# Lines and Angles

There are only a couple of Lines and Angles questions tested on the ACT Math section. The questions are usually easy and should not pose a challenge to you.

August 29, 2020

Here's a quick breakdown of the different types of angles:

## TRANSVERSAL LINES

A transversal line is a line that cuts through two parallel lines.

A transversal line creates eight angles. But by studying the angles, we can see that many of the angles are congruent!

In the diagram below, line R is a transversal line that cuts through lines P and Q,which are parallel to each other.

A transversal line creates eight angles. But by studying the angles, we can see that many of the angles are congruent!

We know that ∠1 is congruent to ∠3 because they are vertical angles. For the same reason, we know that the following angles are also congruent:

∠2 = ∠4

∠5 = ∠7

∠6 = ∠8

But what else do we know? Because P and Q are parallel, the transversal forms **CORRESPONDING ANGLES **- angles that are in the same position in relation to the transversal and therefore congruent. Thus, the following corresponding angles are congruent:

∠1 = ∠5

∠2 = ∠6

∠3 = ∠7

∠4 = ∠8

Furthermore, because P and Q are parallel and cut by the transversal, it means that ∠1 is congruent to ∠7 because they are **ALTERNATE EXTERIOR ANGLES**. Alternate exterior angles are on opposite sides of a transversal and outside the parallel line. Thus, the following alternate exterior angles are congruent:

∠1 = ∠7

∠2 = ∠8

Similarly, because P and Q are parallel, it means that ∠3 is congruent to ∠5 because they are **ALTERNATE INTERIOR ANGLES** - angles that are on opposite sides of a transversal and inside the parallel line. For the same reason, ∠4 is congruent to ∠6.

∠4 = ∠6

∠3 = ∠5

So, putting all information together:

∠1, 3, 5 and 7 are congruent;

∠2, 4, 6 and 8 are congruent.

**The opposite of all of this is also true: if you don't know if two lines are parallel - look at the alternate interior or exterior angles. If they are congruent, the lines are parallel.**

##### Example 1:

Statement A: ∠3 + ∠4 + ∠5 = 131°

Statement B: ∠6 + ∠7 + ∠8 = 131°

Refer to the figure and two statements. Evaluate ∠1 + ∠2.

** A.** 82°

**98°**

*B.***180°**

*C.***262°**

*D.*Let’s consider the first statement by itself. ∠2, ∠3, ∠4, and ∠5 together form a straight angle, so their degree measures a total 180°.

∠2 + ∠3 + ∠4 + ∠5 = 180°

∠2 + 131° = 180°

∠2 = 49°

Without further information, no other angle measures, including that of ∠1, can be found. So, let’s consider the second statement. ∠1, ∠6, ∠7, and ∠8 together form a straight angle, so their degree measures a total 180°.

∠1 + ∠6 + ∠7 + ∠8 = 180°

∠1 + 131° = 180°

∠1 = 49°

We’ve found both ∠1 and ∠2, so all we have to do is add them together to solve the problem:

∠1+ ∠2 = 49° + 49° = 98°

The correct answer is B.

##### Example 2

Statement A: ∠2 = 89°

Statement B: ∠3 = ∠6

Given the above figure and two statements, which of the following is true?

** A. **Line m is perpendicular to line t

**Line m is not perpendicular to line t**

*B.***∠4 = ∠7**

*C.***∠1 = ∠6**

*D.*Statement A alone establishes by definition that line l is not perpendicular to line t, but it does not establish any relationship between line m and line t.

By Statement B alone, we know line l is parallel to line m because alternating interior angles are congruent., line l is parallel to line m. ∠2 and ∠6 are corresponding angles formed by a transversal across parallel lines, so ∠6 = ∠2 = 89°. ∠6 is not a right angle, so line m is not perpendicular to line t and B is the correct answer.

##### Example 3

Consider the line diagram shown at right. Which of the following pieces of information is enough to prove that AB is parallel to CD?

