for the rhombus below, find the measures of $\\angle 1, \\angle 2, \\angle 3$, and $\\angle 4$.\n$m \\angle…

Explanation:

Step1: Identify rhombus diagonal properties

Diagonals of a rhombus bisect the vertex angles.

Step2: Determine measure of angle 4

Since the diagonal bisects the angle, $\angle 4$ equals the given $32^{\circ}$. $$m\angle 4 = 32^{\circ}$$

Step3: Determine measure of angle 3

Opposite angles in a rhombus are congruent and bisected. $$m\angle 3 = 32^{\circ}$$

Step4: Determine measure of angle 2

Consecutive angles are supplementary; the whole angle is $64^{\circ}$. $$180^{\circ} - 64^{\circ} = 116^{\circ}$$

Step5: Calculate bisected angle measure

Angle 2 is half of the supplementary vertex angle. $$m\angle 2 = \frac{116^{\circ}}{2} = 58^{\circ}$$

Step6: Determine measure of angle 1

In a rhombus, the triangle formed is isosceles. $$m\angle 1 = 180^{\circ} - (58^{\circ} + 58^{\circ}) = 64^{\circ}$$

Answer:

$m\angle 1 = 64^{\circ}$ $m\angle 2 = 58^{\circ}$ $m\angle 3 = 32^{\circ}$ $m\angle 4 = 32^{\circ}$