. a. Angles NMQ and MNP are consecutive angles. b. Angles ...

1. NAVIGATION To chart a course, sailors use aparallel ruler. One edge of the ruler is placed alongthe line representing the direction of the course to betaken. Then the other ruler is moved until its edgereaches the compass rose printed on the chart.Reading the compass determines which direction totravel. The rulers and the crossbars of the tool form

.

a. If .b. If .c. If MQ = 4, what is NP?

SOLUTION:

a. Angles NMQ and MNP are consecutive angles.Consecutive angles in a parallelogram aresupplementary. So, Substitute.

b. Angles MQP and MNP are opposite angles.Opposite angles of a parallelogram are congruent. So,

c. are opposite sides. Opposite sidesof a parallelogram are congruent. So,

ANSWER:

a. 148b. 125c. 4

ALGEBRA Find the value of each variable ineach parallelogram.

2.

SOLUTION:

Opposite angles of a parallelogram are congruent. So,

Solve for x.

ANSWER:

38

3.

SOLUTION:

Opposite sides of a parallelogram are congruent. So,

Solve for y.

ANSWER:

15

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6-2 Parallelograms

4.

SOLUTION:

Diagonals of a parallelogram bisect each other. So, a– 7 = 2 and 2b – 6 = 10. Solve for a and b. So, a = 9 and b = 8.

ANSWER:

a = 9, b = 8

5.

SOLUTION:

Diagonals of a parallelogram bisect each other. So,2b + 5 = 3b + 1 and 4w – 7 = 2w + 3. Solve for b. 2b + 5 = 3b + 12b = 3b – 4–b = –4b = 4 Solve for w. 4w – 7 = 2w + 34w = 2w + 102w = 10w = 5

ANSWER:

w = 5, b = 4

6. COORDINATE GEOMETRY Determine thecoordinates of the intersection of the diagonals of

with vertices A(–4, 6), B(5, 6), C(4, –2),and D(–5, –2).

SOLUTION:

Since the diagonals of a parallelogram bisect each

other, their intersection point is the midpoint of or

the midpoint of

Find the midpoint of . Use the Midpoint Formula

.

Substitute.

The coordinates of the intersection of the diagonals ofparallelogram ABCD are (0, 2).

ANSWER:

(0, 2)

CONSTRUCT ARGUMENTS Write theindicated type of proof.


6-2 Parallelograms

7. paragraph Given: is a right angle.Prove: are right angles.(Theorem 6.6)

SOLUTION:

Begin by listing what is known. Since ABCD is aparallelogram, the properties of parallelograms apply.Each pair of opposite sides are parallel andcongruent, and each pair of opposite angles arecongruent. It is given that is a right angle, so

. Given: is a right angle.Prove: are right angles. (Theorem6.6)

Proof: By definition of a parallelogram, .

Because is a right angle, . By the

Perpendicular Transversal Theorem, . is a right angle, because perpendicular lines

form a right angle. becauseopposite angles in a parallelogram are congruent.

are right angles, because all right anglesare congruent.

ANSWER:

Given: is a right angle.Prove: are right angles. (Theorem6.6)

Proof: By definition of a parallelogram, .

Because is a right angle, . By the

Perpendicular Transversal Theorem, . is a right angle, because perpendicular lines

form a right angle. becauseopposite angles in a parallelogram are congruent.

are right angles, because all right anglesare congruent.

8. two-column Given: ABCH and DCGF are parallelograms.Prove:

SOLUTION:

First, list what is known. Since ABCH and DCGF areparallelograms, the properties of parallelograms apply.From the figure, and are verticalangles. Given: ABCH and DCGF are parallelograms.Prove:

Proof:Statements (Reasons)1. ABCH and DCGF are parallelograms. (Given)2. (Vert. )3. (Opp.

.)4. (Subst.)

ANSWER:

Given: ABCH and DCGF are parallelograms.Prove:


6-2 Parallelograms

Proof:Statements (Reasons)1. ABCH and DCGF are parallelograms. (Given)2. (Vert. )3. (Opp.

.)4. (Subst.)

Use to find each measure.

9.

SOLUTION:

Consecutive angles in a parallelogram aresupplementary.So, Substitute.

ANSWER:

52

10. QR

SOLUTION:


ANSWER:

3

11. QP

SOLUTION:


ANSWER:

5

12.

SOLUTION:


ANSWER:

128

13. HOME DECOR The slats on Venetian blinds aredesigned to remain parallel in order to direct the pathof light coming in a widow. In

and .

