Question

find the measures of the numbered angles in rhombus abcd. m∠1 =

m∠2 =

m∠3 =

m∠4 =

Explanation:

Step1: Diagonals of rhombus are perpendicular

In a rhombus, the diagonals are perpendicular to each other. So, $m\angle1 = 90^{\circ}$.

Step2: Diagonal bisects an angle

The diagonal of a rhombus bisects the angles of the rhombus. $\angle BCD=66^{\circ}$, and diagonal $BD$ bisects it. So, $m\angle4=\frac{66^{\circ}}{2}=33^{\circ}$.

Step3: Use angle - sum property of triangle

In right - triangle formed by the diagonals, in the triangle with $\angle1$, $\angle2$ and $\angle4$. Since the sum of angles in a triangle is $180^{\circ}$, and $\angle1 = 90^{\circ}$, $\angle4 = 33^{\circ}$, then $m\angle2=180^{\circ}-90^{\circ}-33^{\circ}=57^{\circ}$.

Step4: Opposite angles are equal in rhombus

$\angle BAD$ and $\angle BCD$ are opposite angles in the rhombus. Diagonal $AC$ bisects $\angle BAD$. Since $\angle BCD = 66^{\circ}$, $\angle BAD=66^{\circ}$ and $m\angle3 = 33^{\circ}$.

Answer:

$m\angle1 = 90^{\circ}$
$m\angle2 = 57^{\circ}$
$m\angle3 = 33^{\circ}$
$m\angle4 = 33^{\circ}$