Lines and Angles

# 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.

## Transversal Lines

∠2 = ∠4
∠5 = ∠7
∠6 = ∠8

∠1 = ∠5
∠2 = ∠6
∠3 = ∠7
∠4 = ∠8

∠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.

#### Solved Example 1Statement A: ∠3 + ∠4 + ∠5 = 131°Statement B: ∠6 + ∠7 + ∠8 = 131°

Refer to the figure and two statements. Evaluate ∠1 + ∠2.
A.
82°
B. 98°
C. 180°
D. 262°

Solution: 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.

#### Solved Example 2Statement 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
B. Line m is not perpendicular to line t
C. ∠4 = ∠7
D. ∠1 = ∠6

Solution: 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.

#### Solved 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
B. ∠4 = 120° and ∠8 = 60°
C. ∠5 = 120° and ∠6 = 60v
D. ∠2 = ∠6 = 100°

Solution: 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.

#### Solved Example 4

∠ABD = 60°. What is ∠DBE?
A.
30°
B. 45°
C. 60°
D. 90°

Solution: 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
$$\frac{1}{180} degree=\frac{1}{\pi}radian$$

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.