Find each measure.

a. JHb. GHc. d.

SOLUTION:

a. Opposite sides of a parallelogram are congruent.So, b. Opposite sides of a parallelogram are congruent.

So,


6-2 Parallelograms

c. Opposite angles of a parallelogram are congruent.So, d. Consecutive angles in a parallelogram aresupplementary. So, Substitute.

ANSWER:

a. 1 in.

b. c. 62d. 118

14. MODELING Wesley is a member of the kennelclub in his area. His club uses accordion fencing likethe section shown to block out areas at dog shows.

a. Identify two pairs of congruent segments.b. Identify two pairs of supplementary angles.

SOLUTION:

a. Opposite sides of a parallelogram are congruent. b. Consecutive angles of a parallelogram aresupplementary.

ANSWER:

a. b. Sample answer:


6-2 Parallelograms

ALGEBRA Find the value of each variable ineach parallelogram.

15.

SOLUTION:

Opposite sides of a parallelogram are congruent. So, and .

Solve for a.

Solve for b.

ANSWER:

a = 7, b = 11

16.

SOLUTION:

Consecutive angles in a parallelogram aresupplementary. So, Solve for x.


ANSWER:

x = 79, y = 101

17.

SOLUTION:

Diagonals of a parallelogram bisect each other. So, y– 7 = 10 and x + 6 = 11. Solve for y. y – 7 = 10y = 17 Solve for x. x + 6 = 11x = 5

ANSWER:

x = 5, y = 17


6-2 Parallelograms

18.

SOLUTION:

Opposite sides of a parallelogram are congruent. So, and .

Solve for a.

Solve for b.

ANSWER:

a = 2, b = 3

19.

SOLUTION:

Consecutive angles in a parallelogram aresupplementary. So, Solve for x.

Substitute in


. Substitute.

ANSWER:

x = 58, y = 63.5


6-2 Parallelograms

20.

SOLUTION:

Diagonals of a parallelogram bisect each other. So, and .

Solve for z.

Solve for y.

ANSWER:

z = 2, y = 5

COORDINATE GEOMETRY Find thecoordinates of the intersection of the diagonalsof with the given vertices.

21. W(–1, 7), X(8, 7), Y(6, –2), Z(–3, –2)

SOLUTION:



the midpoint of Find the midpoint of . Usethe Midpoint Formula

.

Substitute.

The coordinates of the intersection of the diagonals ofparallelogram WXYZ are (2.5, 2.5).

ANSWER:

(2.5, 2.5)

22. W(–4, 5), X(5, 7), Y(4, –2), Z(–5, –4)

SOLUTION:



the midpoint of Find the midpoint of .Use the Midpoint Formula

.

Substitute.

The coordinates of the intersection of the diagonals ofparallelogram WXYZ are (0, 1.5).

ANSWER:

(0, 1.5)

PROOF Write a two-column proof.23. Given: WXTV and ZYVT are parallelograms.

Prove:

SOLUTION:

Begin by listing the known information. It is given thatWXTV and ZYVT are parallelograms, so theproperties of parallelograms apply. From the figure,these two parallelograms share a common side, . Given: WXTV and ZYVT are parallelograms.

Prove:

Proof:


6-2 Parallelograms

Statements (Reasons)1. WXTV and ZYVT are parallelograms. (Given)

2. (Opp. sides of a .)

3. (Trans. Prop.)

ANSWER:

Given: WXTV and ZYVT are parallelograms.

Prove:

Proof:Statements (Reasons)1. WXTV and ZYVT are parallelograms. (Given)

2. (Opp. sides of a .)

3. (Trans. Prop.)

24. Given: Prove:

SOLUTION:

Begin by listing the known information. It is given thatBDHA is a parallelogram, so the properties of aparallelogram apply. It is also given that .From the figure, has sides .Since two sides are congruent, must be anisosceles triangle. Given: Prove:

Proof:Statements (Reasons)

1. (Given)2. (Opp. .)3. (Isos. Thm.)4. (Trans. Prop.)

ANSWER:

Given: Prove:

Proof:Statements (Reasons)

1. (Given)2. (Opp. .)3. (Isos. Thm.)4. (Trans. Prop.)


6-2 Parallelograms

25. FLAGS Refer to the Alabama state flag. Given:

Prove:

SOLUTION:

Begin by listing the known information. It is given that. From CPCTC, ,

, , and . In order to

prove that , show that . Proof:Statements (Reasons):1. (Given)2. (CPCTC)3. (Vert. .)

