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Section 1.1 Angles and Triangles

Historically, trigonometry began as the study of triangles and their properties. Let's review some definitions and facts from geometry.

  • We measure angles in degrees.
  • One full rotation is \(360\degree\text{,}\) as shown below.
  • Half a full rotation is \(180\degree\) and is called a straight angle.
  • One quarter of a full rotation is \(90\degree\) and is called a right angle.
quadrantal angles

Subsection Triangles

If you tear off the corners of any triangle and line them up, as shown below, they will always form a straight angle.

tear corners off triangle

1. The sum of the angles in a triangle is \(180\degree\text{.}\)

Example 1.1

Two of the angles in the triangle at right are \(25\degree\) and \(115\degree\text{.}\) Find the third angle.

triangle with 25 and 115 angles
Solution

To find the third angle, we write an equation.

\begin{equation*} \begin{aligned}[t] x+25+115\amp =180 \amp \amp \blert{\text{Simpify the left side.}}\\ x+140\amp =180 \amp \amp \blert{\text{Subtract 140 from both sides.}}\\ x\amp =40 \amp \amp \end{aligned} \end{equation*}

The third angle is \(40\degree\text{.}\)

Checkpoint 1.2

Find each of the angles in the triangle at right.

triangle with angles x, 2x, and 2x-15
Answer

\(x=39\degree,~~2x=78\degree, ~~2x-15=63\degree\)

Some special categories of triangles are particularly useful. Most important of these are the right triangles.

2. A right triangle has one angle of \(90\degree\text{.}\)

Example 1.3

One of the smaller angles of a right triangle is \(34\degree\text{.}\) What is the third angle?

right triangle with 34 angle
Solution

The sum of the two smaller angles in a right triangle is \(90\degree\text{.}\) So

\begin{equation*} \begin{aligned}[t] x+34\amp =90 \amp \amp \blert{\text{Subtract 34 from both sides.}}\\ x\amp =56 \amp \amp \end{aligned} \end{equation*}

The unknown angle must be \(56\degree\text{.}\)

Checkpoint 1.4

Two angles of a triangle are \(35\degree\) and \(45\degree\text{.}\) Can it be a right triangle?

Answer

No

An equilateral triangle has all three sides the same length.

3. All of the angles of an equilateral triangle are equal.

Example 1.5

All three sides of a triangle are 4 feet long. Find the angles.

equilateral triangle
Solution

The triangle is equilateral, so all of its angles are equal. Thus

\begin{equation*} \begin{aligned}[t] 3x\amp =180 \amp \amp \blert{\text{Divide both sides by 3.}}\\ x\amp =60 \amp \amp \end{aligned} \end{equation*}

Each of the angles is \(60\degree\text{.}\)

Checkpoint 1.6

Find \(x\text{,}\) \(y\text{,}\) and \(z\) in the triangle at right.

equilateral triangle with side 8
Answer

\(x=60\degree, ~y=8, ~z=8\)

An isosceles triangle has two sides of equal length. The angle between the equal sides is the vertex angle. The other two angles are the base angles.

4. The base angles of an isosceles triangle are equal.

Example 1.7

Find \(x\) and \(y\) in the triangle at right.

isosceles triangle with side 12, base angle 38
Solution

The triangle is isosceles, so the base angles are equal. Therefore, \(y=38\degree\text{.}\) To find the vertex angle, we solve

\begin{equation*} \begin{aligned}[t] x+38+38\amp =180 \amp \amp \\ x+76\amp =180 \amp \amp \blert{\text{Subtract 76 from both sides.}}\\ x\amp =104 \amp \amp \end{aligned} \end{equation*}

The vertex angle is \(104\degree\text{.}\)

Checkpoint 1.8

Find \(x\) and \(y\) in the figure at right.

isosceles triangle with side 9, base angle 20
Answer

\(x=140\degree,~y=9\)

Subsection Angles

In addition to the facts about triangles reviewed above, there are several useful properties of angles.

