Download 6•1 Naming and Classifying Angles and Triangles

NAMING AND CLASSIFYING ANGLES AND TRIANGLES
6•1
6•1
Naming and
Classifying Angles
and Triangles
Points, Lines, and Rays
In the world of math, it is sometimes necessary to refer to a
specific point in space. Simply draw a small dot with a pencil
to represent a point. A point has no size; its only function is to
show position.
To name a point, use a single capital letter.
•M
Point M
If you draw two points on a sheet of paper, a line can be used to
connect them. Imagine this line as being perfectly straight and
continuing without end in opposite directions. It has no thickness.
To name a line, use any two points on the line.
.
/
Line MN, or MN
A ray is part of a line that extends without end in one direction.
In MN
, which is read as “ray MN,” M is the endpoint. The
second point that is used to name the ray can be any point other
than the endpoint. You could also name this ray MO
.
.
/
Ray MN, or MO
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Check It Out
Look at the line below.
,
-
1 Name the line in two different ways.
2 What is the endpoint of KL
?
Naming Angles
Imagine two different rays with the same
endpoint. Together they form what is called
an angle. The point they have in common is
called the vertex of the angle. The rays form
the sides of the angle.
1
3
2
3
The angle above is made up of QP
and QR
. Q is the common
endpoint of the two rays. Point Q is the vertex of the angle.
Instead of writing the word angle, you can use the symbol for
an angle, which is ∠.
There are several ways to name an angle. You can name it using
the three letters of the points that make up the two rays with the
vertex as the middle letter (∠PQR, or ∠RQP). You can also use
just the letter of the vertex to name the angle (∠Q). Sometimes
you might want to name an angle with a number (∠3).
When more than one angle is formed at a vertex, you use three
letters to name each of the angles. Because G is the vertex of
three different angles, each angle needs three letters to name it:
∠DGF; ∠DGE; ∠EGF.
&
%
(
'
Naming and Classifying Angles and Triangles
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Measuring Angles
You measure an angle in degrees, using a protractor (p. 379).
The number of degrees in an angle will be greater than 0 and
less than or equal to 180.
EXAMPLE Measuring with a Protractor
Measure ∠ABC.
$
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NAMING AND CLASSIFYING ANGLES AND TRIANGLES
,
Look at the angles formed by the
rays at the right.
3 Name the vertex.
4 Name all the angles.
6•1
Check It Out
#
• Place the center point of the protractor on the vertex of the angle. Align the
0° line on the protractor with one side of the angle.
• Read the number of degrees on the scale where it intersects the second side
of the angle.
So, m∠ABC = 135°.
Check It Out
2
Measure the angles,
using a protractor.
5 ∠PSQ
1
∠QSR
6
7 ∠PSR
4
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Classifying Angles
You can classify angles by their measures.
Acute angle
measures less than 90°
Obtuse angle
measures greater than 90°
and less than 180°
Right angle
measures 90°
Straight angle
measures 180°
Reflex angle
measures greater than 180°
Angles that share a side are called adjacent angles. You can add
measures if the angles are adjacent.
m∠KNL = 25°
m∠LNM = 65°
m∠KNM = 25° + 65° = 90°
Because the sum is 90°, you know that
∠KNM is a right angle.
,
25°
65°
/
.
Check It Out
Use a protractor to measure and classify each angle.
8 ∠SQR
4
9 ∠PQR
10 ∠PQS
1
2
Naming and Classifying Angles and Triangles
3
289
NAMING AND CLASSIFYING ANGLES AND TRIANGLES
Special Pairs of Angles
When the sum of two angles equals 180°, they are called
supplementary angles.
Supplementary
Not Supplementary
%
50°
110°
70°
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+
#
70°
$
∠ABD = 70°
∠DBC = 110°
70° + 110° = 180°
∠ABD + ∠DBC = 180°
(
)
∠JGL = 50°
∠LGH = 70°
50° + 70° = 120°
∠JGL + ∠LGH = 120°
Opposite angles formed by two intersecting lines are called
vertical angles.
1
2
3
6•1
4
∠1 and ∠4 are vertical angles.
∠2 and ∠3 are vertical angles.
Check It Out
Identify each pair of angles as supplementary or vertical.
11
12 ∠6 and ∠8
6
9
8
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When two angles have the same angle measure, they are called
congruent angles. ∠FTS and ∠GPQ are congruent because each
angle measures 45°.
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45°
45°
5
1
2
4
If the sum of the measure of two angles is 90°, then the angles
are complementary angles.
Complementary
Not Complementary
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30°
/
60°
30°
30°
.
3
2
0
∠NMO = 30°
∠LMN = 60°
30° + 60° = 90°
∠NMO + ∠LMN = 90°
4
∠PQR = 30°
∠RQS = 30°
30° + 30° = 60°
∠PQR + ∠RQS = 60°
Check It Out
Identify each pair of angles as complementary or
congruent.
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14
Naming and Classifying Angles and Triangles
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NAMING AND CLASSIFYING ANGLES AND TRIANGLES
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Triangles
Triangles are polygons (p. 298) that have three sides, three
vertices, and three angles.
You name a triangle using the three vertices in any order.
ABC is read “triangle ABC.”
Classifying Triangles
Like angles, triangles are classified by their angle measures. They
are also classified by the number of congruent sides, which are
sides with equal length.
Acute triangle
three acute angles
Obtuse triangle
one obtuse angle
Equilateral triangle
three congruent sides;
three congruent angles
Isosceles triangle
at least two congruent sides;
at least two congruent angles
Right triangle
one right angle
Scalene triangle
no congruent sides
The sum of the measures of the three angles in a triangle is
always 180°.
%
60°
50°
&
70°
'
In DEF, m∠D = 60°, m∠E = 50°, and m∠F = 70°.
60° + 50° + 70° = 180°
So, the sum of the angles of DEF is 180°.
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Finding the Measure of the
Unknown Angle in a Triangle
EXAMPLE
∠S is a right angle, so its measure is 90°. The measure of ∠T
is 35°. Find the measure of ∠U.
4
6
5
90° + 35° = 125°
180° - 125° = 55°
So, ∠U = 55°.
• Add the two known angles.
• Subtract the sum from 180°.
• The difference is the measure of the
third angle.
Check It Out
Find the measure of the third angle of each triangle.
,
15
40°
°
+ 110
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16
#
"
17
45°
$
%
60°
&
60°
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Naming and Classifying Angles and Triangles
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NAMING AND CLASSIFYING ANGLES AND TRIANGLES
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6•1
Exercises
Use the figure to answer Exercises 1–5.
5
1. Give six names for the line that
passes through point P.
2. Name four rays that begin at
point Q.
1
3. Name the right angle.
4. Find m∠PQT.
5. Find m∠PQS.
4
40°
2
3
Use the figure below to answer Exercises 6–8.
2
/
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3 4
7
1
5
6
9
8
6. Identify a pair of complementary angles.
7. Identify a pair of supplementary angles.
8. Identify a pair of vertical angles.
Use the figure to answer Exercises 9 and 10.
.
100°
25°
/
0
9. Is MNO an acute, an obtuse, or a right triangle?
10. Is MNO a scalene, an isosceles, or an equilateral triangle?
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