4. (CPCTC)5. (AAS)

6. (CPCTC)

ANSWER:

Proof:Statements (Reasons):1. (Given)2. (CPCTC)3. (Vert. .)

4. (CPCTC)5. (AAS)

6. (CPCTC)

CONSTRUCT ARGUMENTS Write theindicated type of proof.

26. two-column Given: Prove: and

are supplementary. (Theorem6.5)

SOLUTION:

Begin by listing what is known. It is given the GKLMis a parallelogram, so opposite sides are parallel andcongruent. If two parallel lines are cut by atransversal, consecutive interior angles aresupplementary. Proof:Statements (Reasons)1. (Given)

2. (Opp. sides of a .)3. are supplementary, aresupplementary, are supplementary, and

are supplementary. (Cons. int. aresuppl.)

ANSWER:

Proof:Statements (Reasons)1. (Given)

2. (Opp. sides of a .)3. are supplementary, aresupplementary, are supplementary, and

are supplementary. (Cons. int. aresuppl.)


6-2 Parallelograms

27. two-column Given: Prove: (Theorem 6.8)

SOLUTION:

Begin by listing what is known. It is given the WXYZis a parallelogram, so opposite sides are parallel andcongruent and opposite angles are congruent. Thisinformation can be used to prove . Proof:Statements (Reasons)1. (Given)

2. (Opp. sides of a .)3. (Opp. .)4. (SAS)

ANSWER:


2. (Opp. sides of a .)3. (Opp. .)4. (SAS)

28. two-column Given:

Prove: (Theorem 6.3)

SOLUTION:

Begin by listing what is known. It is given the GKLM

is a parallelogram. We are proving that opposite sidesare congruent, but we can use the other properties ofparallelograms such as opposite sides are parallel. Iftwo parallel lines are cut by a transversal, alternateinterior angles are congruent. If we can show that

, then byCPCTC. Proof:Statements (Reasons)1. (Given)

2. Draw an auxiliary segment and label angles 1,2, 3, and 4 as shown. (Diagonal of PQRS)

3. (Opp. sides of a .)4. (Alt. int. Thm.)

5. (Refl. Prop.)6. (ASA)

7. (CPCTC)

ANSWER:


2. Draw an auxiliary segment and label angles 1,2, 3, and 4 as shown. (Diagonal of PQRS)

3. (Opp. sides of a .)4. (Alt. int. Thm.)

5. (Refl. Prop.)6. (ASA)

7. (CPCTC)


6-2 Parallelograms

29. paragraph Given: is a parallelogram.Prove: (Theorem 6.7)

SOLUTION:

Begin by listing what is known. It is given that ACDEis a parallelogram, so opposite sides are parallel andcongruent. If two parallel lines are cut by atransversal, alternate interior angles are congruent.

To prove that ,

we must show that . Thiscan be done if it is shown that . Proof: It is given that ACDE is a parallelogram.Because opposite sides of a parallelogram are

congruent, . By definition of aparallelogram,

becausealternate interior angles are congruent.

by ASA. byCPCTC. By the definition of segment bisector,

.

ANSWER:

Proof: It is given that ACDE is a parallelogram.Because opposite sides of a parallelogram are

congruent, . By definition of aparallelogram,

becausealternate interior angles are congruent.

by ASA. byCPCTC. By the definition of segment bisector,

.

30. COORDINATE GEOMETRY Use the graphshown.

a. Use the Distance Formula to determine if thediagonals of JKLM bisect each other. Explain.b. Determine whether the diagonals are congruent.Explain.c. Use slopes to determine if the consecutive sidesare perpendicular. Explain.

SOLUTION:

a. Identify the coordinates of each point using thegiven graph. K(2, 5), L(–1, –1), M(–8, –1), J(–5, 5) and P(–3, 2) Use the distance formula.


6-2 Parallelograms

; since JP

= LP and MP = KP, the diagonals bisect each other. b. The diagonals are congruent if they have the samemeasure. Since JP + LP ≠ MP + KP, JL MK sothey are not congruent. c. The slopes of two perpendicular lines are negativereciprocals of each other. Find the slopes of JK andJM. Find the slope of using slope formula.

Find the slope of using slope formula.

The slope of and the slope of . Theslopes are not negative reciprocals of each other. So,the answer is “No”.

