Trig Angles and Triangles

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

α

(alpha),

β

(beta), and

γ

(gamma).

🔗

Checkpoint 1.10.

In the figure,

α,

β,

and

γ

denote the measures of the angles in degrees.

🔗

Find the measure of angle $α .$
Find the measure of angle $β .$
Find the measure of angle $γ .$
What do you notice about the measures of the angles?

🔗

$straight angles with 50 degrees$

Answer.

α = 130^{\circ}, β = 50^{\circ}, γ = 130^{\circ} .

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

α

and

γ

are vertical angles, as are the angles labeled

β

and

50^{\circ} .

🔗

Vertical Angles.

5. Vertical angles are equal.

🔗

Example 1.11.

Explain why

α = β

in the triangle at right.

🔗

$isosceles triangle with alpha and beta$

Solution.

Because they are the base angles of an isosceles triangle,

θ

(theta) and

ϕ

(phi) are equal. Also,

α = θ

because they are vertical angles, and similarly

β = ϕ .

Therefore,

α = β

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.

α = 40^{\circ}, β = 140^{\circ}, γ = 75^{\circ}, δ = 65^{\circ}

🔗

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.

🔗

$transversal$

Paralles lines cut by a 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

A B C D

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.

🔗

Table 1.16. Lower Case Letters in the Greek Alphabet

Greek Alphabet
$α alpha$	$β beta$	$γ gamma$
$δ delta$	$ϵ epsilon$	$ζ zeta$
$η eta$	$θ theta$	$ι iota$
$κ kappa$	$λ lambda$	$μ mu$
$ν nu$	$ξ xi$	$o omicron$
$π pi$	$ρ rho$	$σ sigma$
$τ tau$	$υ upsilon$	$ϕ phi$
$χ chi$	$ψ psi$	$ω omega$

🔗

Review the following skills you will need for this section.

🔗

Algebra Refresher 1.2.

Solve the equation.

🔗

x - 8 = 19 - 2 x

🔗

13 x + 5 = 2 x - 28

🔗

2 x - 9 = 12 - x

🔗

9 + 9 x = - 7 + x

🔗

Solve the system.

🔗

\begin{aligned} 5 x - 2 y & = - 13 \\ 2 x + 3 y & = - 9 \end{aligned}

🔗

\begin{aligned} 4 x + 3 y & = 9 \\ 3 x + 2 y & = 8 \end{aligned}

🔗

\underset{―}{}

🔗

Algebra Refresher Answers

🔗

$9$
$7$
$- 3$
$- 2$
$x = - 3, y = - 1$
$x = 6, y = - 5$

🔗

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^{\circ} .

🔗

2. A right triangle has one angle of

90^{\circ} .

🔗

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

Is it possible to have more than one obtuse angle in a triangle? Why or why not?
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?
What is the difference between a vertex angle and vertical angles?
Can two acute angles be supplementary?
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.

🔗

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

🔗

Exercise Group.

In Problems 21 and 22, the angle labeled

ϕ

is called an exterior angle of the triangle, formed by one side and the extension of an adjacent side. Find

ϕ .

🔗

21.

$ext angle$

🔗

22.

$ext angle$

🔗

23.

In parts (a) and (b), find the exterior angle

ϕ .

🔗

$ext angle$
$ext angle$
Find an algebraic expression for $ϕ .$

$ext angle$
Use your answer to part (c) to write a rule for finding an exterior angle of a triangle.

🔗

24.

Find the three exterior angles of the triangle. What is the sum of the exterior angles?
🔗

$ext angles$
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?
🔗

$ext angles$

🔗

Exercise Group.

In Problems 25 and 26, the figures inscribed are regular polygons, which means that all their sides are the same length, and all the angles have the same measure. Find the angles

θ

and

ϕ .

🔗

25.

$pentagon$

🔗

26.

$hexagon$

🔗

Exercise Group.

In problems 27 and 28,

△ A B C

is equilateral. Find the unknown angles.

🔗

27.

$triangles$

🔗

28.

$triangles$

🔗

29.

$triangles$

$2 θ + 2 ϕ =$
$θ + ϕ =$
$△ A B C$ is

🔗

30.

Find

α

and

β .

🔗

$triangles$

🔗

31.

$circle$

Explain why $∠ O A B$ and $∠ A B O$ are equal in measure.
Explain why $∠ O B C$ and $∠ B C O$ are equal in measure.
Explain why $∠ A B C$ is a right angle. (Hint: Use Problem 29.)

🔗

32.

$circle$

Compare $θ$ with $α + β .$ (Hint: What do you know about supplementary angles and the sum of angles in a triangle?
Compare $α$ and $β .$
Explain why the inscribed angle $∠ B A O$ is half the size of the central angle $∠ B O D .$

🔗

Exercise Group.

33.

Find

α

and

β .

🔗

$equil triangle$

🔗

34.

Find

α

and

β .

🔗

$square$

🔗

Exercise Group.

In Problems 35–44, arrows on a pair of lines indicate that they are parallel. Find

x

$parallel lines$

🔗

45.

Among the angles labeled 1 through 5 in the figure at right, find two pairs of equal angles.
🔗

$parallel lines$
$∠ 4 + ∠ 2 + ∠ 5 =$
Use parts (a) and (b) to explain why the sum of the angles of a triangle is $180^{\circ}$