516 Chapter 10 Measurement, Area, and Volume

LE

SS

ON

10.2Vocabularypolygon, p. 516regular polygon,

p. 516convex, p. 516concave, p. 516pentagon, p. 516hexagon, p. 516heptagon, p. 516octagon, p. 516trapezoid, p. 517parallelogram, p. 517rhombus, p. 517diagonal of a polygon,

p. 518

B E F O R E Now W H Y ?

You classified triangles. You’ll classify polygons andquadrilaterals.

So you can find the length of aside of a clock face, as in Ex. 21.

Polygons and Quadrilaterals

A is a closed plane figure whose sides are segments that intersectonly at their endpoints. In a , all the sides have the samelength and all the angles have the same measure.

Polygons Regular polygons Not polygons

A polygon is if a segment joining any two interior points liescompletely within the polygon. A polygon that is not convex is called

.

You already know that a 3-sided polygon is a triangle and a 4-sidedpolygon is a quadrilateral. Below are names of other polygons.

concave

convex

regular polygonpolygon

Tell whether the figure is a polygon. If it is a polygon, classify it andtell whether it is convex or concave. If not, explain why.

a. b.

The keyhole is not a polygonbecause the top part of the keyhole is round.

Example 1 Identifying and Classifying Polygons

The name n-gon refers to apolygon that has n sides. Forexample, a 15-gon is apolygon that has 15 sides.

Reading Geometry

Polygons Pentagon Hexagon Heptagon Octagon n-gon

Number of sides 5 6 7 8 n

convex concave

The stop sign is an 8-sidedpolygon. So it is an octagon.It is convex and regular.

Lesson 10.2 Polygons and Quadrilaterals 517

Classify the quadrilateral.

a. b.

The quadrilateral is aparallelogram with 4 rightangles. So, it is a rectangle.

The quadrilateral is aparallelogram becauseboth pairs of oppositesides are parallel.

6 in. 6 in.

8 in.

8 in.2.5 m

2.5 m

4 m

4 m

Example 2 Classifying Quadrilaterals

Checkpoint

Tell whether the figure is a polygon. If it is a polygon, classify it andtell whether it is convex or concave. If not, explain why.

1. 2. 3.

You cannot conclude that thequadrilateral in part (a) ofExample 2 is a rectanglebecause no information isgiven about its angles.

Watch Out

For help with identifying parallellines, see pp. 410–411.

Review Help

Quadrilaterals Some quadrilaterals have special names based onwhether they have parallel or congruent sides and whether they haveright angles.

TrapezoidA is a quadrilateral with exactly1 pair of parallel sides.

ParallelogramA is a quadrilateral with both pairs of opposite sides parallel.

RhombusA is a parallelogram with 4 congruent sides.

RectangleA rectangle is a parallelogram with 4 right angles.

SquareA square is a parallelogram with 4 right angles and 4 congruent sides.

rhombus

parallelogram

trapezoid

Quadrilaterals Diagram

518 Chapter 10 Measurement, Area, and Volume

1. How are a trapezoid and a parallelogram different from each other?

Tell whether the figure is a polygon. If it is a polygon, classify it and tellwhether it is convex or concave. If not, explain why.

2. 3. 4.

In Exercises 5 and 6, use the quadrilateral shown.

5. Classify the quadrilateral.

6. Find the value of y.

Skill Check

Vocabulary Check

Guided Practice

ExercisesMore Practice, p. 812

10.2 eWorkbook PlusCLASSZONE.COM

INTERNET

Angle Measures in Quadrilaterals A is a segmentthat joins two vertices that are not adjacent. You can use a diagonal of aquadrilateral to show that the sum of the angle measures in a quadrilateralis 360�.

diagonal of a polygon

Segments that connectadjacent vertices of a polygonare the sides of the polygon.These segments are notconsidered to be diagonals.

Reading Geometry

Find the value of x.

x� � (2x � 1)� � 90� � 68� � 360�

3x � 159 � 360 Combine like terms.

3x � 201 Subtract 159 from each side.

x � 67 Divide each side by 3.

Sum of angle measures inquadrilateral is 360�.

Example 3 Finding an Unknown Angle Measure

x�

68�

(2x � 1)�

y�

137� 97�

(2y � 3)�

Draw diagonal FH&*, which divides quadrilateral FGHI into two triangles.

The sum of the angle measures in each triangle is 180�.

The sum of the angle measures in a quadrilateral is 180� � 180� � 360�.

3

2

1F

G

H

I

Lesson 10.2 Polygons and Quadrilaterals 519

Tell whether the figure is a polygon. If it is a polygon, classify it and tellwhether it is convex or concave. If not, explain why.

7. 8. 9.

10. Error Analysis Describe and correct the error in solving the following problem.

A quadrilateral has 4 congruent sides, and the opposite sides of thequadrilateral are parallel. Sketch and classify the quadrilateral.

Classify the quadrilateral.

11. 12. 13.

Copy and complete the statement using always, sometimes, or never.

14. A square is _?_ a rectangle. 15. A square is _?_ a rhombus.

16. A rhombus is _?_ a square. 17. A trapezoid is _?_ a parallelogram.

Find the value of x.

18. 19. 20.

21. Extended Problem Solving The Allen-Bradley Clock Tower inMilwaukee, Wisconsin, has four faces. Each face is a regular octagon.The perimeter of one octagonal face is approximately 133 feet.

a. Calculate Find the length of one side of one of the octagonal faces.

b. Visual Thinking Your friend says that you can find the area of one of the clocks by dividing one of the octagons into 8 congruenttriangles. Sketch a regular octagon and show how to divide it into 8 congruent triangles.

c. Critical Thinking What additional information would you need inpart (b) to find the area of the clock face? Assume that you had thisinformation. What would your next steps be?

22. For the trapezoid shown, the ratiomaA : maC is 2 : 1. Write and solve anequation to find the value of x.

x�x�

(2x � 18)�

(2x � 18)�x�x�

x�x�

x�

59�

(2x � 17)�

Practice and Problem Solving

Example Exercises1 7–92 11–173 18–20

Homework Help

Online ResourcesCLASSZONE.COM

• More Examples• eTutorial Plus

The figure is a square.

x�

A B

D CAllen-Bradley Clock Tower

520 Chapter 10 Measurement, Area, and Volume

23. In mathematics, a kite is a special type ofquadrilateral. Two pairs of sides are congruent,but opposite sides are not congruent. Exactly onepair of opposite angles are congruent. In kiteABCD shown, the measure of aA is twice themeasure of aC, and aB has a measure of 114�.Find the measures of aA, aC, and aD.

24. Challenge Use the figure shown to findmaWXY and maXYZ. Explain yourreasoning.

Solve the linear system by graphing. (Lesson 8.8)

25. y � x � 5 26. y � �3x � 7 27. x � y � 6y � 2x � 1 y � 3x � 4 2x � 8y � �11

28. Find the midpoint of the segment with endpoints (�7, 5) and (4, �20). (Lesson 9.5)

29. The ratio of the angle measures of a triangle is 2 : 3 : 7. Find the anglemeasures. Then classify the triangle by its angle measures. (Lesson 10.1)

30. Extended Response The top of the picnic table shown has the shape ofa regular polygon.

a. Sketch and classify the polygon.Is it convex or concave?

b. Draw a single segment that divides the polygon in your sketch into two trapezoids.

c. Find the sum of the measures of the angles of the polygon.

Standardized TestPractice

Mixed Review

A

DB

C

114�

ZW

YX

60�45�

Move exactly two toothpicks inthe figure below to make 4congruent squares instead of 5. Each toothpick must be used asa side of a square.

Toothpick Task