ANSWER:

a. ; sinceJP = LP and MP = KP, the diagonals bisect eachother.b. No; JP + LP ≠ MP + KP.c. No; the slope of and the slope of .The slopes are not negative reciprocals of each other.

ALGEBRA Use to find each measureor value.

31. x

SOLUTION:

Opposite sides of a parallelogram are congruent. So, .

Solve for x.

ANSWER:

3


6-2 Parallelograms

32. y

SOLUTION:

Opposite sides of a parallelogram are congruent. So, .

Solve for y.

ANSWER:

6

33.

SOLUTION:

In the figure, angle AFB and the angle form alinear pair. So,

ANSWER:

131

34.

SOLUTION:

Since vertical angles are congruent, .

The sum of the measures of interior angles in atriangle is . Here, Substitute.

and represent the same angle. So,

ANSWER:

72


6-2 Parallelograms

35.

SOLUTION:

Since vertical angles are congruent, .


So, By the Alternate Interior Angles Theorem,

So, In , Substitute.

ANSWER:

29

36.

SOLUTION:


ANSWER:

101


6-2 Parallelograms

37. COORDINATE GEOMETRY hasvertices A(–3, 5), B(1, 2), and C(3, –4). Determinethe coordinates of vertex D if it is located inQuadrant III. Explain.

SOLUTION:

Opposite sides of a parallelogram are parallel. Since the slope of , the slope of

must also be .

To locate vertex D, make a line from A, starting fromvertex A and moving down 6 and right 2. The slope of is . The slope of

must also be . Draw a line from vertex C. The intersection of the two lines is (–1, –1). This isvertex D.

ANSWER:

(–1, –1); Sample answer: opposite sides of aparallelogram are parallel. Since the slope of

, the slope of must also be . Tolocate vertex D, start from vertex A and move down6 and right 2.

38. MECHANICS Scissor lifts are variable elevationwork platforms. In the diagram, ABCD and DEFGare congruent parallelograms.

a. List the angle(s) congruent to . Explain yourreasoning.

b. List the segment(s) congruent to . Explainyour reasoning.c. List the angle(s) supplementary to . Explainyour reasoning.

SOLUTION:

Use the properties of parallelograms that you learnedin this lesson. a. The angle(s) congruent to are

is congruent to because opposite angles ofparallelograms are congruent. is congruent to

because the parallelograms are congruent, and is congruent to because opposite angles of

parallelograms are congruent and congruent to by the Transitive Property. b. The segment(s) congruent to are

is congruent to because

opposite sides of parallelograms are congruent.

is congruent to because the parallelograms are

congruent, and is congruent to becauseopposite sides of parallelograms are congruent and

congruent to by the Transitive Property. c. The angle(s) supplementary to are

Angles ABC and


6-2 Parallelograms

ADC are supplementary to because consecutiveangles of parallelograms are supplementary.is supplementary to because it is congruent to

by the Vertical Angles Theorem andsupplementary to by substitution. iscongruent to because opposite angles ofparallelograms are congruent and it is supplementaryto by substitution.

ANSWER:

a. sample answer: is congruent to because opposite angles of parallelograms are

congruent. is congruent to because theparallelograms are congruent, and is congruentto because opposite angles of parallelograms arecongruent and congruent to by the TransitiveProperty.

b. ; sample answer: is congruent to

because opposite sides of parallelograms are

congruent. is congruent to because the

parallelograms are congruent, and is congruent

to because opposite sides of parallelograms are

congruent and congruent to by the TransitiveProperty.c. sample answer:Angles ABC and ADC are supplementary to because consecutive angles of parallelograms aresupplementary. is supplementary to because it is congruent to by the VerticalAngles Theorem and supplementary to bysubstitution. is congruent to becauseopposite angles of parallelograms are congruent and itis supplementary to by substitution.

PROOF Write a two-column proof.

39. Given: Prove:

SOLUTION:

Begin by listing what is known. It is given that

. Opposite sides andangles are congruent in parallelograms.

ANSWER:

40. MULTIPLE REPRESENTATIONS In thisproblem, you will explore tests for parallelograms.


6-2 Parallelograms

a. Geometric Draw three pairs of segments thatare both congruent and parallel and connect theendpoints to form quadrilaterals. Label onequadrilateral ABCD, one MNOP, and one WXYZ.Measure and label the sides and angles of thequadrilaterals. b. Tabular Copy and complete the table below.

c. Verbal Make a conjecture about quadrilateralswith one pair of segments that are both congruentand parallel.

