Angles and Triangles

Section 1.4 Angles and Triangles

Angles
Measuring Angles
Using a Protractor
Triangles
Sides and Angles

Subsection 1.4.1 Angles

When two line segments meet at a point, they form an angle. The point where the two sides meet is called the vertex. Here are some angles.

$Acute angles$

You are probably familiar with a right angle, such as the corner of a square. Each of the corners of the tabletop below is a right angle. The two arms of a plus sign form four right angles. We say that the sides of a right angle are perpendicular.

$~~~~~~~~~~~~~~~~~~~~\blert{\text{Right angles: a square table}}~~~~~~~~~~\blert{\text{Perpendicular lines make a plus sign}}$

Now imagine slowly opening a laptop computer until the two sides lie flat on the table. The two sides form an angle that gradually opens until the two covers form a straight line. We give different names to angles, depending on how open they are.

$~~~~~~~~~~~~~~~\blert{\text{Partially open}}~~~~~~~~~~\blert{\text{More than half-way open}}~~~~~~~~~~~~~\blert{\text{All the way open}}$

$~~~~~~~~~~~~~~~\blert{\text{An acute angle}}~~~~~~~~~~~~~~~~\blert{\text{An obtuse angle}}~~~~~~~~~~~~~~~~~~~~~~\blert{\text{A straight angle}}$

Kinds of Angles.

If the sides are less open than a right angle, we call the angle acute.
If the sides are more open than a right angle, we call the angle obtuse.
If the two sides are totally open to form a straight line, we call the angle a straight angle.

$Acute, obtuse, and straight angles$

Example 1.4.1.

In the pictures below, the handles of the hedge clipper form an acute angle. The hands of the clock form an obtuse angle.

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\blert{\text{An acute angle}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\blert{\text{An obtuse angle}}$

Checkpoint 1.4.2.

Acute, right, or obtuse? Choose the best answer for each angle in the photograph.

The ballerina’s arms form a(n) angle.
The ballerina’s legs form a(n) angle.
The male dancer’s legs form a(n) angle.

Answer.

obtuse
right
acute

Subsection 1.4.2 Measuring Angles

We use degrees to measure how open an angle is. The symbol for a degree is $\degree\text{.}$ A right angle is 90$\degree\text{,}$ and a straight angle is 180$\degree\text{.}$

$Acute, obtuse, and straight angles$

The degree measure of an acute angle is less than 90$\degree\text{,}$ and the degree measure of an obtuse angle is between 90$\degree$ and 180$\degree\text{.}$

$Acute angles$

$\qquad\quad\blert{\text{Some acute angles}}$

$Obtuse angles$

$\qquad\quad\blert{\text{Some obtuse angles}}$

Question.

What do you think an angle of 0$\degree$ looks like?

Note 1.4.3.

The measure of an angle does not depend on the orientation of the angle, or on how long the sides are, or whether the angle is combined with other angles. It only depends on how open the sides are. Here are some pictures of 30$\degree$ angles.

$Thirty degree angles in different orientations$

We use the symbol $\angle$ to stand for "angle."

Example 1.4.4.

The picture shows two angles that together make a right angle. If the measure of $\angle~A$ is 25$\degree\text{,}$ what is the measure of $\angle~B\text{?}$

$Complementary angles$

Solution.

A right angle measures 90$\degree\text{,}$ so the sum of $\angle~A$ plus $\angle~B$ must be 90$\degree\text{.}$ Therefore,

\begin{equation*} \angle~B = 90\degree - \angle~A = 90\degree - 25\degree = 65\degree \end{equation*}

The measure of $\angle~B$ is 65$\degree\text{,}$ or $\angle~B = 65\degree\text{.}$

Checkpoint 1.4.5.

Find the measure of the unknown angle. Parts (a) and (b) show right angles.

$Complementary angles$

$\displaystyle \angle~A = 38\degree, ~~ \angle~B =~?$
$Complementary angles$

$\displaystyle \angle~A = 20\degree, ~~ \angle~B = \angle~C =~?$
$Supplementary angles$

$\displaystyle \angle~A =~?, ~~ \angle~B = 85\degree$
$Supplementary angles$

$\displaystyle \angle~B = 27\degree, ~~ \angle~A =~?, ~~ \angle~C =~?$

Answer.

$\displaystyle 52 \degree$
$\displaystyle 35 \degree$
$\displaystyle 95 \degree$
$\displaystyle 153 \degree, ~ 153 \degree$

Subsection 1.4.3 Using a Protractor

We use a protractor to measure angles.

$Protractor$

Using a protractor to measure an angle.

Place the center dot of the protractor at the vertex of the angle.
Line up the bottom line of the protractor along one of the sides of the angle.
Read the number where the other side of the angle meets the scale on the circular edge of the protractor.

Note 1.4.6.

Notice that the protractor has one scale that starts at the right edge and increases in the counterclockwise direction, and a second scale that starts on the left edge. That’s so you can line up the first side of the angle in either direction from the center.

Example 1.4.7.

