Similarly, as , we have
Recall that these triangles are considered to be similar because they have three pair of congruent angles. This is just one of three ways to determine that two triangles are similar. The table below summarizes criteria for determining if two triangles are similar.
Criteria Description Example
AAA Two triangles are similar of they have three pair of congruent angles
SSS Two triangles are similar if all three pair of corresponding sides are in the same proportion
SAS Two triangles are similar if two pair of corresponding sides are in the same proportion, and the included angles are congruent.
A special case of SSS is “HL,” or “hypotenuse leg.” This is the case of two right triangles being similar. This case is examined in example 5 below.
Example 5: Determine if the triangles are similar.
Solution: The triangles are similar
Recall that for every right triangle, we can use the Pythagorean Theorem to find the length of a missing side. In we have:
Similarly, in triangle we have:
Therefore the sides of the triangles are proportional, with a ratio of .
Because we will always be able to use the Pythagorean Theorem in this way, two right triangles will be similar if the hypotenuse and one leg of one triangle are in proportion with the hypotenuse and one leg of the second triangle. This is the HL criteria.
Applications of similar triangles
Similar triangles can be used to solve problems in which lengths or distances are proportional. The following example will show you how to solve such problems.
Example 6: Use similar triangles to solve the problem:
A tree casts a shadow that is long. A person who is tall is standing in front of the tree, and his shadow is long. Approximately how tall is the tree?
Solution:
The picture shows us similar right triangles: the person and his shadow are the legs of one triangle, and the tree and its shadow form the legs of the larger triangle. The triangles are similar because of their angles: they both have a right angle, and they share one angle. Therefore the third angles are also congruent, and the triangles are similar.
The ratio of the triangles’ lengths is . If we let represent the height of the tree, we have:
Lesson Summary
In this lesson we have reviewed key aspects of triangles, including the names of different types of triangles, the triangle angle sum, and criteria for similar triangles. In the last example, we used similar triangles to solve a problem involving an unknown height. In general, triangles are useful for solving such problems, but notice that we did not use the angles of the triangles to solve this problem. This technique will be the focus of problems you will solve later in the chapter.
Points to Consider
Why is it impossible for a triangle to have more than one right angle?
Why is it impossible for a triangle to have more than one obtuse angle?
How big can the measure of an angle get?
Review Questions
Triangle is an isosceles triangle. If side is long, and side is long, how long is side ?
Can a right triangle be an obtuse triangle? Explain.
A triangle has one angle that measures and a second angle that measures . What is the measure of the third angle in the triangle?
Claim: the two non-right angles in any right triangle are complements. Explain why this claim is true
Use this claim to find the measure of the third angle in the triangle below.
In triangle , the measure of angle is twice the measure of angle , and the measure of angle is three times the measure of angle . What are the measures of the three angles?
Triangles and shown below are similar. What is the length of ?
In triangles and above, if angle measures , what is the measure of angle ?
Determine if the triangles are similar:
A building casts a foot shadow, while a foot flagpole next to the building casts a foot shadow. How tall is the building?
Explain in your own words what it means for triangles to be similar.
Review Answers
Either or .
A right triangle cannot be an obtuse triangle. If a triangle is right triangle, one angle measures . If a triangle is obtuse, one angle measures greater than . Therefore the sum of the two angles would be greater than , which is not possible.
The angle sum in the triangle is . If you subtract the degree angle, you have , which is the sum of the remaining angles.
No
Yes, by SSS or HL
Answers will vary. Responses should include (1) three pairs of congruent angles and (2) sides in proportion, or some other notion of “scaling up” or “scaling down”
Vocabulary
Acute angle
An acute angle has a measure of less than .
Alternate interior angles of parallel lines
In the diagram show below, lines and are parallel, and they are intersected by a transversal . Angles and are alternate interior angles. Angles and are also alternate interior angles.
Congruent
Two angles are congruent if they have the same measure. Two segments are congruent if they have the same lengths.
Acute triangle
A triangle with all acute angles.
Isosceles triangle
A triangle with two congruent sides, and, consequentially, two congruent angles.
Equilateral triangle
A triangle with all sides congruent, and, consequently, all angles congruent.
