ACT Math - How to find an angle of a line

Varsity_question

AB and CD are two parrellel lines intersected by line EF. If the measure of angle 1 is , what is the measure of angle 2?

$48^{\circ}$

$132^{\circ}$

$64^{\circ}$

$180^{\circ}$

$24^{\circ}$

Explanation

The angles are equal. When two parallel lines are intersected by a transversal, the corresponding angles have the same measure.

A student creates a challenge for his friend. He first draws a square, the adds the line for each of the 2 diagonals. Finally, he asks his friend to draw the circle that has the most intersections possible.

How many intersections will this circle have?

Explanation

The answer to this problem is 12. This can be drawn as shown below (intersections marked in red).

Square

We can also be sure that this is the maximal case because it is the largest answer selection. Were it not given as a multiple choice question, however, we could still be sure this was the largest. This is because no line can intersect a circle in more than 2 points. Keeping this in mind, we look at the construction of our initial shape. The square has 4 lines, and then each diagonal is an additional 2. We have thus drawn in 6 lines. The maximum number of intersections is therefore going to be twice this, or 12.

$\dpi{100} \small \overline{AB}$ is a straight line. $\dpi{100} \small \overline{CD}$ intersects $\dpi{100} \small \overline{AB}$ at point $\dpi{100} \small E$ . If $\dpi{100} \small \angle AEC$ measures 120 degrees, what must be the measure of $\dpi{100} \small \angle BEC$ ?

None of the other answers

$\dpi{100} \small 70$ degrees

$\dpi{100} \small 65$ degrees

$\dpi{100} \small 75$ degrees

$\dpi{100} \small 60$ degrees

Explanation

$\dpi{100} \small \angle AEC$ & $\dpi{100} \small \angle BEC$ must add up to 180 degrees. So, if $\dpi{100} \small \angle AEC$ is 120, $\dpi{100} \small \angle BEC$ (the supplementary angle) must equal 60, for a total of 180.

Angle $\angle ABC$ measures $20^{\circ}$

$\overrightarrow{BD}$ is the bisector of $\angle ABC$

$\overrightarrow{BE}$ is the bisector of $\angle CBD$

What is the measure of $\angle ABE$ ?

$5^{\circ}$

$10^{\circ}$

$15^{\circ}$

$30^{\circ}$

$40^{\circ}$

Explanation

Angle pic

Let's begin by observing the larger angle. $\angle ABC$ is cut into two 10-degree angles by $\overrightarrow{BD}$ . This means that angles $\angle ABD$ and $\angle CBD$ equal 10 degrees. Next, we are told that $\overrightarrow{BE}$ bisects $\angle CBD$ , which creates two 5-degree angles. $\angle ABE$ consists of $\angle ABD$ , which is 10 degrees, and $\angle DBE$ , which is 5 degrees. We need to add the two angles together to solve the problem.

$\angle ABE=\angle ABD+\angle DBE$

$\angle ABE=10^{\circ}+5^{\circ}$

$\angle ABE=15^{\circ}$

Two parallel lines are intersected by a transversal. If the minor angle of intersection between the first parallel line and the transversal is $36^{\circ}$ , what is the minor angle of intersection between the second parallel line and the transversal?

$36^{\circ}$

$66^{\circ}$

$144^{\circ}$

$54^{\circ}$

$18^{\circ}$

Explanation

When a line intersects two parallel lines as a transversal, it always passes through both at identical angles (regardless of distance or length of arc).

If $\angle A$ measures $(40-10x)^{\circ}$ , which of the following is equivalent to the measure of the supplement of $\angle A$ ?

$(10x+140)^{\circ}$

$(10x+50)^{\circ}$

$(50-10x)^{\circ}$

$(140-10x)^{\circ}$

$(100x)^{\circ}$

Explanation

When the measure of an angle is added to the measure of its supplement, the result is always 180 degrees. Put differently, two angles are said to be supplementary if the sum of their measures is 180 degrees. For example, two angles whose measures are 50 degrees and 130 degrees are supplementary, because the sum of 50 and 130 degrees is 180 degrees. We can thus write the following equation:

$\dpi{100} measure\ of\ \angle A+ measure\ of\ supplement\ of\ \angle A=180$

$\dpi{100} 40-10x+ measure\ of\ supplement\ of\ \angle A=180$

Subtract 40 from both sides.

$\dpi{100} -10x+ measure\ of\ supplement\ of\ \angle A=140$

Add $\dpi{100} 10x$ to both sides.

$\dpi{100} measure\ of\ supplement\ of\ \angle A=140+10x=10x+140$

The answer is $(10x+140)^{\circ}$ .

Parallellines

In the diagram, AB || CD. What is the value of a+b?

None of the other answers.

140°

80°

60°

160°

Explanation

Refer to the following diagram while reading the explanation:

Parallellines-answer

We know that angle b has to be equal to its vertical angle (the angle directly "across" the intersection). Therefore, it is 20°.

Furthermore, given the properties of parallel lines, we know that the supplementary angle to a must be 40°. Based on the rule for supplements, we know that a + 40° = 180°. Solving for a, we get a = 140°.

Therefore, a + b = 140° + 20° = 160°

In the following diagram, lines and are parallel to each other. What is the value for ?

Sat_math_166_03

It cannot be determined

Explanation

When two parallel lines are intersected by another line, the sum of the measures of the interior angles on the same side of the line is 180°. Therefore, the sum of the angle that is labeled as 100° and angle y is 180°. As a result, angle y is 80°.

Another property of two parallel lines that are intersected by a third line is that the corresponding angles are congruent. So, the measurement of angle x is equal to the measurement of angle y, which is 80°.

Angles

Figure not drawn to scale.

In the figure above, APB forms a straight line. If the measure of angle APC is eighty-one degrees larger than the measure of angle DPB, and the measures of angles CPD and DPB are equal, then what is the measure, in degrees, of angle CPB?

114

Explanation

Let x equal the measure of angle DPB. Because the measure of angle APC is eighty-one degrees larger than the measure of DPB, we can represent this angle's measure as x + 81. Also, because the measure of angle CPD is equal to the measure of angle DPB, we can represent the measure of CPD as x.

Since APB is a straight line, the sum of the measures of angles DPB, APC, and CPD must all equal 180; therefore, we can write the following equation to find x:

x + (x + 81) + x = 180

Simplify by collecting the x terms.

3x + 81 = 180

Subtract 81 from both sides.

3x = 99

Divide by 3.

x = 33.

This means that the measures of angles DPB and CPD are both equal to 33 degrees. The original question asks us to find the measure of angle CPB, which is equal to the sum of the measures of angles DPB and CPD.

measure of CPB = 33 + 33 = 66.

The answer is 66.

Two pairs of parallel lines intersect:

Screen_shot_2013-03-18_at_10.29.11_pm

If A = 135o, what is 2*|B-C| = ?

140°

150°

160°

170°

180°

Explanation

By properties of parallel lines A+B = 180o, B = 45o, C = A = 135o, so 2*|B-C| = 2* |45-135| = 180o

How to find an angle of a line

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