PSAT Math : Lines

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #591 : Psat Mathematics

Pythagoras

Refer to the above figure. You are given that \(\displaystyle AB = 5\) and \(\displaystyle BC = 12\). Which of the following statements would be sufficient to prove that \(\displaystyle \angle 2\) is a right angle, given what is already known?

I) \(\displaystyle AC = 13\)

II) \(\displaystyle \angle BAC\) and \(\displaystyle \angle BCA\) are both acute

III) \(\displaystyle \angle 1\) is a right angle

Possible Answers:

I, II, and III

I and II only

None of these

I and III only

II and III only

Correct answer:

I and III only

Explanation:

If \(\displaystyle AC = 13\), then \(\displaystyle \Delta ABC\) has short sides \(\displaystyle AB = 5 , BC = 12\) and long side \(\displaystyle AC = 13\). Since

\(\displaystyle 5^{2} + 12^{2} = 25 + 144 = 169 = 13^{2}\),

then, by the converse of the Pythagorean Theorem, \(\displaystyle \Delta ABC\) is a right triangle with right angle \(\displaystyle \angle 2\). Statement I is sufficient.

 

If \(\displaystyle \angle BAC\) and \(\displaystyle \angle BCA\) are both acute,we know nothing about \(\displaystyle \angle 2\); every triangle has at least two acute angles regardless of type. Statement II tells us nothing.

 

\(\displaystyle \angle 1\) and \(\displaystyle \angle 2\) form a linear pair and are therefore supplementary. If one is a right angle, so is the other. Therefore, if  \(\displaystyle \angle 1\) is a right angle, so is \(\displaystyle \angle 2\). Statement III is sufficient.

 

The correct response is Statement I and III only.

Example Question #2 : Lines

Thingy_5

Note: \(\displaystyle m\parallel n\)

Refer to the above diagram. \(\displaystyle \angle DCG\) and which other angle form a pair of corresponding angles?

Possible Answers:

\(\displaystyle \angle ACB\)

\(\displaystyle \angle HGJ\)

\(\displaystyle \angle BCG\)

\(\displaystyle \angle CGH\)

\(\displaystyle \angle CGF\) 

Correct answer:

\(\displaystyle \angle HGJ\)

Explanation:

Two angles formed by a transversal line crossing two other lines are corresponding angles if, relative to the points of intersection, they are in the same position. \(\displaystyle \angle DCG\) is formed by the intersection of transversal \(\displaystyle t\) and \(\displaystyle m\); the angle in the same relative position where \(\displaystyle t\) intersects \(\displaystyle n\) is \(\displaystyle \angle HGJ\).

Example Question #591 : Psat Mathematics

Thingy_5

Refer to the above diagram. \(\displaystyle \angle DCG\) and which other angle form a pair of alternate interior angles?

Possible Answers:

\(\displaystyle \angle CGF\) 

\(\displaystyle \angle BCG\)

\(\displaystyle \angle CGH\)

\(\displaystyle \angle HGJ\)

\(\displaystyle \angle ACB\)

Correct answer:

\(\displaystyle \angle CGF\) 

Explanation:

Two angles formed by a transversal line crossing two other lines are alternate interior angles if:

I) Both angles have their interiors between the lines crossed

II) The angles have their interiors on the opposite sides of the transversal.

Of the given choices, only \(\displaystyle \angle CGF\) fits the description; the interior of each is between \(\displaystyle m\) and \(\displaystyle n\), and the interiors are on the opposite sides of \(\displaystyle t\).

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

Thingy_5

Refer to the above diagram. 

\(\displaystyle \angle ABC\) and which other angle form a pair of vertical angles?

Possible Answers:

\(\displaystyle \angle BFG\)

\(\displaystyle \angle ZBF\)

\(\displaystyle \angle ACB\)

\(\displaystyle \angle ZBA\)

\(\displaystyle \angle AFY\)

Correct answer:

\(\displaystyle \angle ZBF\)

Explanation:

Two angles are vertical angles if they share a vertex, anf if their union is a pair of intersecting lines. Of the five choices, only \(\displaystyle \angle ZBF\) fits both descriptions with \(\displaystyle \angle ABC\).

Example Question #1 : How To Find A Ray

Thingy

Refer to the above figure.

Which of the following segments is a diagonal of Pentagon \(\displaystyle MNOQK\) ?

Possible Answers:

\(\displaystyle \overline{QK}\)

\(\displaystyle \overline{RN}\)

\(\displaystyle \overline{OK}\)

\(\displaystyle \overline{ML}\)

\(\displaystyle \overline{OR}\)

Correct answer:

\(\displaystyle \overline{OK}\)

Explanation:

A diagonal of a polygon is a segment whose endpoints are nonconsecutive vertices of the polygon. Of the five choices, only \(\displaystyle \overline{OK}\) fits this description.

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