  • Two angles that add to \(180\degree\) are called supplementary.
  • Two angles that add to \(90\degree\) are called complementary.
  • Angles between \(0\degree\) and \(90\degree\) are called acute.
  • Angles between \(90\degree\) and \(180\degree\) are called obtuse.
types of angles
Example 1.9

In the figure at right,

  • \(\angle\)\(AOC\) and \(\angle\)\(BOC\) are supplementary.
  • \(\angle\)\(DOE\) and \(\angle\)\(BOE\) are complementary.
  • \(\angle\)\(AOC\) is obtuse,
  • and \(\angle\)\(BOC\) is acute.
types of angles

In trigonometry we often use lower-case Greek letters to represent unknown angles (or, more specifically, the measure of the angle in degrees). In the next Exercise, we use the Greek letters \(\alpha\) (alpha), \(\beta\) (beta), and \(\gamma\) (gamma).

Checkpoint 1.10

In the figure, \(\alpha\text{,}\) \(\beta\text{,}\) and \(\gamma\) denote the measures of the angles in degrees.

  1. Find the measure of angle \(\alpha\text{.}\)
  2. Find the measure of angle \(\beta\text{.}\)
  3. Find the measure of angle \(\gamma\text{.}\)
  4. What do you notice about the measures of the angles?
straight angles with 50 degrees
Answer

\(\alpha=130\degree, ~\beta=50\degree, ~\gamma=130\degree.\) The non-adjacent angles are equal.

Non-adjacent angles formed by the intersection of two straight lines are called vertical angles. In the previous exercise, the angles labeled \(\alpha\) and \(\gamma\) are vertical angles, as are the angles labeled \(\beta\) and \(50\degree\text{.}\)

5. Vertical angles are equal.

Example 1.11

Explain why \(\alpha=\beta\) in the triangle at right.

isosceles triangle with alpha and beta
Solution

Because they are the base angles of an isosceles triangle, \(\theta\) (theta) and \(\phi\) (phi) are equal.

Also, \(\alpha=\theta\) because they are vertical angles, and similarly \(\beta=\phi\text{.}\)

Therefore, \(\alpha=\beta\) because they are equal to equal quantities.

Checkpoint 1.12

Find all the unknown angles in the figure at right. (You will find a list of all the Greek letters and their names at the end of this section.)

triangle with external angle 150
Answer

\(\alpha=40\degree,~ \beta=140\degree,~ \gamma=75\degree, \delta=65\degree\)

A line that intersects two parallel lines forms eight angles, as shown in the figure below. There are four pairs of vertical angles, and four pairs of corresponding angles, or angles in the same position relative to the transversal on each of the parallel lines.

For example, the angles labeled 1 and 5 are corresponding angles, as are the angles labeled 4 and 8. Finally, angles 3 and 6 are called alternate interior angles, and so are angles 4 and 5.

transversal

6. If parallel lines are intersected by a transversal, the alternate interior angles are equal. Corresponding angles are also equal.

Example 1.13

The parallelogram \(ABCD\) shown at right is formed by the intersection of two sets of parallel lines. Show that the opposite angles of the parallelogram are equal.

parallelogram
Solution

Angles 1 and 2 are equal because they are alternate interior angles, and angles 2 and 3 are equal because they are corresponding angles. Therefore angles 1 and 3, the opposite angles of the parallelogram, are equal. Similarly, you can show that angles 4, 5, and 6 are equal.

Checkpoint 1.14

Show that the adjacent angles of a parallelogram are supplementary. (You can use angles 1 and 4 in the parallelogram of the previous example.)

Answer

Note that angles 2 and 6 are supplementary because they form a straight angle. Angle 1 equals angle 2 because they are alternate interior angles, and similarly angle 4 equals angle 5. Angle 5 equals angle 6 because they are corresponding angles. Thus, angle 4 equals angle 6, and angle 1 equals angle 2. So angles 4 and 1 are supplementary because 2 and 6 are.

