FeatureLesson Geometry Lesson Main 1. If AB = 3x + 11, BC = 2x + 19, and CD = 7x – 17, find x. 2....

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1. If AB = 3x + 11, BC = 2x + 19, and CD = 7x – 17, find x.

2. If m BAD = y and m ADC = 4y – 70, find y.

3. If m ABC = 2x + 100 and m ADC = 6x + 84, find m BCD.

4. If m BCD = 80 and m CAD = 34, find m ACD.

5. If AP = 3x, BP = y, CP = x + y, and DP = 6x – 40, find x and y.x = 10, y = 20

Use parallelogram ABCD for Exercises 1–5

Properties of ParallelogramsProperties of Parallelograms

Lesson 6-2

Lesson Quiz

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GeometryGeometry

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(For help, go to Lessons 1-8 and 3-7.)

Proving That a Quadrilateral is a Parallelogram Proving That a Quadrilateral is a Parallelogram

Lesson 6-3

Check Skills You’ll Need

1. Find the coordinates of the midpoints of AC and BD. What is the relationship between AC and BD?

2. Find the slopes of BC and AD. How do they compare?

3. Are AB and DC parallel? Explain.

4. What type of figure is ABCD?

Use the figure at the right for Exercises 1–4.

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GeometryGeometry

LessonMain

1. For A(1, 2) and C(4, 1),

or (2.5, 1.5). For B(1, 0) and D(4, 3),

or (2.5, 1.5). They bisect one another.

1 + 42

2 + 12

x1 + x2

y1 + y2

2, = , = ,

x1 + x2

y1 + y2

21 + 4

20 + 3

= , 52

Solutions

Lesson 6-3

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LessonMain

Lesson 6-3

Solutions (continued)

2. For BC, the endpoints are B(1, 0) and C(4, 1);

For AD, the endpoints are A(1, 2) and D(4, 3);

The slopes of BC and AD are equal.

3. Yes; they are vertical lines.

4. From Exercise 2, the slopes of BC and AD are the same, so the lines are parallel. From Exercise 3, AB and DC are parallel. Thus ABCD is a parallelogram.

y2 – y1

x2 – x1

1 – 04 – 1

= =13m =

y2 – y1

x2 – x1

3 – 24 – 1

= = 13

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GeometryGeometry

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GeometryGeometry

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Lesson 6-3

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Lesson 6-3

Statements Reasons

;WX ZY XY WZ

WXZ YZX ;WXZ YZX WZX YXZ

& are AIAs;

are AIAs

WXZ YZX

WZX YXZ

;WX ZY XY WZ

Reflexive POC

Defn of AIA

Converse of AIA Thm

Defn of WXYZ is a

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Lesson 6-3

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Lesson 6-3

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Lesson 6-3

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Lesson 6-3

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Lesson 6-3

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Find values of x and y for which ABCD must be a parallelogram.

If the diagonals of quadrilateral ABCD bisect each other, then ABCD is a parallelogram by Theorem 6-5. Write and solve two equations to find values of x and y for which the diagonals bisect each other.

If x = 18 and y = 89, then ABCD is a parallelogram.

10x – 24 = 8x + 12 Diagonals of parallelograms 2y – 80 = y + 9bisect each other.

Lesson 6-3

x = 182x = 36

y = 89 Solve.

2x – 24 = 12 y – 80 = 9Collect the variable terms on one side.

Quick Check

Additional Examples

Finding Values for Parallelograms

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GeometryGeometry

LessonMain

Determine whether the quadrilateral is a parallelogram. Explain.

a. All you know about the quadrilateral is that only one pair of opposite sides is congruent.

b. The sum of the measures of the angles of a polygon is (n – 2)180, where n represents the number of sides, so the sum of the measures of the angles of a quadrilateral is (4 – 2)180 = 360.

Therefore, you cannot conclude that the quadrilateral is a parallelogram.

If x represents the measure of the unmarked angle, x + 75 + 105 + 75 = 360, so x = 105.

Lesson 6-3

Because both pairs of opposite angles are congruent, the quadrilateral is a parallelogram by Theorem 6-6. Quick Check

Additional Examples

Is the Quadrilateral a Parallelogram?

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GeometryGeometry

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Lesson 6-3

Quick Check

Additional Examples

The crossbars and the sections of the rulers are congruent no matter how they are positioned. So, ABCD is always a parallelogram. Since ABCD is a parallelogram, the rulers are parallel. Therefore, the direction the ship should travel is the same as the direction shown on the chart’s compass.

FeatureLesson

GeometryGeometry

LessonMain

The captain of a fishing boat plots a course toward a

school of bluefish. One side of a parallel rule connects the boat

with the school of bluefish. The other side makes a 36° angle

north of due east on the chart’s compass. Explain how the

captain knows in which direction to sail to reach the bluefish.

Because both sections of the rulers and the crossbars are congruent, the

rulers and crossbars form a parallelogram.

Therefore, the angle shown on the chart’s compass is congruent to the angle the boat should travel, which is 36° north of due east.

Lesson 6-3

Quick Check

Additional Examples

Real-World Connection

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GeometryGeometry

LessonMain

Find the values of the variables for which GHIJ must be a parallelogram.

x = 6, y = 0.75 a = 34, b = 26

Lesson 6-3

Determine whether the quadrilateral must be a parallelogram. Explain.

3. 4. 5.

No; both pairs of opposite sides are not necessarily congruent.

Yes; the diagonals bisect each other.

Yes; one pair of opposite sides is both congruent and parallel.

Lesson Quiz

FeatureLesson Geometry Lesson Main 1. If AB = 3x + 11, BC = 2x + 19, and CD = 7x – 17, find x. 2....

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