Math  /  Geometry

QuestionIn the figure below, m4=112m \angle 4=112^{\circ}. Find m<1,m2m<1, m \angle 2, and m<3m<3.

Studdy Solution

STEP 1

1. The sum of angles around a point is 360360^\circ.
2. Vertical angles are equal.
3. Adjacent angles on a straight line sum up to 180180^\circ.

STEP 2

1. Identify the vertical angles.
2. Calculate the measures of the adjacent angles.
3. Verify the calculations.

STEP 3

Identify the vertical angles. Since angles 1 and 4 are vertical angles, they are equal. Therefore:
m1=m4=112 m \angle 1 = m \angle 4 = 112^\circ

STEP 4

Calculate the measures of the adjacent angles. Angles 2 and 4 are on a straight line, so they are supplementary. Therefore:
m2+m4=180 m \angle 2 + m \angle 4 = 180^\circ

STEP 5

Substitute the known value of m4 m \angle 4 into the equation:
m2+112=180 m \angle 2 + 112^\circ = 180^\circ

STEP 6

Solve for m2 m \angle 2 :
m2=180112 m \angle 2 = 180^\circ - 112^\circ m2=68 m \angle 2 = 68^\circ

STEP 7

Since angles 2 and 3 are vertical angles, they are equal. Therefore:
m3=m2=68 m \angle 3 = m \angle 2 = 68^\circ
The measures of the angles are:
m1=112 m \angle 1 = 112^\circ m2=68 m \angle 2 = 68^\circ m3=68 m \angle 3 = 68^\circ

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