Scalene triangle
A triangle with no pairs of sides congruent.
Leg
One of the two shorter sides of a right triangle.
Hypotenuse
The longest side of a right triangle, opposite the right angle.
Obtuse angle
An angle that measures more than .
Parallel lines
Lines that never intersect.
Right angle
An angle that measures .
Transversal
A line that intersects parallel lines.
Measuring Rotation
Learning objectives
A student will be able to:
Determine if an angle is acute, right, obtuse, or straight.
Express the measure of angles in degrees, minutes, and seconds.
Express the measure of angles in decimal degrees.
Identify and draw angles of rotation in standard position.
Identify quadrantal angles.
Identify co-terminal angles.
Introduction
In this lesson you will learn about angles of rotation, which are found in many different real phenomena. Consider, for example, a game that is played with a spinner. When you spin the spinner, how far has it gone?
You can answer this question in several ways. You could say something like “the spinner spun around 3 times.” This means that the spinner made complete rotations, and then landed back where it started.
We can also measure the rotation in degrees. In the previous lesson we worked with angles in triangles, measured in degrees. You may recall from geometry that a full rotation is , usually written as . Half a rotation is then and a quarter rotation is . Each of these measurements will be important in this lesson, as well as in the remainder of the chapter.
Acute, Right, Obtuse, and Straight Angles
In general, angles are categorized by their size. The table below summarizes the categories, which might be familiar from the previous lesson.
Name Description
Acute An angle whose measure is less than
Right An angle whose measure is exactly
Obtuse An angle whose measure is more than , but less than .
Straight An angle whose measure is exactly
You should make sure that you can visually determine which category an angle belongs to.
Example 1: Determine if the angle is ac
ute, right, obtuse, or straight.
a.
b.
c.
Solution:
a. This angle is an acute angle
If it is difficult to categorize the angle visually, you can compare it to a right angle. Doing this will help you see that the angle is smaller than a right angle.
b. This angle is an obtuse angle
Again, you can compare the angle to a right angle, if needed.
c. This angle is a right angle.
It is important to note that usually a right angle is marked with a small square.
It is also important to note that you can determine the measure of an angle using a protractor. This measure will of course be an approximation, as no protractor is perfect and the person measuring cannot perfectly line up the protractor or hold it steady.
Example 2: Use a protractor to measure the angle in example 1a.
Solution: The angle is about .
When working with angles measured in degrees, we often report our answers using a decimal, such as . However, in some contexts, angles are measured using fractional parts.
Measuring angles
Example 3: Two wheels are in direct contact. The radius of one is . The radius of the other is . The smaller one rotates four full turns. How many rotations does the larger wheel make? How many degrees does the larger wheel rotate through?
Solution: Every time the small wheel rotates once, its entire circumference passes along the larger wheel, Since the circumference of the large wheel is the large wheel rotates half way around. So if the small wheel rotates or the large wheel rotates , or .
We can measure angles in much the same way we measure time. A minute is of a degree. A second is of a minute, so it is of a degree. For example, is the way we write , , and . We can write this angle using fraction notation, as well as decimal notation:
We can also write a decimal degree using degrees, minutes, and seconds. For example, we can rewrite if we write the decimal part as a fraction:
Now solve for :
Now we have . We need to write . as seconds:
Therefore .
Notice that the angle is an obtuse angle. Its measure is less than . What does angle look like that is more than ? More than ?
Next you will learn about a particular way to represent angles that will allow you to represent , , or any other angle.
Angles of rotation in standard position
We can use our knowledge of graphing to represent any angle. The figure below shows an angle in what is called standard position.
The initial side of an angle in standard position is always on the positive axis. The terminal side always meets the initial side at the origin. Notice that the rotation goes in a counterclockwise direction. This means that if we rotate clockwise, we are generating a negative angle. Below are several examples of angles in standard position.
The degree angle is one of four quadrantal angles. A quadrantal angle is one whose terminal side lies on an axis. Along with , , and are quadrantal angles.
These angles are referred to as quadrantal because each angle defines a quadrant. Notice that without the arrow indicating the rotation, looks as if it is a , defining the fourth quadrant. Notice also that would look just like . The difference is in the action of rotation. This idea of two angles actually being the same angle is discussed next.
