Lesson 2 ~ Angles • Polygons ~ Triangles • Quadrilaterals

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Lesson 2Angles Polygons Triangles
Quadrilaterals
Example 2.2
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Angles
Lesson 2.A
If two lines cross, we say they intersect and the
point where they cross is called the point of
intersection.
Two lines which are in the same plane must either
intersect or not intersect. When two lines in
the same plane do not intersect they are called
parallel lines.
Intersecting lines
Parallel lines
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Lesson 2.APage 2
Angles (Continued)
If two lines make square corners at the point of
intersection, they are called perpendicular lines
and the angles formed are called right angles.
4 right angles
We can draw a little square at the point of
intersection to indicate that all four angles
formed are right angles.
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Lesson 2.APage 3
Angles (Continued)
Two right angles form a straight angle.
Straight angle
An angle smaller than a right angle is called an
acute angle.
An angle greater than a right angle is called an
obtuse angle.
obtuse
acute
acute
obtuse
2 acute angles and 2 obtuse angles
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Lesson 2.APage 4
Angles (Continued)
A right angle has a measure of 90 degrees.
90 in a right angle
A straight angle has a measure of 180 degrees.
180 in a straight angle
A circle has a measure of 360 degrees.
360 in a circle
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Polygons
Lesson 2.B
A polygon is a special type of geometric figure.
The word polygon is formed from the Greek roots
poly, which means more than one or many, and
gonon, which means angle. Therefore, the word
polygon literally means more than one angle.
The modern understanding of a polygon is a
simple, closed, flat geometric figure whose sides
are line segments. The following are examples of
figures that are not polygons.
not a closed figure
has a curved side
lines cross not simple
lines cross not simple
a) not a polygon
b) not a polygon
c) not a polygon
d) not a polygon
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Polygons (continued)
Lesson 2.BPage 2

• Quadrilateral
• 4 sides
• 4 vertices

• Triangle
• 3 sides
• 3 vertices

• Pentagon
• 5 sides
• 5 vertices

• Hexagon
• 6 sides
• 6 vertices

• Heptagon
• 7 sides
• 7 vertices

• Octagon
• 8 sides
• 8 vertices

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Polygons (continued)
Lesson 2.BPage 3
Each segment of a polygon is called a side. Each
endpoint of a side and also the points where two
sides meet is called a vertex. The plural of
vertex is vertices.
For each polygon, the number of sides is always
equal to the number of vertices.
Polygons are named according to the number of
sides they have

• The polygon with the fewest number of sides (3)
  is the triangle.
• A polygon with 4 sides is called a quadrilateral.
  
• A polygon with 5 sides is called a pentagon.
• A polygon with 6 sides is called a hexagon.
• A polygon with 7 sides is called a heptagon.
• A polygon with 8 sides is called a octagon.
• A polygon with 9 sides is called a nonagon.
• A polygon with 10 sides is called a decagon.
• A polygon with 11 sides is called a undecagon.
• A polygon with 12 sides is called a dodecagon.

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Lesson 2.BPage 4
Concave and ConvexPolygons
If a polygon has an indentation (or cave), the
polygon is called a concave polygon. Any polygon
that does not have an indentation is called a
convex polygon.
Any two points in the interior of a convex
polygon can be connected by a line segment that
does not cut or cross a side of the polygon.
Convex polygon
Concave polygon
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Lesson 2.BPage 5
Regular Polygons
If all the sides of a polygon have the same
length, the polygon is called an equilateral
polygon.
If all the angles of a polygon have the same
measure, the polygon is called an equiangular
polygon.
Polygons in which all sides are the same length
and all angles have the same measure are called
regular polygons.
Regular polygon
Equilateral polygon
Equiangular polygon
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Lesson 2.BPage 6
Regular Polygons
The following are examples of regular polygons.
Regular Quadrilateral (Square)
Regular Triangle (Equilateral Triangle)
Regular Hexagon
Regular Pentagon
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Lesson 2.CPage 1
Triangles
Remember that the polygon with the fewest number
of sides is the triangle. A triangle has three
sides and three angles.
The sum of the measures of the three angles in
any triangle is 180.
If the triangle has a right angle (90), the
triangle is a right triangle.
If all angles have a measure less than 90, the
triangle is an acute triangle.
If one angle has a measure greater than 90, the
triangle is an obtuse triangle.
If all three angles have the same measure, the
triangle is an equiangular triangle.
Right Triangle
Acute Triangle
Equiangular Triangle
Obtuse Triangle
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Lesson 2.CPage 2
Triangles
Triangles are also classifed according to the
relative lengths of their sides.
An isosceles triangle is a triangle that has at
least two sides of equal length.
An equilateral triangle is a triangle in which
the lengths of all sides are equal.
If all the sides of a triangle have different
lengths, the triangle is called a scalene
triangle.
Equilateral Triangle
Scalene Triangle
Isoceles Triangle
The lengths of the sides of a triangle and the
measures of the angles opposite these sides are
related. In any triangle, the angles opposite
sides of equal lengths have equal measures.
Also, the sides opposite angles of equal measures
have equal lengths.
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Lesson 2.DPage 1
Quadrilaterals
Remember that the polygon with four (4) sides is
the quadrilateral. We will discuss five
different types of quadrilaterals.
A trapezoid is a quadrilateral that has exactly
two parallel sides (1 pair).
A parallelogram is a quadrilateral that has at
two pairs parallel sides (2 pair).
A rectangle is a parallelogram with four right
angles.
Rectangle
Trapezoid
Parallelogram
A rhombus is an equilateral parallelogram.
A square is an rhombus with four right angles.
Square
Rhombus
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Example 2.1
Find x.
130
20
x
30
Remember the sum of the measures of the three
angles in any triangle is 180. Therefore,
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Example 2.2
Find x and y.
80
50
Remember the sides opposite angles of equal
measure are equal. Therefore,
and
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Example 2.3
Find x and y.
35
35
Remember the sides opposite angles of equal
measure are equal. Therefore,
and
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Are there any questions?