Download 7•2 Naming and Classifying Polygons and Polyhedrons

7•2
Naming and
Classifying Polygons
and Polyhedrons
Quadrilaterals
Four-sided figures are called quadrilaterals. Some quadrilaterals
have specific names based on the relationship between their sides
and/or angles.
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To name a quadrilateral, list the
four vertices, either clockwise or
counterclockwise. One name for the
figure at the right is quadrilateral ISHF.
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Angles of a Quadrilateral
The sum of the angles of a quadrilateral is 360°. If you know the
measures of three angles in a quadrilateral, you can find the
measure of the fourth angle.
EXAMPLE
Finding the Measure of the
Unknown Angle in a Quadrilateral
Find m∠A in quadrilateral ABCD.
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90° + 90° + 115° = 295°
360° - 295° = 65°
90°
115°
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90°
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• Add the measures of the three
known angles.
• Subtract the sum from 360°. The
difference is the measure of the
fourth angle.
So, m∠A = 65°.
Naming and Classifying Polygons and Polyhedrons
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NAMING AND CLASSIFYING POLYGONS AND POLYHEDRONS
7•2
Check It Out
Use the figure below to answer Exercises 1–3.
3
2
4
60°
95°
100°
1
1 Name the quadrilateral in two ways.
2 What is the sum of the angles of a quadrilateral?
3 Find m∠P.
Types of Quadrilaterals
A rectangle is a quadrilateral with four right angles. ABCD is a
rectangle. Its length is 10 centimeters and its width is 6 centimeters.
10 cm
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6 cm
6 cm
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10 cm
Opposite sides of a rectangle are congruent. If all four sides of
the rectangle are congruent, the rectangle is a square. A square is
a regular polygon because all of the sides are congruent and all
of the interior angles are congruent. All squares are rectangles,
but not all rectangles are squares.
A parallelogram is a quadrilateral with opposite sides parallel.
In a parallelogram, opposite sides are congruent, and opposite
angles are congruent. WXYZ is a parallelogram.
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10 cm
120°
60°
6 cm
;
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9
6 cm
60°
120°
10 cm
:
Not all parallelograms are rectangles, but all rectangles are
parallelograms. Therefore, all squares are also parallelograms.
If all four sides of a parallelogram are congruent, the
parallelogram is a rhombus. EFGH is a rhombus.
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8 cm
)
8 cm
'
110° 70°
70° 110°
8 cm
8 cm
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Every square is a rhombus, but not every rhombus is a square
because a square must also have congruent angles.
In a trapezoid, two sides are parallel and two are not. A trapezoid
is a quadrilateral, but it is not a parallelogram. ACKJ is a trapezoid.
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+
,
Check It Out
Complete Exercises 4–7.
4 Is quadrilateral RSTU a rectangle? a parallelogram? a
square? a rhombus? a trapezoid?
3
6 cm
6 cm
6
4
6 cm
6 cm
5
5 Is a square a rhombus? Why or why not?
6 Is a rectangle always a square? Why or why not?
7 Is a parallelogram always a rectangle? Why or why not?
Naming and Classifying Polygons and Polyhedrons
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NAMING AND CLASSIFYING POLYGONS AND POLYHEDRONS
7•2
Polygons
A polygon is a closed figure that has three or more sides. Each
side is a line segment, and the sides meet only at the endpoints,
or vertices.
This figure is a polygon.
These figures are not.
A rectangle, a square, a parallelogram, a rhombus, a trapezoid,
and a triangle are all examples of polygons.
A regular polygon is a polygon with congruent sides and angles.
A polygon always has an equal number of sides, angles, and
vertices.
For example, a polygon
with three sides has
three angles and three
vertices. A polygon
with eight sides has
eight angles and eight
vertices, and so on.
1
8
2
3
2
1
8
7
3
6
1
A line segment connecting two vertices
of a polygon is either a side or a diagonal.
−−
−−
AE is a side of polygon ABCDE. AD is
a diagonal.
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1 2
6 5
2
3
3
4
4
5
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&
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Triangle
3 sides
Quadrilateral
4 sides
Pentagon
5 sides
Hexagon
6 sides
Octagon
8 sides
A seven-sided polygon is called a heptagon, a nine-sided polygon
is called a nonagon, and a ten-sided polygon is called a decagon.
328 HotTopics
Check It Out
State whether the figure is a polygon. If it is a polygon,
classify it according to its number of sides.
