PSAT Math : Geometry

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #2 : 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:

360 – x + y + z

180 – 1/2(x + z)

180 – y

180 – (x + y + z)

180 – 2(x + 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 : Intersecting Lines And Angles

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:

180°

170°

140°

160°

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:

(100x)^{\circ}

(10x+140)^{\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 #11 : Intersecting Lines And Angles

 

 

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

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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 #2 : Geometry

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:

140

50

40

190

90

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 : How To Find An Angle Of A Line

 \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 75 degrees

None of the other answers

\dpi{100} \small 70 degrees

\dpi{100} \small 65 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.

Example Question #11 : Plane Geometry

Two parallel lines are intersected by a transversal. If the minor angle of intersection between the first parallel line and the transversal is , what is the minor angle of intersection between the second parallel line and the transversal?

Possible Answers:

Correct answer:

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).

Example Question #251 : New Sat Math Calculator

If  , , and , what is the measure, in degrees, of 

Alternate interior angles   

 

Possible Answers:

32

122

62

148

58

Correct answer:

148

Explanation:

The question states that . The alternate interior angle theorem states that if two parallel lines are cut by a transversal, then pairs of alternate interior angles are congruent; therefore, we know the following measure:

The sum of angles of a triangle is equal to 180 degrees. The question states that ; therefore we know the following measure:

Use this information to solve for the missing angle:

The degree measure of a straight line is 180 degrees; therefore, we can write the following equation:

The measure of  is 148 degrees. 

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