1. Find the measure of the third angle of a triangle2. Classify triangles as acute, obtuse, or right3. Classify triangles as equilateral, isosceles, or scalene4. Recognize similar triangles5. Apply similar triangles to find an unknown length.
Now that you know something about angles, it is interesting to again look at triangles. Whyis this shape called a triangle?
Literally, triangle means “three angles.”The same classifications we used for angles can be used for triangles. If a triangle has a
right angle, we call it a right triangle.
If it has three acute angles, it is called an acute triangle.
If it has an obtuse angle, it is called an obtuse triangle.
Example 1
Identifying an Acute Triangle
Which of the following triangles is acute?
Only �DEF is an acute triangle. Both �ABC and �XYZ have one obtuse angle.
We can also classify triangles based on how many angles have the same measure.A triangle is called an equilateral triangle if all three angles have the same measure.
A triangle is called an isosceles triangle if two angles have the same measure.
A triangle is called a scalene triangle if no two angles have the same measure.
C H E C K Y O U R S E L F 1
Which of the following triangles is obtuse?
YZ
X
E
FDCA
B
Example 2
Labeling Types of Triangles
Of the following triangles, which are equilateral? Isosceles? Scalene?
�ABC is an isosceles triangle because two of the angles have the same measure. And�DEF is an equilateral triangle because all three angles have the same measure. �XYZ isa scalene triangle because no two angles have the same measure.
Z
XY
60�
30�
60�
60�
D F
E
60�
40�
70� 70�
CA
B
NOTE Notice that each ofthese triangles can be classifiedin different ways. XYZ is a righttriangle, but it is also scalene.
Go back and look at the sum of the angles inside each of the triangles in the last exam-ple. You will note that they always add up to 180°. No matter how we draw a triangle, thesum of the three angles inside the triangle will always be 180°.
Here is an experiment that might convince you that this is always the case.
1. Using a straight edge, draw any triangle you wish on a sheet of paper.
2. Use scissors to cut out the triangle.
3. Rip the three vertices off of the triangle.
4. Lay the three vertices (with the points of the triangle touching) together. They willalways form a straight angle, which we saw in the previous section has a measure of180°.
C H E C K Y O U R S E L F 2
Label each triangle as equilateral, isosceles, or scalene.
100�40�40�
(3)
60�
60�60�
(2)(1)
70�
80�
30�
For any triangle ABC,
m�A � m�B � m�C � 180°
Rules and Properties: Angles of a Triangle
Example 3
Finding an Angle Measure
Find the measure of the third angle in this triangle.
We need the three measurements to add to 180°, so we add the two given measurements(53° � 68° � 121°). Then we subtract that from 180° (180° � 121° � 59°). This gives usthe measure of the third angle, 59°.
C H E C K Y O U R S E L F 3
Find the measure of �ABC.
B
A C
70� 35�
If the measurements of the three angles in two different triangles are the same,we say the two triangles are similar triangles.
Definitions: Similar Triangles
Example 4
Identifying Similar Triangles
Which two triangles are similar?
Although they are of different size, �ABC and �XYZ are similar because they have thesame angle measurements.
The most common use of this property of similar triangles occurs when you wish to find theheight of a tall object. Example 5 illustrates this application of similar triangles.
If two triangles are similar, their corresponding sides have the same ratio.
Rules and Properties: Similar Triangles
Example 5
Finding the Height of a Tree
If a man who is 180 cm tall casts a shadow that is 60 cm long, how tall is a tree that casts ashadow that is 9 m long?
Note that, because of the sun, the man and his shadow forms a similar triangle to the treeand its shadow. Because of this, we can use the common ratio to find the height of the tree.
x � 27 m
The tree is 27 m tall.
x �(180 ⋅ 9)
60 m
180 cm
60 cm�
x m
9 m
180 cm
60 cm 9 m
x m
Finally, let’s return to an idea that we first saw in Chapter 5.
43. A light pole casts a shadow that measures 4 ft. At the same time, a yardstick casts ashadow that is 9 in. long. How tall is the pole?
44. A tree casts a shadow that measures 5 m. At the same time, a meter stick casts ashadow that is 0.4 m long. How tall is the tree?
45. Use the ideas of similar triangles to determine the height of a pole or tree on yourcampus. Work with one or two partners.
In exercises 46 to 48, one side of the triangle has been extended, forming what is called an“exterior angle.” In each case, find the measure of the indicated exterior angle.
49. What do you observe from exercises 46 to 48? Write a general conjecture about anexterior angle of a triangle.
50. Write an argument to show that an equilateral triangle cannot have a right angle.
51. Argue that, given an equilateral triangle, the measure of each angle must be 60°.
52. Argue that a triangle cannot have more than one obtuse angle.
53. Is it possible to have an isosceles right triangle? If such a triangle exists, what can besaid about the angles? Defend your statements.
If the sum of the measures of two angles is 90°, the two angles are said to becomplementary. If �A and �B are complementary angles, �A is said to be thecomplement of �B, and �B is said to be the complement of �A.
54. Create an argument to support the following statement:
If �ABC is a right triangle, with m�C � 90°, then �A and �B must be acute andcomplementary.
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