Basic Geometry : How to find an angle of a line

Study concepts, example questions & explanations for Basic Geometry

varsity tutors app store varsity tutors android store

Example Questions

Example Question #1 : Plane Geometry

One-half of the measure of the supplement of angle ABC is equal to the twice the measure of angle ABC. What is the measure, in degrees, of the complement of angle ABC?

Possible Answers:

36

54

18

72

90

Correct answer:

54

Explanation:

Let x equal the measure of angle ABC, let y equal the measure of the supplement of angle ABC, and let z equal the measure of the complement of angle ABC.

Because x and y are supplements, the sum of their measures must equal 180. In other words, x + y = 180. 

We are told that one-half of the measure of the supplement is equal to twice the measure of ABC. We could write this equation as follows:

(1/2)y = 2x.

Because x + y = 180, we can solve for y in terms of x by subtracting x from both sides. In other words, y = 180 – x. Next, we can substitute this value into the equation (1/2)y = 2x and then solve for x.

(1/2)(180-x) = 2x.

Multiply both sides by 2 to get rid of the fraction.

(180 – x) = 4x.

Add x to both sides.

180 = 5x.

Divide both sides by 5.

x = 36.

The measure of angle ABC is 36 degrees. However, the original question asks us to find the measure of the complement of ABC, which we denoted previously as z. Because the sum of the measure of an angle and the measure of its complement equals 90, we can write the following equation:

x + z = 90.

Now, we can substitute 36 as the value of x and then solve for z.

36 + z = 90.

Subtract 36 from both sides.

z = 54.

The answer is 54. 

Example Question #1 : Geometry

Parallellines

 

 

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

Possible Answers:

60°

None of the other answers.

160°

140°

80°

Correct answer:

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°

Example Question #1 : How To Find The Angle Of Two Lines

In rectangle ABCD, both diagonals are drawn and intersect at point E.  

Let the measure of angle AEB equal x degrees.

Let the measure of angle BEC equal y degrees.

Let the measure of angle CED equal z degrees.

Find the measure of angle AED in terms of x, y, and/or z.

Possible Answers:

180 – 1/2(x + z)

180 – 2(x + z)

180 – y

360 – x + y + z

180 – (x + y + z)

Correct answer:

180 – 1/2(x + z)

Explanation:

Intersecting lines create two pairs of vertical angles which are congruent. Therefore, we can deduce that y = measure of angle AED.

Furthermore, intersecting lines create adjacent angles that are supplementary (sum to 180 degrees). Therefore, we can deduce that x + y + z + (measure of angle AED) = 360.

Substituting the first equation into the second equation, we get

x + (measure of angle AED) + z + (measure of angle AED) = 360

2(measure of angle AED) + x + z = 360

2(measure of angle AED) = 360 – (x + z)

Divide by two and get:

measure of angle AED = 180 – 1/2(x + z)

Example Question #1 : Plane Geometry

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?

Possible Answers:

Correct answer:

Explanation:

Example Question #1 : Geometry

Two pairs of parallel lines intersect:

 Screen_shot_2013-03-18_at_10.29.11_pm

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

Possible Answers:

160°

180°

170°

140°

150°

Correct answer:

180°

Explanation:

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

Example Question #1 : How To Find An Angle Of A Line

Slide1

Lines  and  are parallel.  , , is a right triangle, and  has a length of 10.  What is the length of 

 

 

Possible Answers:

Not enough information.

Correct answer:

Explanation:

Since we know opposite angles are equal, it follows that angle  and .  

Imagine a parallel line passing through point .  The imaginary line would make opposite angles with  & , the sum of which would equal .  Therefore, .

\cos (60)=.5=\frac{EG}{EF}\rightarrow EF=\frac{EG}{.5}=20

Example Question #1 : Geometry

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

Possible Answers:

(10x+140)^{\circ}

(100x)^{\circ}

(10x+50)^{\circ}

(140-10x)^{\circ}

(50-10x)^{\circ}

Correct answer:

(10x+140)^{\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}.

Example Question #2 : Geometry

 

 

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

Sat_math_166_03

Possible Answers:

It cannot be determined

Correct answer:

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°.

Example Question #11 : Lines

The measure of the supplement of angle A is 40 degrees larger than twice the measure of the complement of angle A. What is the sum, in degrees, of the measures of the supplement and complement of angle A?

Possible Answers:

90

140

50

190

40

Correct answer:

190

Explanation:

Let A represent the measure, in degrees, of angle A. By definition, the sum of the measures of A and its complement is 90 degrees. We can write the following equation to determine an expression for the measure of the complement of angle A.

A + measure of complement of A = 90

Subtract A from both sides.

measure of complement of A = 90 – A

Similarly, because the sum of the measures of angle A and its supplement is 180 degrees, we can represent the measure of the supplement of A as 180 – A.

The problem states that the measure of the supplement of A is 40 degrees larger than twice the measure of the complement of A. We can write this as 2(90-A) + 40.

Next, we must set the two expressions 180 – A and 2(90 – A) + 40 equal to one another and solve for A:

180 – A = 2(90 – A) + 40

Distribute the 2:

180 - A = 180 – 2A + 40

Add 2A to both sides:

180 + A = 180 + 40

Subtract 180 from both sides:

A = 40

Therefore the measure of angle A is 40 degrees. 

The question asks us to find the sum of the measures of the supplement and complement of A. The measure of the supplement of A is 180 – A = 180 – 40 = 140 degrees. Similarly, the measure of the complement of A is 90 – 40 = 50 degrees.

The sum of these two is 140 + 50 = 190 degrees.

 

Example Question #11 : Plane Geometry

 \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?

Possible Answers:

\dpi{100} \small 60 degrees

\dpi{100} \small 65 degrees

None of the other answers

\dpi{100} \small 70 degrees

\dpi{100} \small 75 degrees

Correct answer:

\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.

Learning Tools by Varsity Tutors