SOLUTION:

a. Sample answer:

b.

c. Sample answer: If a quadrilateral has a pair ofsides that are ≅ and ‖, then the quadrilateral is aparallelogram.

ANSWER:

a. Sample answer:

b.

c. Sample answer: If a quadrilateral has a pair ofsides that are ≅ and ‖, then the quadrilateral is aparallelogram.

41. CHALLENGE ABCD is a parallelogram with sidelengths as indicated in the figure. The perimeter ofABCD is 22. Find AB.

SOLUTION:


6-2 Parallelograms

We know that opposite sides of a parallelogram arecongruent, so 2y + 1 = 3 – 4w and 3x – 2 = x – w +1. The perimeter is 22. First, solve this equation for x in terms of w:

.

Next, set up an equation for the perimeter. Sinceopposite sides are congruent, AB = DC and AD =BC.

Substitute in CD.

Since the opposite sides of a parallelogram arecongruent, AB = CD = 7.

ANSWER:

7

42. WRITING IN MATH Explain why parallelogramsare always quadrilaterals, but quadrilaterals aresometimes parallelograms.

SOLUTION:

A parallelogram is a polygon with four sides in whichthe opposite sides and angles are congruent.Quadrilaterals are defined as four-sided polygons.Since a parallelogram always has four sides, it isalways a quadrilateral. A quadrilateral is only aparallelogram when the opposite sides and angles ofthe polygon are congruent. A quadrilateral that is also a parallelogram:

A quadrilateral that is not a parallelogram:

ANSWER:

A parallelogram is a polygon with four sides in whichthe opposite sides and angles are congruent.Quadrilaterals are defined as four-sided polygons.Since a parallelogram always has four sides, it isalways a quadrilateral. A quadrilateral is only aparallelogram when the opposite sides and angles ofthe polygon are congruent.


6-2 Parallelograms

43. OPEN-ENDED Provide a counterexample to showthat parallelograms are not always congruent if theircorresponding sides are congruent.

SOLUTION:

These parallelograms each have correspondingcongruent sides, but the corresponding angles are notcongruent. So, the parallelograms are not congruent.

ANSWER:

44. REASONING Find in the figure.Explain.

SOLUTION: ∠10 is supplementary to the 65 degree anglebecause consecutive angles in a parallelogram aresupplementary, so m∠10 is 180 − 65 or 115. ∠1 is supplementary to ∠8 because consecutiveangles in a parallelogram are supplementary, so m∠8= 64 because alternate interior angles are congruent.Therefore, m∠1 is 116.

ANSWER: m∠1 = 116, m∠10 = 115; sample answer: m∠8 = 64because alternate interior angles are congruent. ∠1is supplementary to ∠8 because consecutive anglesin a parallelogram are supplementary, so m∠1 is 116.∠10 is supplementary to the 65° angle becauseconsecutive angles in a parallelogram aresupplementary, so m∠10 is 180 − 65 or 115.


6-2 Parallelograms

45. WRITING IN MATH Summarize the properties ofthe sides, angles, and diagonals of a parallelogram.

SOLUTION:

Sample answer: In a parallelogram, the opp. sides and.

Two consecutive supplementary. Ifone angle of a is right, then all the angles areright.

The diagonals of a parallelogram bisect each other.

ANSWER:

Sample answer: In a parallelogram, the opp. sides and. Two consecutive ∠s in a

are supplementary. If one angle of a is right,then all the angles are right. The diagonals of aparallelogram bisect each other.

46. The park near Karla’s home is shaped like aparallelogram, as shown in the figure. Side is 3times as long as side , and the perimeter of thepark is 1200 feet.

What is the length of ?

A 450 ftB 400 ftC 300 ftD 150 ft

SOLUTION: Since is three times as long as , if the lengthof is x, then the length of is 3x. The oppositesides of a parallelogram are congruent, therefore wecan label all four sides of the parallelogram as shownin the figure below.

To find the perimeter, add up all the sides of theparallelogram or use the formula .

, so the correct answer is choice D.

ANSWER: D


6-2 Parallelograms

47. In parallelogram ABCD, ∠A and ∠B areconsecutive angles. The measure of ∠A is 40 morethan the measure of ∠B. What is the measure of∠A?

A 65B 70C 110D 140

SOLUTION: The measure of is 40 degrees more than themeasure of , so let and .Consecutive angles of a parallelogram aresupplementary. Therefore,

The measure of is 40 more than the measure of

, so . The correct answer is choice C.

