Question

find ( mangle b ) and ( mangle c ) in rhombus ( abcd ).
( mangle b = square^circ )
( mangle c = square^circ )

Explanation:

Step1: Recall rhombus angle property

In a rhombus, adjacent angles are supplementary, so \( m\angle B + m\angle C = 180^\circ \).

Step2: Substitute angle expressions

Given \( m\angle B = 2x - 6^\circ \) and \( m\angle C = 2x - 82^\circ \), substitute into the supplementary equation:
\( (2x - 6) + (2x - 82) = 180 \)

Step3: Solve for x

Simplify the left side: \( 4x - 88 = 180 \)
Add 88 to both sides: \( 4x = 180 + 88 = 268 \)
Divide by 4: \( x = \frac{268}{4} = 67 \)

Step4: Find \( m\angle B \)

Substitute \( x = 67 \) into \( 2x - 6 \):
\( 2(67) - 6 = 134 - 6 = 128^\circ \)

Step5: Find \( m\angle C \)

Substitute \( x = 67 \) into \( 2x - 82 \):
\( 2(67) - 82 = 134 - 82 = 52^\circ \)

Answer:

\( m\angle B = \boxed{128}^\circ \)
\( m\angle C = \boxed{52}^\circ \)