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9
10
Angles of a Polygon
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You know that the sum of the angles of
a triangle is 180°. To find the sum of the
interior angles of any polygon, add another
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180° for each additional side to the
measure of the first three angles. Look
%
at pentagon ABCDE.
−−
−−
Diagonals EB and EC show that the sum of the angles of a
pentagon is equal to the sum of the angles in three triangles.
3 · 180° = 540°
So, the sum of the interior angles of a pentagon is 540°.
You can use the formula (n - 2) · 180° to find the sum of the
interior angles of a polygon. Let n equal the number of sides of a
polygon. The solution is equal to the sum of the measures of all
the angles of the polygon.
EXAMPLE Finding the Sum of the Angles of a Polygon
Find the sum of the interior angles of an octagon.
• Use the formula.
(n - 2) · 180°
• Substitute the number of sides.
= (8 - 2) · 180°
• Simplify, using the order of operations.
= 6 · 180°
= 1,080°
So, the sum of the angles of an octagon is 1,080°.
Naming and Classifying Polygons and Polyhedrons
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NAMING AND CLASSIFYING POLYGONS AND POLYHEDRONS
7•2
You can use what you know about finding the sum of the angles of
a polygon to find the measure of each angle of a regular polygon.
Begin by finding the sum of all the angles, using the formula
(n - 2) · 180°. For example, a hexagon has 6 sides, and so
substitute 6 for n.
(6 - 2) · 180° = 4 · 180° = 720°
Then divide the sum of the angles by the total number of angles.
Because a hexagon has 6 angles, divide by 6.
720° ÷ 6 = 120°
Therefore, each angle of a regular hexagon measures 120°.
Check It Out
Use the formula (n - 2) · 180°.
11 Find the sum of the angles of a decagon.
12 Find the measure of each angle in a regular pentagon.
Polyhedrons
Some solid shapes are curved. These shapes are not polyhedrons.
Sphere
Cylinder
Cone
Some solid shapes have flat surfaces. Each of the figures below is
a polyhedron.
Cube
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Prism
Pyramid
A polyhedron is a solid with flat surfaces that are polygons.
Triangles, quadrilaterals, and pentagons make up the faces of
the common polyhedrons below.
A prism has two bases, or “end” faces. The bases of a prism are
polygons that are congruent and parallel to each other. The other
faces are parallelograms. The bases of the prisms shown below
are shaded. When all six faces of a rectangular prism are square,
the figure is called a cube.
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Triangular
prism
Rectangular
prism
Cube
Pentagonal
prism
A pyramid is a structure that has one polygonal base. It has
triangular faces that meet at a point called the apex. The base of
each pyramid shown below is shaded. A triangular pyramid is a
tetrahedron. A tetrahedron has four faces. Each face is triangular.
1ZSBNJET
Triangular
pyramid
(tetrahedron)
Rectangular
pyramid
Square
pyramid
Pentagonal
pyramid
Check It Out
Identify each polyhedron.
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14
Naming and Classifying Polygons and Polyhedrons
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Exercises
1. Give two other names for
quadrilateral MNPQ.
2. Find m∠M.
/
100°
3. Give two other names for
quadrilateral RSTU.
4. Find m∠U.
5. Give two other names for
quadrilateral VWXY.
6. Find m∠W.
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64°
6
7
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110°
1
70°
.
63°
118°
Identify each polygon.
14.
15.
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16.
4
8
115°
65°
2
5
115°
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Tell whether each statement below is true or false.
7. A square is a parallelogram.
8. Every rectangle is a parallelogram.
9. Not all rectangles are squares.
10. Some trapezoids are parallelograms.
11. Every square is a rhombus.
12. All rhombuses are quadrilaterals.
13. A quadrilateral cannot be both a rectangle and a rhombus.
7•2
NAMING AND CLASSIFYING POLYGONS AND POLYHEDRONS
7•2
17.
18.
Find the sum of the angles for each polygon.
19. pentagon
20. nonagon
21. heptagon
22. What is the measure of each angle in a regular octagon?
Identify each polyhedron.
23.
24.
25.
Identify each real-world polygon or polyhedron.
26. The infield of a
baseball diamond.
27. Home plate on a
baseball diamond.
Infield
Home plate
28.
29.
STOP
YIELD
30.
Naming and Classifying Polygons and Polyhedrons
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