Use a protractor to measure this angle:

$Acute angle$

Solution.

Place the protractor as shown below, with the vertex O in the center and side OB lined up along the bottom of the protractor.

$Angle under protractor$

Now follow the scale that increases from side OB up to side OA. In this case it is the inner scale. We see that side OA meets the inner scale at 40$\degree\text{.}$ The measure of $\angle~AOB$ is 40$\degree\text{.}$

Checkpoint 1.4.8.

What is the measure of the obtuse angle shown below?

$Obtuse angle under protractor$

Answer.

$135 \degree$

Activity 1.4.1. Measuring Angles.

Use your straightedge to draw an acute angle. Use your protractor to find the degree measure of your angle.
Use your straightedge to draw an obtuse angle. Use your protractor to find the degree measure of your angle.
Use your protractor to find the degree measure of each angle.

$Acute and obtuse angles$
1. On another sheet of paper, use your straightedge to draw a triangle of any shape you like. Label the angles of your triangle A, B, and C.
2. Now cut or tear off the three angles and place the vertex of angle A on the dot below.
3. Next place the vertex of angle B at the dot so that one of its sides touches one of the sides of angle A. Repeat with angle C.
  
  \begin{equation*} \LARGE{\cdot} \end{equation*}
4. What do you notice about the sum of the three angles? Write it below:

Subsection 1.4.4 Triangles

Triangles come in various shapes, depending on the size of their angles or the lengths of their sides. Here are the names of some triangles classified by the size of their angles.

Triangles.

A right triangle has one right angle.
In an acute triangle, all three angles are acute.
In an obtuse triangle, one of the angles is obtuse.

Example 1.4.9.

Some right triangles

$Three right triangles$
Some acute triangles

$Three acute triangles$
Some obtuse triangles

$Three obtuse triangles$

Checkpoint 1.4.10.

Identify each triangle as right, acute, or obtuse. The little square indicates that the angle is $90 \degree\text{.}$

$30-60-90 triangle$
$Obtuse triangle$
$Acute triangle$
$Right triangle$
$Acute triangle$
$Acute triangle$

Answer.

right
obtuse
acute
right
obtuse
acute

Here are the names of some triangles classified by the lengths of their sides.

Triangles.

All three sides of an equilateral triangle are equal in length.
In an isosceles triangle, two sides are equal in length.
In a scalene triangle, all three sides have different lengths.

Example 1.4.11.

An equilateral triangle.

$Two acute triangles$
Some isosceles triangles.

$Three isosceles triangles$
Some scalene triangles.

$Three scalene triangles$

Checkpoint 1.4.12.

Identify each triangle as equilateral, isosceles, or scalene. Sides that have the same number of little hatch marks are equal in length.

$Triangle with two equal sides$
$Triangle with three different length sides$
$Triangle with three different length sides$
$Triangle with three equal sides and equal angles$
$Right triangle with two equal sides and two equal angles$
$Triangle with two equal sides and two equal angles$

Answer.

isosceles
scalene
scalene
equilateral
isosceles
isosceles

Subsection 1.4.5 Sides and Angles

You may have noticed that the longest side in a triangle is always opposite the largest angle.

Example 1.4.13.

In the triangle below, side $c$ is opposite the 120$\degree$ angle, and it is the longest side. Side $b\text{,}$ which is the shortest side, is opposite the 20$\degree$ angle.

$Triangle with angles 20 deg, 40 deg, 120 deg and opposite sides b, a, c$

In fact, because $120\degree \gt 40\degree \gt 20\degree\text{,}$ we know that $c \gt a \gt b\text{.}$

Note 1.4.14.

The symbol $\blert{\gt}$ means "is greater than." So, $c \gt a \gt b$ means "$c$ is greater than $a\text{,}$ which is greater than $b\text{.}$"
Similarly, the symbol $\blert{\lt}$ means "is less than."
These symbols are called inequality symbols (as opposed to equals signs), and sentences such as $c \gt a \gt b$ are called inequalities (as opposed to equations).

Inequality Symbols.

$~~~~~~~~\blert{\gt}~~~~$ means "is greater than"

$~~~~~~~~\blert{\lt}~~~~$means "is less than"

Checkpoint 1.4.15.

Write an inequality about the angles in the triangle below.

$Triangle with sides 10, 14, and 16$

Answer.

$\angle A \gt \angle C \gt \angle B$

Sides and Angles.

In any triangle, the angle opposite the longest side is the largest angle, and the angle opposite the shortest side is the smallest angle.

What about triangles that have equal sides, namely isosceles and equilateral triangles? The angles opposite equal sides must also be equal.

Definition.

In an isosceles triangle, the two equal angles (the ones opposite the equal sides) are called the base angles, and the third angle (the one included between the equal sides) is called the vertex angle.

$Triangle with two equal sides and two equal angles$

Example 1.4.16.

The triangle below is isosceles. What is the measure of the other base angle, and what is the measure of the vertex angle?

$Triangle with two sides of length 12 and base angle 38 deg$

Solution.

