Examkin

Question

Difficulty: HardLines and Angles

Lines L1L_1 and L2L_2 intersect at point PP to form an acute angle of 54∘54^\circ. Line MM passes through point PP and is perpendicular to line L1L_1. Ray RR originates at point PP and lies in the interior of one of the obtuse angles formed by L1L_1 and L2L_2. If Ray RR bisects the angle formed between line MM and line L2L_2, what is the measure, in degrees, of the acute angle formed by Ray RR and line L1L_1?

Answer: 72 degrees

Answer

The measure of the acute angle formed by Ray RR and line L1L_1 is 72∘72^\circ.
Lines L1L_1 and L2L_2 form an obtuse angle of 180∘−54∘=126∘180^\circ - 54^\circ = 126^\circ. Perpendicular line MM consumes 90∘90^\circ of this angle relative to L1L_1, leaving 36∘36^\circ between line MM and line L2L_2. Bisecting this remaining angle gives 18∘18^\circ. Ray RR therefore lies 18∘18^\circ away from perpendicular line MM, making an acute angle of 90∘−18∘=72∘90^\circ - 18^\circ = 72^\circ with line L1L_1.

Step-by-Step Solution

1
Determine the measure of the obtuse angle between lines L1L_1 and L2L_2.
The obtuse angle measures 180∘−54∘=126∘180^\circ - 54^\circ = 126^\circ.
Adjacent angles along intersecting straight lines are supplementary and sum to 180∘180^\circ.
2
Find the angle between perpendicular line MM and line L2L_2.
The angle between line MM and line L2L_2 within the obtuse region is 126∘−90∘=36∘126^\circ - 90^\circ = 36^\circ.
Line MM is perpendicular to line L1L_1, taking up 90∘90^\circ of the 126∘126^\circ obtuse angle.
3
Determine the angle formed by Ray RR after bisecting the 36∘36^\circ angle.
The angle between Ray RR and line MM is 36∘/2=18∘36^\circ / 2 = 18^\circ.
An angle bisector divides an angle into two equal parts.
4
Calculate the acute angle between Ray RR and line L1L_1.
The acute angle formed between Ray RR and line L1L_1 is 90∘−18∘=72∘90^\circ - 18^\circ = 72^\circ.
Line MM forms a 90∘90^\circ angle with line L1L_1. Subtracting the 18∘18^\circ offset created by Ray RR yields the acute angle of 72∘72^\circ.

Key Concept

Supplementary angles, perpendicular lines, and angle bisectors
Estimated Time:2m 0s
Rate this question

Topics

Lines and Angles

More questions from this topic