Note 1.15

In the Section 1.1 Summary, you will find a list of vocabulary words and a summary of the facts from geometry that we reviewed in this section. You will also find a set of study questions to test your understanding, and a list of skills to practice in the homework problems.

Greek Alphabet
\(\alpha~~~~\text{alpha}\) \(\beta~~~~\text{beta}\) \(\gamma~~~~\text{gamma}\)
\(\delta~~~~\text{delta}\) \(\epsilon~~~~\text{epsilon}\) \(\gamma~~~~\text{gamma}\)
\(\eta~~~~\text{eta}\) \(\theta~~~~\text{theta}\) \(\iota~~~~\text{iota}\)
\(\kappa~~~~\text{kappa}\) \(\lambda~~~~\text{lambda}\) \(\mu~~~~\text{mu}\)
\(\nu~~~~\text{nu}\) \(\xi~~~~\text{xi}\) \(\omicron~~~~\text{omicron}\)
\(\pi~~~\text{pi}\) \(\rho~~~~\text{rho}\) \(\sigma~~~~\text{sigma}\)
\(\tau~~~~\text{tau}\) \(\upsilon~~~~\text{upsilon}\) \(\phi~~~~\text{phi}\)
\(\chi~~~\text{chi}\) \(\psi~~~\text{psi}\) \(\omega~~~\text{omega}\)

Subsection Algebra Refresher

Subsubsection Exercises

Solve the equation.

1

\(x-8=19-2x\)

2

\(2x-9=12-x\)

3

\(13x+5=2x-28\)

4

\(4+9x=-7+x\)

Solve the system.

5

\(\begin{aligned}[t] 5x-2y\amp =-13 \amp \amp \\ 2x+3y\amp =-9 \amp \amp \\ \end{aligned}\)

6

\(\begin{aligned}[t] 4x+3y\amp =9 \amp \amp \\ 3x+2y\amp =8 \amp \amp \\ \end{aligned}\)

Subsection Algebra Refresher Answers

  1. \(9\)
  2. \(2\)
  3. \(-3\)
  4. \(-2\)
  5. \(x=-3,y=-1\)
  6. \(x=6, y=-5\)

Subsection Section 1.1 Summary

Subsubsection Vocabulary

  • Right angle
  • Straight angle
  • Right triangle
  • Equilateral triangle
  • Isosceles triangle
  • Vertex angle
  • Base angle
  • Supplementary
  • Complementary
  • Acute
  • Obtuse
  • Vertical angles
  • Transversal
  • Corresponding angles
  • Alternate interior angles

Subsubsection Concepts

Facts from Geometry

1. The sum of the angles in a triangle is \(180\degree\text{.}\)

2. A right triangle has one angle of \(90\degree\text{.}\)

3. All of the angles of an equilateral triangle are equal.

4. The base angles of an isosceles triangle are equal.

5. Vertical angles are equal.

6. If parallel lines are intersected by a transversal, the alternate interior angles are equal. Corresponding angles are also equal.

Subsubsection Study Questions

  1. Is it possible to have more than one obtuse angle in a triangle? Why or why not?
  2. Draw any quadrilateral (a four-sided polygon) and divide it into two triangles by connecting two opposite vertices by a diagonal. What is the sum of the angles in your quadrilateral?
  3. What is the difference between a vertex angle and vertical angles?
  4. Can two acute angles be supplementary?
  5. Choose any two of the eight angles formed by a pair of parallel lines cut by a transversal. Those two angles are either equal or _______ .

Subsubsection Skills

Practice each skill in the Homework Problems listed.

  1. Sketch a triangle with given properties #1–6
  2. Find an unknown angle in a triangle #7–12, 17–20
  3. Find angles formed by parallel lines and a transversal #13–16, 35–44
  4. Find exterior angles of a triangle #21–24
  5. Find angles in isosceles, equilateral, and right triangles #25–34
  6. State reasons for conclusions #45–48

Subsection Homework 1.1

For Problems 1–6, sketch and label a triangle with the given properties.