ANSWER: C

48. In the figure, quadrilateral MNPQ is a parallelogram.Which of the following statements about the figuremust be true?

I. II. III.

A I onlyB II onlyB II onlyC III onlyD I and II onlyE I, II, and III

SOLUTION: In a parallelogram, the opposite angles are congruentby Theorem 6.4. Therefore, I is always true.Additionally, Theorem 6.7 states that the diagonalsbisect each other. So, II is always true. However, theconsecutive sides of a parallelogram are not alwayscongruent. Therefore, III is not always true. So, thecorrect answer is choice D.

ANSWER: D


6-2 Parallelograms

49. In parallelogram ABCD, the measure of ∠A is 55°,the measure of ∠C is (4x + 11)°. a. What is the value of x?b. What is the measure of ∠B?

SOLUTION: a. Angles A and C are congruent in the parallelogram,so (4x + 11) = 55. Solve for x.

b. The sum of the measures of the interior angles ofa parallelogram is 360. Because we know that anglesA and C are congruent, we can write the equation 55+ x + 55 + x = 360, where x represents the measuresof congruent angles B and D. Solve for x.

So, the measure of ∠B is 125°.

ANSWER: a. 11b. 125

50. What is the x-coordinate of the intersection of thediagonals of a parallelogram with vertices A(–5, 0),B(3, 4), C(6, 3), and D(–2, –1)?

SOLUTION: The diagonals of a parallelogram intersect at theirmidpoints. Find the x-coordinate by finding themidpoint of the either diagonal. For example, the x-coordinate of the midpoint of diagonal is

. This answer can be confirmed bygraphing the parallelogram and locating theapproximate intersection of the diagonals, as shownbelow.

ANSWER: 0.5

51. MULTI-STEP Quadrilateral PQRS is aparallelogram.

a. Which Theorem defines a relationship between

and ?b. Find SQ.c. Prove PST ≅ QRT.d. What mathematical practice did you use to solvethis problem.

SOLUTION: a. Theorem 6.7 states that if a quadrilateral is aparallelogram, then its diagonals bisect each other.


6-2 Parallelograms

b. In the figure, ST and TQ are congruent. So, setboth expressions equal to one another and solve for y.

Substitute 6 for y in both expressions, and then addthe lengths together to find SQ. 3(6) – 1 = 172(6) + 5 = 17 17 + 17 = 34 So, SQ = 34. c. In a parallelogram, opposite sides are congruent sowe know PS and QR are congruent. Since PR andSQ bisect, we also know that angles PTS and QTRare congruent. Using this information, we can provethat PST ≅ QRT. d. See students' work.

ANSWER: a. Theorem 6.7b. 34c. In a parallelogram, opposite sides are congruent sowe know PS and QR are congruent. Since PR andSQ bisect, we also know that angles PTS and QTRare congruent. Using this information, we can provethat PST ≅ QRT.d. See students' work.

52. The coordinates of a quadrilateral are J(–2, 4), K(–1,–1), L(9, 1), and M(8, 6). Which statements are true?Select all that apply. A B C JML ≅ LKJ D JMK ≅ LKM E JKLM is a rhombus.

SOLUTION: Sketch a graph of the quadrilateral, and then considereach option. In choice A, segments JM and LM are notcongruent.In choice B, segments JL and KM bisect thequadrilateral and are congruent.In choice C, the triangles are created by a diagonal sothey are congruent.In choice D, the triangles are created by the otherdiagonal so they are congruent.In choice E, the sides are not all equal so JKLM isnot a rhombus. So, the correct answers are choices B, C, and D.

ANSWER: B, C, D


6-2 Parallelograms

53. In parallelogram EFGH, the measure of ∠E is (4x +18)° and the measure of ∠G is (2x + 54)°. Whatmust be true about the parallelogram? A EFGH is a rhombus.B EFGH is a trapezoid.C The measure of ∠E is 18°. D The measure of ∠F is 90°.

SOLUTION: Because we know that the sum of the measures ofthe interior angles of a parallelogram is 360, andopposite angles are congruent, we can write andsolve an equation.

Since x = 18, we can determine the measures of both∠E and ∠G.

Since, both angles measure 90° we know that allangles will equal 90° and the parallelogram is either arectangle or square. So, the correct answer is choice D.

ANSWER: D


6-2 Parallelograms

. a. Angles NMQ and MNP are consecutive angles. b. Angles ...

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