** A. **∠1 = ∠3

**∠4 = 120° and ∠8 = 60°**

*B.***∠5 = 120° and ∠6 = 60v**

*C.***∠2 = ∠6 = 100°**

*D.*By learning that ∠4 = 120° and ∠8 = 60°, we have enough information to determine that ∠1 = 60° (because it forms a straight angle with ∠4 ) and that ∠7 = 120° (since it forms a straight angle with ∠8 ). At this point, we can see that the pairs of corresponding angles, angles 1 and 8 and angles 4 and 7, match in value. This means that AB is parallel to CD. The correct answer choice is B.

##### Example 4

∠ABD = 60°. What is ∠DBE?

** A.** 30°

**45°**

*B.***60°**

*C.***90°**

*D.*The sum of ∠ABD, ∠DBE, and ∠EBC equally 180° because they are all on the straight-line AC.

∠ABD + ∠DBE + ∠EBC = 180°

We know that ∠ABD = 60° and that ∠EBC = 90°, so we can substitute in those values.

60° + ∠DBE + 90° = 180°

∠DBE = 30°

The correct answer is A.

# Standard Position of Angles

An angle is said to be in **STANDARD POSITION **if its vertex is positioned at origin and one line is at the positive -axis. The line on the -axis is called the **INITIAL SIDE**, and the other line is called the **TERMINAL SIDE.**

The rotation from the initial side to the terminal side is a measure of an angle. The counterclockwise rotation is considered positive. An angle in the clockwise direction is taken as negative. **CO-TERMINAL** angles are the two angles that have the same terminal side.

# Radians and Degrees

**RADIANS** and **DEGREES** are units to measure angles. The conversion between these two units can be done using the following equation.

#### \(\frac{1}{360} degree=\frac{1}{2\pi}radian\) *OR \(\frac{1}{180} degree=\frac{1}{\pi}radian\)*

*OR \(\frac{1}{180} degree=\frac{1}{\pi}radian\)*

To convert from degrees to radians and vice versa, we need the following:

Deg. = Rad. x \(\frac{180}{\pi}\)

Rad. = Deg. x \(\frac{\pi}{180}\)

For example, if we need to convert 50 degrees to radians, we need to multiply the degree measure by \(\frac{\pi}{180}\) , i.e.

50°= 50 x \(\frac{\pi}{180}rad = \frac{5\pi}{18}rad\)

Similarly, if we need to convert radians into degrees, we need to multiply the radian measure by \(\frac{\pi}{180}\)

Some important conversions are:

\(30^{\circ}=\frac{\pi}{6}rad.\)

\(45^{\circ}=\frac{\pi}{4}rad.\)

\(60^{\circ}=\frac{\pi}{3}rad.\)

\(90^{\circ}=\frac{\pi}{2}rad.\)

\(180^{\circ}=\pi rad.\)

EXAMPLE: In the xy-plane, O is the center of the circle, and the measure of radians. What is the value of a?

By the distance formula, the length of radius OA is \(\sqrt{\sqrt{3^{2}}+1^{2}}=2\)

Thus, SIN (∠AOB) = \(\frac{1}{2}\)

Therefore ∠AOB is 30°, which is equal to \(30\frac{(\pi)}{180}=\frac{\pi}{6}radians\)

Hence, the value of a is 6.

EXAMPLE: In a circle with center O, central angle AOB has a measure of \(\frac{5\pi}{4}radians\). The area of the sector formed by central angle AOB is what fraction of the area of the circle?

A complete rotation around a point is 360° or 2π radians. Since the central angle AOB has measure \(\frac{5\pi}{4}radians\), it represents

\(\frac{5\pi}{4}\) x \(\frac{1}{2\pi}\)

\(\frac{5}{8}of\: a\: complete\: rotation\: around\: point\: 0\)

Therefore, the sector formed by central angle AOB has area equal to 5/8 the area of the entire circle. The answer is 5/8 or in decimal form .625.