Name———————————————————————— Lesson
1.6
Date —————————————
Study Guide
For use with the lesson “Classify Polygons”
goal
Classify polygons.
Vocabulary
A polygon is a closed plane figure formed by three or more line
segments called sides. Each side intersects exactly two sides, one
at each endpoint, so that no two sides with a common endpoint
are collinear. Each endpoint of a side is a vertex of the polygon.
A polygon is convex if no line that contains a side of the polygon
contains a point in the interior of the polygon.
A polygon that is not convex is called nonconvex or concave.
The term n-gon, where n is the number of a polygon’s sides, can be
used to name a polygon.
In an equilateral polygon, all sides are congruent.
In an equiangular polygon, all angles in the interior of the polygon
are congruent.
A polygon is regular if it is convex and both equilateral and
­equiangular.
Identify polygons
Lesson 1.6
Tell whether the figure is a polygon and whether it is convex
or concave.
a.
b.
c.
d.
a. The figure is a concave polygon.
b. Part of the figure is not a segment, so it is not a polygon.
c. Some segments intersect more than two segments, so it is not a polygon.
d. The figure is a convex polygon.
Exercises for Example 1
Tell whether the figure is a polygon and whether it is convex
or concave.
1.
1-84
2.
3.
4.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
example 1
Geometry
Chapter Resource Book
CS10_CC_G_MECR710761_C1L06SG.indd 84
4/27/11 2:33:03 PM
Name———————————————————————— Lesson
1.6
Date —————————————
Study Guide continued
For use with the lesson “Classify Polygons”
example 2
Classify polygons
Classify the polygon by the number of sides. Tell whether the polygon
is equilateral, equiangular, or regular. Explain your reasoning.
a.
b.
c.
a. The polygon has 4 sides, so it is a quadrilateral. The angles in the
interior of the polygon are congruent, so it is equiangular. Not all
of the sides are congruent, so it is not equilateral. So, the polygon
is not regular.
b. The polygon has 8 sides. It is equilateral and equiangular, so it is
a regular octagon.
c. The polygon has 3 sides, so it is a triangle. It is not equilateral or
equiangular, so it is not regular.
Find side lengths
(2x 2 1) cm
The figure shown at the right is a regular
pentagon. Find the length of a side.
First, write and solve an equation to find the value of x.
Use the fact that the sides of a regular pentagon are congruent.
2x 2 1 5 x 1 3
Write equation.
x 2 1 5 3
Subtract x from each side.
x 5 4
Lesson 1.6
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
example 3
(x 1 3) cm
Add 1 to each side.
Then find a side length. Evaluate one of the expressions when x 5 4.
2x 2 1 5 2(4) 2 1 5 7
The length of a side of the pentagon is 7 centimeters.
Exercises for Examples 2 and 3
5. Classify the polygon by the number of sides. Tell whether the polygon is
equilateral, equiangular, or regular.
Explain your reasoning.
6. The figure shown at the right is a regular (3x 1 1) mm
hexagon. Find the length of a side.
(2x 1 6) mm
CS10_CC_G_MECR710761_C1L06SG.indd 85
Geometry
Chapter Resource Book
1-85
4/27/11 2:33:03 PM
answers
Lesson 1.6 Classify Polygons,
continued
concave polygon. 3. The figure is a convex
polygon. 4. regular pentagon; It has 5 sides, and
it is both equilateral and equiangular.
5. equilateral quadrilateral; It has 4 sides. It is
equilateral, but not equiangular. 6. triangle;
It has 3 sides. It is neither equilateral nor
equiangular, so it is not regular.
7. hexagon; It has 6 sides. It is neither equilateral
nor equiangular, so it is not regular. 8. 50 ft
9. 1448 10. 23 km 11. sometimes
12. sometimes 13. never 14. always
15. Sample answer: 16. Sample answer:
4. Quadrilateral; regular; the figure is equilateral
and equiangular. 5. Dodecagon; equilateral; the
figure is not convex so it is not regular.
6. Octagon; equiangular; the sides are not
congruent so it is not regular. 7. Heptagon;
regular; the figure is equilateral and equiangular.
8. 23 m 9. 41.5 in. 10. 114.5 ft
11. Sample answer:
12. Sample answer:
13. Sample answer: 14. Sample answer:
15. never 16. always 17. sometimes
17. Sample answer: 18. Sample answer:
18. never 19. 9 20. 7 21. 8 22. 4, –4
23. 6, –6 24. 4.5
25.
y
C
D
B
C (2, 3)
1
A
1
x
E
F
Hexagon;
}}}
}
2
1 (2 2 (21))2 ​5 Ï
​ 10 ​ ,
AB 5 Ï
​ (21 2 (22))   
D(4, 3)
1
}}
}
}}
}
}}
2
}
}}
}
}}
AC 5 Ï
​ (3 2 2) 1 (0  
2 3)  ​5 Ï
​ 10 ​, 
2
2
AD 5 Ï
​ (3 2 4)2 1 (0  
2 3)2 ​5 ​Ï 10 ​, 
BC 5 ​Ï (3 2 2) 1 (6  
2 3)  ​5 Ï
​ 10 ​, 
2
}
BC 5 ​Ï (2 2 (21))2 1 (3
  
2 2)2 ​5 ​Ï10 ​, 
A(3, 0) x
}}
1
BD 5 ​Ï (3 2 4)2 1 (6  
2 3)2 ​5 Ï
​ 10 ​; 
AC 5 AD 5 BC 5 BD
26. 5.5 ft; The sides of the triangle are congruent
because the triangle is regular. Using this fact,
write and solve an equation to find x. Then
determine the perimeter of the triangle in feet.
27. a. convex b. equiangular octagon c. 2; This
octagon has only 2 lines of symmetry while a
regular octagon has 8.
Practice Level C
}
CD 5 ​Ï (5 2 2)2 1 (2  
2 3)2 ​5 Ï
​ 10 ​ ,
}}
}
DE 5 ​Ï (4 2 5)2 1 (21  
2 2)2 ​5 ​Ï10 ​, 
}}}
}
EF 5 ​Ï (1 2 4)2 1 (22 2  
(21))2 ​5 Ï
​ 10 ​, 
}}}
}
FA 5 ​Ï (22 2 1)2 1   
(21 2 (22))2 ​5 Ï
​ 10 ​; 
AB 5 BC 5 CD 5 DE 5 EF 5 FA
26. a. 1: 18-gon, 2: decagon, 3: heptagon,
4: quadrilateral b. 1: concave, 2: convex,
3: convex, 4: convex c. The notches allow the paper to be folded into right or straight angles with a
minimal amount of overlap.
Study Guide
1. not a polygon 2. polygon; convex 3. not a
polygon 4. polygon; concave 5. quadrilateral;
regular 6. 16 mm
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
1
19. 30 20. }
​ 2 ​ 21. 67 22. 61 23. 6 24. 5
y
25.
quadrilateral;
B(3, 6)
1. Yes; concave 2. No; some segments
intersect more than two segments. 3. Yes; convex
A10
Geometry
Chapter Resource Book
CS10_CC_G_MECR710761_C1AK.indd 10
4/28/11 2:35:36 PM