Question

find ( mangle c ) and ( mangle b ) in rhombus ( abcd ).
( mangle c = \boxed{64}^circ )
( mangle b = square^circ )

Explanation:

Step1: Use rhombus consecutive angle property

Consecutive angles in a rhombus are supplementary, so:
$$(w+56) + 8w = 180$$

Step2: Simplify and solve for $w$

Combine like terms and isolate $w$:
$$9w + 56 = 180$$
$$9w = 180 - 56$$
$$9w = 124$$
$$w = \frac{124}{9} \approx 13.78$$
(Note: Using the given $m\angle C=64^\circ$ to verify: $w+56=64 \implies w=8$, which is consistent with $8w=64$, correcting the earlier equation error: in a rhombus, opposite angles are equal, so $\angle C = \angle A$, so $w+56=8w$)

Step3: Correct equation using opposite angles

Opposite angles in a rhombus are equal:
$$w + 56 = 8w$$
$$56 = 7w$$
$$w = 8$$

Step4: Calculate $m\angle B$

Consecutive angles are supplementary to $\angle C$:
$$m\angle B = 180^\circ - m\angle C$$
$$m\angle B = 180 - 64$$

Answer:

$m\angle C = 64^\circ$
$m\angle B = 116^\circ$