Chapter 1: Triangles and Circles

1.1 Triangles and Angles

Algebra Refresher

Algebra Refresher for Triangles and Angles

Solve the equation.

1. [latex]x-8 = 19 -2x[/latex]
2. [latex]2x-9 = 12 -x[/latex]
3. [latex]13x+5 = 2x -28[/latex]
4. [latex]4+9x = -7+x[/latex]

Solve the system

5. [latex]5x - 2y = -13, 2x+3y = -9[/latex]
6. [latex]4x+3y=9, 3x+2y= 8[/latex]

 

Algebra Refresher Answers:
  1. [latex]9[/latex]
  2. [latex]2[/latex]
  3. [latex]-3[/latex]
  4. [latex]-2[/latex]
  5. [latex]x=-3, y= -1[/latex]
  6. [latex]x = 6, y = -5[/latex]

 

Triangles and Angles

Learning Objectives

  • Sketch a triangle with given properties
  • Find an unknown angle in a triangle
  • Find angles formed by parallel lines and a transversal
  • Find exterior angles of a triangle
  • Find angles in isosceles, equilateral, and right triangles
  • State reasons for conclusions

 

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 [latex]360^{\circ}[/latex] as shown below.
  • Half a full rotation is [latex]180^{\circ}[/latex] and is called a straight angle.
  • One-quarter of a full rotation is [latex]90^{\circ}[/latex] and is called a right angle.

 

quadrantal angles
Figure 5

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

 

Sum of angles in a triangle

The sum of the angles in a triangle is [latex]180^{\circ}[/latex].

 

Example 1.1

Two of the angles in the triangle at right are [latex]25^{\circ}[/latex] and [latex]115^{\circ}[/latex]. Find the third angle.

triangle with 25 and 115 angles

 

Solution

To find the third angle, we write an equation:
[latex]\begin{align*} x+25+115 = & 180 & \text{Simplify the left side.} \\ x+140 = & 180 & \text{Subtract 140 from both sides} \\ x= 40 \end{align*}[/latex]

The third angle is [latex]40^{\circ}[/latex]

 

 

Checkpoint 1.2.

Find each of the angles in the triangle at right.

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

Answer.

[latex]x=39^{\circ}[/latex], [latex]2x=78[/latex], [latex]2x-15=63[/latex].  [latex]x=40^{\circ}[/latex]

 

 

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

Right Triangle

A right triangle has one angle of [latex]90^{\circ}[/latex]

 

Example 1.3.

One of the smaller angles of a right triangle is [latex]34^{\circ}[/latex]. What is the third angle?
right triangle with 34 angle
Solution

The sum of the two smaller angles in a right triangle is [latex]90^{\circ}[/latex]. So

[latex]\begin{align*} x+34 = & 90 & \text{Subtract 34 from both sides.} \\ x =56 \end{align*}[/latex]

The unknown angle must be [latex]56^{\circ}[/latex]

Checkpoint 1.4.

Two angles of a triangle are [latex]35^{\circ}[/latex]and [latex]65^{\circ}[/latex]. Can it be a right triangle?
Answer.

No

 

 

An equilateral  triangle has all three sides the same length.

Angles of equilateral triangle

 

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

[latex]\begin{align*} 3x= & 180 & \text{Divide both sides by 3}.\\ x = 60 \end{align*}[/latex]

Each of the angles is [latex]60^{\circ}[/latex]

 

Checkpoint 1.6.

Find [latex]x[/latex], [latex]y[/latex], and [latex]z[/latex]  in the triangle at right.

equilateral triangle with side 8

Answer.

[latex]x=60^{\circ}[/latex],[latex]y=8[/latex],[latex]z=8[/latex]

 

 

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.

 

Base angles of an isosceles triangle

The base angles of an isosceles triangle are equal.

 

Example 1.7.

Find [latex]x[/latex] and [latex]y[/latex] 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, [latex]y=38^{\circ}[/latex]. To find the vertex angle, we solve
[latex]\begin{align*} x+38+38=180 \\ x+76= & 180 & \text{Subtract 76 from both sides.}\\ x=104 \end{align*}[/latex]
The vertex angle is [latex]104^{\circ}[/latex]

 

Checkpoint 1.8.

Find [latex]x[/latex] and [latex]y[/latex] in the figure at right.

isosceles triangle with side 9, base angle 20

Answer.