1

An isosceles triangle with vertex angle 30°

2

A scalene triangle with one obtuse angle (Scalene means three unequal sides.)

3

A right triangle with legs 4 and 7

4

An isosceles right triangle

5

An isosceles triangle with one obtuse angle

6

A right triangle with one angle 20°

For Problems 7–20, find each unknown angle.

7
triangle theta
8
triangle phi
9
triangle alpha
10
triangle gamma
11
triangle beta
12
triangle omega
13
triangle alpha
14
triangle beta
15
triangle theta
16
triangle phi
17
triangle theta
18
triangle alpha
19
triangle psi
20
triangle beta

In Problems 21 and 22, the angle labeled \(\phi\) is called an exterior angle of the triangle, formed by one side and the extension of an adjacent side. Find \(\phi\text{.}\)

21
ext angle
22
ext angle
23

In parts (a) and (b), find the exterior angle \(\phi\text{.}\)

  1. ext angle
  2. ext angle
  3. Find an algebraic expression for \(\phi\text{.}\)

    ext angle
  4. Use your answer to part (c) to write a rule for finding an exterior angle of a triangle.
24
  1. Find the three exterior angles of the triangle. What is the sum of the exterior angles?

    ext angles
  2. Write an algebraic expression for each exterior angle in terms of one of the angles of the triangle. What is the sum of the exterior angles?

    ext angles

In Problems 25 and 26, the figures inscribed are regular polygons, which means that all their sides are the same length. Find the angles \(\theta\) and \(\phi\text{.}\)

25
pentagon
26
hexagon

In problems 27 and 28, \(\triangle ABC\) is equilateral. Find the unknown angles.

27
triangles
28
triangles
29
triangles
  1. \(2 \theta + 2 \phi =~\)

  2. \(\theta + \phi =~\)

  3. \(\triangle ABC\) is\(~~~\)

30

Find \(\alpha\) and \(\beta\text{.}\)

triangles
31
circle
  1. Explain why \(\angle OAB\) and \(\angle ABO\) are equal in measure.
  2. Explain why \(\angle OBC\) and \(\angle BCO\) are equal in measure.
  3. Explain why \(\angle ABC\) is a right angle. (Hint: Use Problem 29.)
32
circle
  1. Compare \(\theta\) with \(\alpha + \beta\text{.}\) (Hint: What do you know about supplementary angles and the sum of angles in a triangle?
  2. Compare \(\alpha\) and \(\beta\text{.}\)
  3. Explain why the inscribed angle \(\angle BAO\) is half the size of the central angle \(\angle BOD\text{.}\)
33

Find \(\alpha\) and \(\beta\text{.}\)

equil triangle
34

Find \(\alpha\) and \(\beta\text{.}\)

square

In Problems 35–44, arrows on a pair of lines indicate that they are parallel. Find \(x\) and \(y\) .

35
parallel lines
36
parallel lines
37
parallel lines
38
parallel lines
39
parallel lines
40
parallel lines
41
parallel lines
42
parallel lines
43
parallel lines
44
parallel lines
45
  1. Among the angles labeled 1 through 5 in the figure below, find two pairs of equal angles.

    parallel lines
  2. \(\angle 4 + \angle 2 + \angle 5 =~\)
  3. Use parts (a) and (b) to explain why the sum of the angles of a triangle is \(180 \degree \)
46
  1. In the figure below, find \(\theta\text{,}\) and justify your answer.

    parallel lines
  2. Write an algebraic expression for \(\theta\) in the figure below.

    parallel lines
47

\(ABCD\) is a rectangle. The diagonals of a rectangle bisect each other. In the figure, \(\angle AQD = 130\degree\text{.}\) Find the angles labeled 1 through 5 in order, and give a reason for each answer.

rectangle
48

A tangent meets the radius of a circle at a right angle. In the figure,\(\angle AOB = 140\degree\text{.}\) Find the angles labeled 1 through 5 in order, and give a reason for each answer.

circle with tangents