e . B D C Exercises 37, 38

38. To test the accuracy of the foundation’s measurements,

lines strings are joined from opposite corners of the building’s foundation. How should the strings that are represented by and be related?

39. For quadrilateral ABCD, the measures of its angles are

, , , and . Determine the measure of each angle of ABCD and whether ABCD is a parallelogram.

40. Prove: In a parallelogram, the sum of squares of the

lengths of its diagonals is equal to the sum of squares of the lengths of its sides. m ∠D = 7 3 x - 16 m ∠C = 3 2 x - 11 m ∠B = 2x + 1 m ∠A = x + 16 BD AC 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