[latex]x=140^{\circ},~y=9[/latex]

 

Angles

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

  • Two angles that add to [latex]180^{\circ}[/latex] are called supplementary.
  • Two angles that add to [latex]90^{\circ}[/latex] are called complementary.
  • Angles between [latex]0^{\circ}[/latex] and [latex]90^{\circ}[/latex] are called acute.
  • Angles between [latex]90^{\circ}[/latex] and [latex]180^{\circ}[/latex] are called obtuse.

types of angles

 

Example 1.9.

In the figure at right,types of angles

  • [latex]\angle AOC[/latex] and [latex]\angle BOC[/latex] are supplementary.
  • [latex]\angle DOE[/latex] and [latex]\angle BOE[/latex] are complementary.
  • [latex]\angle AOC[/latex] is obtuse.
  • and [latex]\angle BOC[/latex] is acute.

 

In trigonometry we often use lowercase Greek letters to represent unknown angles (or, more specifically, the measure of the angle in degrees).

 

Table 1.16. Lowercase Letters in the Greek Alphabet
Greek Alphabet
[latex]\alpha[/latex](alpha) [latex]\beta[/latex](beta) [latex]\gamma[/latex](gamma)
[latex]\delta[/latex](delta) [latex]\epsilon[/latex](epsilon)
[latex]\eta[/latex](eta) [latex]\theta[/latex](theta) [latex]\iota[/latex](iota)
[latex]\kappa[/latex](kappa) [latex]\lambda[/latex](lambda) [latex]\mu[/latex](mu)
[latex]\nu[/latex](nu) [latex]\xi[/latex](xi) [latex]\omicron[/latex](omicron)
[latex]\pi[/latex](pi) [latex]\rho[/latex](rho) [latex]\sigma[/latex](sigma)
[latex]\tau[/latex](tau) [latex]\upsilon[/latex](upsilon) [latex]\phi[/latex](phi)
[latex]\chi[/latex](chi) [latex]\psi[/latex](psi) [latex]\omega[/latex](omega)

In the next exercise, we use the Greek letters [latex]\alpha[/latex] (alpha), [latex]\beta[/latex] (beta), and [latex]\gamma[/latex] (gamma).

 

Checkpoint 1.10.

In the figure, [latex]\alpha[/latex],[latex]\beta[/latex], and [latex]\gamma[/latex] denote the measures of the angles in degrees.

  1. Find the measure of angle [latex]\alpha[/latex]
  2. Find the measure of angle [latex]\beta[/latex]
  3. Find the measure of angle [latex]\gamma[/latex]
  4. What do you notice about the measures of the angles?

straight angles with 50 degrees

Answer.

[latex]\alpha=130^{\circ},~\beta=50^{\circ}, ~\gamma=130^{\circ}[/latex]. 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 [latex]\alpha[/latex] and [latex]\gamma[/latex]are vertical angles, as are the angles labeled [latex]\beta[/latex] and [latex]50^{\circ}[/latex].

 

 

Vertical Angles.

 

Example 1.11.

isosceles triangle with alpha and beta

Explain why [latex]\alpha=\beta[/latex] in the triangle at right.

Solution.

Because they are the base angles of an isosceles triangle, [latex]\theta[/latex](theta) and [latex]\phi[/latex](phi) are equal. Also,[latex]\alpha=\theta[/latex] because they are vertical angles, and similarly [latex]\beta=\phi[/latex]. Therefore,[latex]\alpha=\beta[/latex] 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 prior to Checkpoint 1.10.)

triangle with external angle 150

Answer.

[latex]\alpha=40^{\circ},~\beta=140^{\circ},~\gamma=75^{\circ},~\delta=65^{\circ}[/latex]

 

 

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 [latex]1[/latex] and [latex]5[/latex] are corresponding angles, as are the angles labeled [latex]4[/latex] and [latex]8[/latex]. Finally, angles [latex]3[/latex] and [latex]6[/latex] are called alternate interior angles, and so are angles [latex]4[/latex] and [latex]5[/latex].

transversal

 

Parallel lines cut by a transversal.

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

 

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

 

Checkpoint 1.14.

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

Answer.

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

 

 

Section 1.1 Summary

VOCABULARY

 

CONCEPTS

FACTS FROM GEOMETRY

 

1. The sum of the angles in a triangle is [latex]180^{\circ}[/latex].
2. A right triangle has one angle of [latex]90^{\circ}[/latex].
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.

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 ___________ .

 

definition

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Trigonometry Copyright © 2024 by Bimal Kunwor; Donna Densmore; Jared Eusea; and Yi Zhen. All Rights Reserved.

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