The base angles of an isosceles triangle are equal, so the other base angle is 38$\degree\text{.}$ To find the vertex angle, we recall that the sum of all the angles in a triangle is 180$\degree\text{,}$ so the vertex angle must be

\begin{align*} \text{vertex angle} \amp = 180 \degree - (38 \degree + 38 \degree)\\ ~~~~~~~~~~\amp = 180 \degree - 76 \degree = 104 \degree \end{align*}

Checkpoint 1.4.17.

The vertex angle of the isosceles triangle below is 120$\degree\text{.}$ What is the measure of each base angle?

$Triangle with two sides of length 12 and base angle 38 deg$
What is the measure of each angle of an equilateral triangle?

Answer.

$\displaystyle 30\degree$
$\displaystyle 60\degree$

Angles in Triangles.

The sum of the angles in any triangle is 180$\degree\text{.}$
The base angles of an isosceles triangle are equal.
All the angles of an equilateral triangle are equal.

Subsection 1.4.6 Vocabulary

angle
vertex
right angle
perpendicular
acute angle
obtuse angle
straight angle
protractor
degrees
equilateral
isosceles
scalene
inequality symbols
base angles
vertex angle

Exercises 1.4.7 Practice 1.4

1.

At what time do the hands of a clock form a 90$\degree$ angle? At what time do they form a straight angle?

2.

A rectangle has four right angles. What is the sum of the angles in a rectangle?

3.

Is it possible for a triangle to have more than one obtuse angle? Why or why not?

4.

What can you say about the sum of the two acute angles in a right triangle?

5.

The word "vertex" has several meanings in mathematics. Describe two ways it is used in this lesson.

6.

Can two acute angles add up to a straight angle? Why or why not?

Exercise Group.

For Problems 7–12, sketch and label a triangle with the given properties.

7.

An isosceles triangle with vertex angle 30$\degree$

8.

A right triangle with one angle 20$\degree$

9.

A scalene triangle with one obtuse angle

10.

An isosceles right triangle

11.

An isosceles triangle with one obtuse angle

12.

An equilateral triangle

Exercise Group.

For Problems 13–14, use a protractor to measure the angles in the triangle. Find the sum of the angles to check your work.

13.

14.

Exercise Group.

For Problems 15–16, find the angle made by the hands of the clock, without using a protractor.

15.

$clock$

16.

$clock$

Exercise Group.

For Problems 17–22, find the measure of the unknown angles. Show your work.

17.

$clock$

$~~~~~~~~~\angle A = 38\degree$

18.

$clock$

$~~~~~~~~~\angle A = 74\degree$

19.

$angle in first quadrant$

$~~~~~~~~~\angle B = 57\degree$

20.

$angle in first quadrant$

$~~~~~~~~~\angle A = 158\degree$

21.

$intersecting lines$

$~~~~~~~~~\angle A = 39\degree$

22.

$intersecting lines$

$\angle B = 86\degree$

Exercise Group.

For Problems 23–24, the angles between consecutive radii are equal. Give the measure of each angle.

23.

$semicircle divided into six sectors$

$\displaystyle \angle AOB =$
$\displaystyle \angle AOC =$
$\displaystyle \angle AOD =$
$\displaystyle \angle AOE =$
$\displaystyle \angle AOF =$

24.

$semicircle divided into four sectors$

$\displaystyle \angle AOB =$
$\displaystyle \angle AOC =$
$\displaystyle \angle AOD =$

Exercise Group.

For Problems 25–36, find the measure of the unknown angle. Show your work.

25.

$Triangle with 23 and 48 deg angles$

26.

$Triangle with 131 and 17 deg angles$

27.

$Right triangle with 26 deg angle$

28.

$Right triangle with 26 deg angle$

29.

$Isosceles triangle with 26 deg vertex angle$

30.

$Isosceles triangle with 32 deg base angle$

31.

$External angle for triangle with 80 and 40 deg opposite angles$

32.

$External angle for triangle with 50 and 110 deg opposite angles$

33.

$External angle for triangle with 30 deg angle and other external angle 122 deg$

34.

$External angle for triangle with adjacent external angle of 140 deg angle and other external angle 110 deg$

35.

$Equilateral triangle with altitude$

36.

$Square with diagonal$

Exercise Group.

For Problems 37–40, explain why the measurements shown in the figure cannot be accurate.

37.

$27-58-103 degree angles$

38.

$27-58-103 degree angles$

39.

$Two 23 degree angles, scalene$

40.

$20 and 85 degree angles, scalene$

41.

What is the measure of each base angle in a right isosceles triangle?

42.

In a particular right triangle, the largest angle is three times the smallest angle. What are the measures of the three angles?

43.

Why is the longest side in a right triangle always opposite the right angle?

44.

Is it possible for the vertex angle in an isosceles triangle to be obtuse? Explain your reasoning.

45.

Find the measure of each angle in the octagon. (There are eight identical isosceles triangles around the center of the octagon.)

$octagon$

46.

Find the measure of each unknown angle.

$20 and 85 degree angles, scalene$

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