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

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 Two additional theorems involving the kite are found in Exercises 27 and 28 of this section. When observing an old barn or shed, we often see that it has begun to lean. Unlike a triangle, which is rigid in shape [Figure 4.15a] and bends only when broken, a quadrilateral [Figure 4.15b] does not provide the same level of strength and stability. In the construction of a house, bridge, building, or swing set [Figure 4.15c], note the use of wooden or metal triangles as braces. EXAMPLE 2 Complete the proof of Theorem 4.2.4. GIVEN: Kite ABCD with congruent sides as marked. [See Figure 4.14a.] PROVE: ∠B ⬵ ∠D A C B D a Figure 4.14 A C B D b PROOF Statements Reasons 1. Kite ABCD 2. and 3. Draw [Figure 4.14b] 4. 5. 6. ? 䉭ACD ⬵ 䉭ACB AC ⬵ AC AC AB ⬵ AD BC ⬵ CD 1. ? 2. A kite has two pairs of adjacent sides 3. Through two points, there is exactly one line 4. ? 5. ? 6. CPCTC ⬵ 쮿 Discover From a sheet of construction paper, cut out kite ABCD so that and . a When you fold kite ABCD along the diagonal , are two congruent triangles formed? b When you fold kite ABCD along diagonal , are two congruent triangles formed? ANSWERS BD AC BC = DC AB = AD a Yes b No Exs. 5–10 a Figure 4.15 b c The brace in the swing set in Figure 4.15c suggests the following theorem. The segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to one-half the length of the third side. THEOREM 4.2.5 Refer to Figure 4.16a; Theorem 4.2.5 claims that and . We will prove the first part of this theorem but leave the second part as an exercise. MN = 1 2 BC MN ‘ BC The line segment that joins the midpoints of two sides of a triangle is parallel to the third side of the triangle. GIVEN: In Figure 4.16a, with midpoints M and N of and , respectively. PROVE: MN ‘ BC AC AB 䉭ABC Discover Sketch regular hexagon ABCDEF. Draw diagonals and . What type of quadrilateral is ABCE? ANSWER CE AE Kite B a C A M N Figure 4.16 B b C A M N 2 3 D E

4 1

PROOF Statements Reasons 1. , with midpoints M and N of and , respectively 2. Through C, construct , as in Figure 4.16b 3. Extend to meet at D , as in Figure 4.16b 4. and 5. and 6. 7. 8. 9. Quadrilateral BMDC is a 10. MN ‘ BC ⵥ MB ⬵ DC AM ⬵ DC 䉭ANM ⬵ 䉭CND ∠4 ⬵ ∠3 ∠1 ⬵ ∠2 AN ⬵ NC AM ⬵ MB Í CE MN Í CE ‘ AB AC AB 䉭ABC 1. Given 2. Parallel Postulate 3. Exactly one line passes through two points 4. The midpoint of a segment divides it into segments 5. If two lines are cut by a transversal, alternate interior are 6. AAS 7. CPCTC 8. Transitive both are to 9. If two sides of a quadrilateral are both and , the quadrilateral is a parallelogram 10. Opposite sides of a are ‘ ⵥ ‘ ⬵ AM ⬵ ⬵ ∠s ‘ ⬵ Technology Exploration Use computer software if available. 1. Construct any triangle. 2. Where M is the midpoint of and N is the midpoint of , draw . 3. Measure and B. 4. Show that , which shows that . 5. Now measure and . 6. Show that . Measures may not be “perfect.” MN = 1 2 BC BC MN MN ‘ BC m ∠AMN = m∠B ∠ ∠AMN MN AC AB 䉭 ABC In the preceding proof, we needed to show that a quadrilateral having certain char- acteristics is a parallelogram. The line segment that joins the midpoints of two sides of a triangle has a length equal to one-half the length of the third side. General Rule: Methods for proof include the definition of parallelogram as well as Theorems 4.2.1, 4.2.2, and 4.2.3. Illustration: In the proof of Theorem 4.2.5, statements 2 and 8 allow the conclusion in statement 9 used Theorem 4.2.1. STRATEGY FOR PROOF 왘 Proving That a Quadrilateral Is a Parallelogram Theorem 4.2.5 also asserts the following: EXAMPLE 3 In in Figure 4.17, M and N are the midpoints of and , respectively. a If , find MN . b If , find ST . MN = 15.8 ST = 12.7 RT RS 䉭RST Solution a , so . b , so . Multiplying by 2, we find that . ST = 31.6 15.8 = 1 2 ST MN = 1 2 ST MN = 1 2 12.7 = 6.35 MN = 1 2 ST R M N S T Figure 4.17 쮿 Discover Draw a triangle with midpoints D of and E of . Cut out and place it at the base . By sliding along , what do you find? ANSWER AB DE AB 䉭 CDE CB CA 䉭 ABC or AB =2 DE DE= 1 2 AB EXAMPLE 4 GIVEN: in Figure 4.18, with D the midpoint of and E the midpoint of ; ; FIND: x , DE, and AB AB = 5x - 1 DE = 2x + 1 BC AC 䉭ABC A B C D E Figure 4.18 In the final example of this section, we consider the design of a product. Also see related Exercises 17 and 18 of this section. Solution By Theorem 4.2.5, so Multiplying by 2, we have Therefore, . Similarly, . NOTE: In Example 4, a check shows that . DE = 1 2 AB AB = 5 3 - 1 = 14 DE = 2 3 + 1 = 7 3 = x 4x + 2 = 5x - 1 2x + 1 = 1 2 5x - 1 DE = 1 2 AB 쮿 Exs. 11–15 EXAMPLE 5 In a studio apartment, there is a bed that folds down from the wall. In the vertical position, the design shows drop-down legs of equal length; that is, [see Figure 4.19a]. Determine the type of quadrilateral ABDC, shown in Figure 4.19b, that is formed when the bed is lowered to a horizontal position. AB = CD A B C D a Figure 4.19 B D A C b Solution See Figure 4.19a. Because , it follows that ; here, BC was added to each side of the equation. But and . Thus, by substitution. In Figure 4.19b, we see that and . Because both pairs of opposite sides of the quadrilateral are congruent, ABDC is a parallelogram. NOTE: In Section 4.3, we will also show that ABDC of Figure 4.19b is a rectangle a special type of parallelogram. 쮿 AC = BD AB = CD AC = BD BC + CD = BD AB + BC = AC AB + BC = BC + CD AB = CD Exercises 4.2