Coterminal angles
Consider the angle , in standard position.
Now consider the angle . We can think of this angle as a full rotation , plus an additional .
Notice that looks the same as . Formally, we say that the angles share the same terminal side. Therefore we call the angles co-terminal. Not only are these two angles co-terminal, but there are infinitely many angles that are co-terminal with these two angles. For example, if we rotate another , we get the angle . Or, if we create the angle in the negative direction (clockwise), we get the angle . Because we can rotate in either direction, and we can rotate as many times as we want, we can keep generating angles that are co-terminal with .
Example 3. Which angles are co-terminal with ?
a.
b.
c.
d.
Solution: b. and c. are co-terminal with .
Notice that terminal side of the first angle, , is in the quadrant. The last angle, is in the quadrant. Therefore neither angle is co-terminal with .
Now consider . This is a full rotation, plus an additional . So this angle is co-terminal with . The angle can be generated by rotating clockwise. To determine where the terminal side is, it can be helpful to use quadrantal angles as markers. For example, if you rotate clockwise (for a total of ), the terminal side of the angle is on the positive axis. For a total clockwise rotation of , we have more to rotate. This puts the terminal side of the angle at the same position as .
Lesson Summary
In this lesson we have categorized angles according to their size, and we have extended our knowledge of angles to include angles of rotation. We have defined what it means for an angle to be in standard position, and we have looked at angles in standard position, including the quadrantal angles. We have also defined the concept of co-terminal angles. All of the ideas in this lesson will be used in the following lesson, to define the trigonometric functions that are the focus of this chapter.
Points to Consider
How can one angle look exactly the same as another angle?
Where might you see angles of rotation in real life?
Review Questions
Determine if the angle is acute, right, obtuse, or straight.
Approximate the measure of the angle. Explain how you approximated.
Rewrite the measure of each angle in degrees, minutes, and seconds.
Rewrite the measure of each angle in decimal degrees.
Determine the measure of the angle between the clock hands at the given time.
Through what angle does the minute hand of a clock rotate between 12:00am and 1am?
A car goes around a degree circular curve in a racetrack. The diameter of an automobile’s wheel is . The distance between the wheels is . The radius of the curve the car is following is measured at the closest wheel to the track. What is the difference in number of rotations that the outer wheel must turn compared with the inner wheel?
State the measure of an angle that is co-terminal with
Name two angles that are co-terminal with An angle that is negative
An angle that is greater than
A drag racer goes around a degree circular curve in a racetrack in a path of radius . Its front and back wheels have different diameters. The front wheels are in diameter. The rear wheels are much larger; they have a diameter of . The axles of both wheels are long. Which wheel has more rotations going around the curve. How many more degrees does that wheel rotate compared with the wheel that rotates the least making that curve?
Review Answers
Acute
Straight
The angle is about . You can approximate the measure of the angle using a protractor, or by using other angles, such as and .
Answers will vary. Examples:
Answers will vary. Examples:
The front wheel rotates more. It rotates revolutions versus revolutions for the back wheel, which is a degree difference.
Vocabulary
Acute angle
An acute angle is an angle with measure between and .
Co-terminal angles
Angles of rotation in standard position are co-terminal of they share the same terminal side.
Minutes
A minute is of a degree.
Obtuse angle
An obtuse angle is an angle with measure between and .
Protractor
A protractor is a tool used to measure angles.
Quadrantal angle
A quadrantal angle is an angle in standard position whose terminal side lies on an axis.
Right angle
A right angle i
s an angle with measure exactly .
Seconds
A second is of a minute, or of a degree.
Standard position
An angle in standard position has its initial side on the positive axis, its vertex at the origin, and its terminal side anywhere in the plane. A positive angle means a counterclockwise rotation. A negative angle means a clockwise rotation.
Straight angle
A straight angle is an angle with measure . A straight angle makes a straight line.
Defining Trigonometric Functions
Learning objectives
A student will be able to:
Find the values of the six trig functions for angles in right triangles.
Find the values of the six trig functions for angles of rotation.
Work with angles in the unit circle.
Introduction
CK-12 Trigonometry Page 3
