Math / Geometry

QuestionFind the measure of each numbered angle. $m \angle 1=$ $\square$ , $m \angle 2=$ $\square$ : $m \angle 3=$

Studdy Solution

STEP 1

1. The sum of angles in a triangle is $180^\circ$ .
2. The angles at the intersection of the two triangles are supplementary.

STEP 2

1. Find the measure of angle 1 using triangle MPQ.
2. Find the measure of angle 2 using the supplementary angle relationship at point Q.
3. Find the measure of angle 3 using triangle NOQ.

STEP 3

To find $m \angle 1$ in triangle MPQ, use the fact that the sum of angles in a triangle is $180^\circ$ . We have:
$m \angle M + m \angle P + m \angle 1 = 180^\circ$
Substitute the known values:
$66^\circ + 58^\circ + m \angle 1 = 180^\circ$

STEP 4

Solve for $m \angle 1$ :
$124^\circ + m \angle 1 = 180^\circ$
$m \angle 1 = 180^\circ - 124^\circ$
$m \angle 1 = 56^\circ$

STEP 5

To find $m \angle 2$ , use the supplementary angle relationship at point Q. Angles 1 and 2 are supplementary because they form a straight line. Thus:
$m \angle 1 + m \angle 2 = 180^\circ$
Substitute the known value of $m \angle 1$ :
$56^\circ + m \angle 2 = 180^\circ$

STEP 6

Solve for $m \angle 2$ :
$m \angle 2 = 180^\circ - 56^\circ$
$m \angle 2 = 124^\circ$

STEP 7

To find $m \angle 3$ in triangle NOQ, use the sum of angles in a triangle:
$m \angle N + m \angle 3 + m \angle 2 = 180^\circ$
Substitute the known values:
$50^\circ + m \angle 3 + 124^\circ = 180^\circ$

STEP 8

Solve for $m \angle 3$ :
$174^\circ + m \angle 3 = 180^\circ$
$m \angle 3 = 180^\circ - 174^\circ$
$m \angle 3 = 6^\circ$
The measures of the angles are:
$m \angle 1 = 56^\circ, \quad m \angle 2 = 124^\circ, \quad m \angle 3 = 6^\circ$

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