Lines l_1 and l_2 intersect at point P to form an acute angle measuring 44^\circ. Line b_1… — GRE General Test

Lines $l_1$ and $l_2$ intersect at point $P$ to form an acute angle measuring $44^\circ$ . Line $b_1$ is the angle bisector of this acute angle. A third line, $l_3$ , is drawn perpendicular to $b_1$ and intersects line $l_1$ at point $Q$ (where $Q \neq P$ ). What is the measure, in degrees, of the acute angle formed by the intersection of line $l_2$ and line $l_3$ ?

A
$22^\circ$
B
$46^\circ$
$68^\circ$ Answer
D
$78^\circ$
E
$112^\circ$

Answer

The measure of the acute angle formed by the intersection of line

l_2

and line

l_3

68^\circ

The correct answer is

68^\circ

. Line

b_1

divides the

44^\circ

angle between

l_1

and

l_2

into two

22^\circ

angles. Because line

l_3

is perpendicular to

b_1

, it forms a right triangle with

l_1

and

b_1

, making the angle between

l_1

and

l_3

equal to

90^\circ - 22^\circ = 68^\circ

. Considering the large triangle formed by lines

l_1

l_2

, and

l_3

, two of its angles are

44^\circ

and

68^\circ

. Thus, the third interior angle at the intersection of

l_2

and

l_3

180^\circ - (44^\circ + 68^\circ) = 68^\circ

, which is acute.

Step-by-Step Solution

Determine the angle formed by the angle bisector

b_1

with line

l_1

and line

l_2

Since line

b_1

bisects the

44^\circ

acute angle between

l_1

and

l_2

, the angle between

l_1

and

b_1

\frac{44^\circ}{2} = 22^\circ

, and the angle between

l_2

and

b_1

is also

22^\circ

An angle bisector divides an angle into two equal congruent parts.

Find the measure of the interior angle between line

l_1

and line

l_3

Line

l_3

is perpendicular to

b_1

, forming a right triangle with

l_1

and

b_1

. The interior angle between

l_1

and

l_3

180^\circ - 90^\circ - 22^\circ = 68^\circ

The sum of interior angles in any triangle is

180^\circ

Calculate the interior angle at the intersection of line

l_2

and line

l_3

in the main triangle formed by

l_1

l_2

, and

l_3

The interior angle at the intersection of

l_2

and

l_3

180^\circ - 44^\circ - 68^\circ = 68^\circ

The interior angles of the triangle formed by lines

l_1

l_2

, and

l_3

must sum to

180^\circ

Verify that the calculated angle is acute.

Since

68^\circ < 90^\circ

, the acute angle formed by lines

l_2

and

l_3

68^\circ

An angle measuring strictly less than

90^\circ

is defined as an acute angle.

Key Concept

Angle bisector properties, perpendicular lines, and interior angle sum theorem for triangles.

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