1. a As shown, must

quadrilateral ABCD be a parallelogram? b Given the lengths of the sides as shown, is the measure of unique? 2. a As shown, must RSTV be a parallelogram? b With measures as indicated, is it necessary that ? RS = 8 ∠A

8. In kite WXYZ, the measures of selected angles are shown.

Which diagonal of the kite has the greater length? D C A B 3 3 7 7 S R T V 5 8 W X Z Y Exercises 3, 4 A C B 6

6 4

4 D

3. In the drawing, suppose that and

bisect each other. What type of quadrilateral is WXYZ? XZ WY

4. In the drawing, suppose that is the perpendicular

bisector of . What type of quadrilateral is WXYZ?

5. A carpenter lays out boards of lengths 8 ft, 8 ft, 4 ft, and

4 ft by placing them end to end. a If these are joined at the ends to form a quadrilateral that has the 8-ft pieces connected in order, what type of quadrilateral is formed? b If these are joined at the ends to form a quadrilateral that has the 4-ft and 8-ft pieces alternating, what type of quadrilateral is formed?

6. A carpenter joins four boards of lengths 6 ft, 6 ft, 4 ft, and

4 ft, in that order, to form quadrilateral ABCD as shown. a What type of quadrilateral is formed? b How are angles B and D related? WY ZX

7. In parallelogram ABCD not shown, ,

, and . Which diagonal has the greater length? BC =

5 m

∠B = 110° AB = 8 X W Y Z 55 50 55 50 A M N C B Exercises 9, 10 S Y X Z R T Exercises 11–14

9. In ,

M and N are midpoints of and , respectively. If , how long is ? MN AB = 12.36 BC AC 䉭ABC

10. In ,

M and N are midpoints of and , respectively. If , how long is ? In Exercises 11 to 14, assume that X, Y, and Z are midpoints of the sides of .

11. If , , and ,

find: a XY b XZ c YZ RT = 16 ST = 14 RS = 12 䉭RST AB MN = 7.65 BC AC 䉭ABC

12. If , , and ,

find: a RS b ST c RT

13. If the perimeter sum of the lengths of all three sides of

is 20, what is the perimeter of ?

14. If the perimeter sum of the lengths of all three sides of

is 12.7, what is the perimeter of ?

15. Consider any kite.

a Does it have line symmetry? If so, describe an axis of symmetry. b Does it have point symmetry? If so, describe the point of symmetry. 䉭RST 䉭XYZ 䉭XYZ 䉭RST XZ = 10 YZ = 8 XY = 6 A B D C

16. Consider any parallelogram.

a Does it have line symmetry? If so, describe an axis of symmetry. b Does it have point symmetry? If so, describe the point of symmetry.

17. For compactness, the drop-down wheels of a stretcher or

gurney are folded under it as shown. In order for the board’s upper surface to be parallel to the ground when the wheels are dropped, what relationship must exist between and ? CD AB

18. For compactness, the

drop-down legs of an ironing board fold up under the board. A sliding mechanism at point A and the legs being connected at common midpoint M cause the board’s upper surface to be parallel to the floor. How are and related? In Exercises 19 to 24, complete each proof.

19. Given: and

Prove: MNPQ is a kite ∠3 ⬵ ∠4 ∠1 ⬵ ∠2 CD AB M D B C A N M P Q R T S Z Y W X M N M P Q 2 1 4 3 PROOF Statements Reasons

1. and

1. ? 2. 2. ? 3. ? 3. ASA 4. and 4. ? 5. ? 5. If a quadrilateral has two pairs of adjacent sides, it is a kite

20. Given: Quadrilateral ABCD, with

midpoints E, F, G, and H of the sides Prove: EF ‘ HG ⬵ MQ ⬵ PQ MN ⬵ PN NQ ⬵ NQ ∠3 ⬵ ∠4 ∠1 ⬵ ∠2 PROOF Statements Reasons 1. ? 1. Given 2. Draw 2. Through two points, there is one line 3. In , and 3. ? in , 4. ? 4. If two lines are to the same line, these lines are to each other

21. Given: M-Q-T

and P-Q-R such that MNPQ and QRST are Prove: ∠N ⬵ ∠S ⵥs ‘ ‘ HG ‘ AC 䉭ADC EF ‘ AC 䉭ABC AC In Exercises 25 to 28, write a formal proof of each theorem or corollary.

25. If both pairs of opposite sides of a quadrilateral are

congruent, then the quadrilateral is a parallelogram.

24. Given: , with T the midpoint of

and S the midpoint of Prove: , and MSPT is a ⵥ 䉭QMS ⬵ 䉭NPT QP MN ⵥMNPQ

23. Given: Kite HJKL with diagonal

Prove: bisects ∠LHJ HK HK

22. Given: with diagonals

and Prove: 䉭WMX ⬵ 䉭YMZ XZ WY ⵥWXYZ A E B D H F C G L J H K Q S P M T N

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.