Download Introduction to Geometry Angles

Math 952 (Spring ’09)
10.1 "Angles and Triangles"
Skills Objectives:
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Recognize the terms point, line, and plane.
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Know the de…nition of angles and learn how to classify angles by their measure as: acute, right, obtuse, or straight.
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Know the meaning of the terms complementary angles, and supplementary angles.
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Learn how to classify a triangle by its sides and by its angles.
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Know that for any triangle:
- The sum of the measures of its angles is 180
- The sum of the lengths of any two sides must be greater than the length of the third side.
Preliminaries:
Plane Geometry is the study of the properties of …gures in a plane. The most basic ideas in plane geometry are point, line,
and plane. These simple terms are then used to de…ne higher-level ideas such as rays, angles, triangles, circles, and so on.
Introduction to Geometry
De…nitions:
Term
1.Point
"Point", "Line", and "Plane"
Representation
Discussion
A dot represents a point. Points are labeled
with capital letters.
2. Line
A line has no beginning or end. Lines are labeled
with small letters or two points on the line.
3. Plane
Flat surfaces, such as table tops or walls, represent
planes. Planes are labeled with capital letters.
Angles
De…nitions:
Term
1. ray
2. angle
3. vertex
4. side
"Ray", "Angle", "Vertex", and "Sides"
Representation
Discussion
A ray consists of a point (called the endpoint) and
all the points on one side of that point.
An angle consists of two rays (called the sides) with
a common endpoint. The common endpoint is called
the vertex.
Page: 1
Notes by Bibiana Lopez
Prealgebra by Franklin Wright
Three Ways of Labeling Angles:
M ethod
1. Three Capital letters
10.1
Illustration :
2. Using Numbers
3. One Capital Letter
Classifying Angles by Their Measures:
Name
Measure
Illustration
Acute
Obtuse
Right
straight
Example 1: (classifying angles by their measures)
Use the …gure below to determine whether each of the following angles is acute, right, obtuse, or straight:
b) 6 XOZ
c) 6 W OY
a) 6 1
De…nition:
1. Two angles are complementary if the sum of their measures is
2. Two angles are supplementary if the sum of their measures is
3. Two angles are equal if they have the same measure.
Question:
What is the supplement of a right angle?
Question:
What is the complement of an acute angle?
Page: 2
Notes by Bibiana Lopez
Prealgebra by Franklin Wright
10.1
Example 2: (Naming complementary and supplementary angles)
In the …gure shown on the right:
a. Name two pairs of supplementary angles.
b. Name two pairs of complementary angles.
Example 3: (Finding complementary and supplementary angles)
If 6 3 and 6 4 are supplementary and m6 3 = 45 , what is m6 4?
De…nitions:
"vertical angles", "adjacent angles"
Term
Representation
1. vertical angles
Discussion
If two lines intersect, then two pairs of vertical angles
are formed. Vertical angles are also called opposite
angles.
NOTE: Vertical angles have the same measure.
2. adjacent angles
Two angles are adjacent if they have a common side.
Example 4: (Using properties of vertical angles)
In the …gure shown below, l, m, and n are straight lines with m6 1 = 20 and m6 6 = 90
a. Find the measures of the other four angles.
b. Which angle is supplementary to 6 6?
Page: 3
Notes by Bibiana Lopez
Prealgebra by Franklin Wright
10.1
Triangles
De…nition:
Term
"triangle"
Representation
1. triangle
Discussion
A triangle consists of three line segments which
join three points. The line segments are called the
the sides of the triangle, and the points are
the vertices.
Properties of Triangles:
- The sum of the measures of its angles is 180
- The sum of the lengths of any two sides must be greater than the length of the third side.
Triangles Classi…ed by Sides
Type
Property
Illustration
Scalene
Isosceles
Equilateral
Triangles Classi…ed by Angles
Type
Property
Illustration
Acute
Obtuse
Right
Equiangular
Page: 4
Notes by Bibiana Lopez
Prealgebra by Franklin Wright
10.1
Example 5: (Naming triangles)
Name the following triangles in two ways:
a.
b.
Example 6: (Determining the existence of a triangle)
Suppose that the lengths of the sides of a triangle are 13; 6; 19. Is this possible?
Example 7:
In 4ST U below, m6 T = 50 and m6 U = 40 :
a. What is m6 S?
b. What kind of triangle is 4ST U ?
c. Which side is opposite m6 T ?
d. Which sides include m6 T ?
e. Is 4ST U a right triangle? Why or why not?
Page: 5
Notes by Bibiana Lopez