Question

In the diagram, rhombus ABCD has a diagonal line DB¯¯¯¯¯¯¯¯ , and m∠A=114° . What is m∠B of rhombus ABCD ? (1 point) Responses 57° 57° 33° 33° 114° 114° 66°

Answer 1

In a rhombus, opposite angles are equal and the sum of the angles in any quadrilateral is 360°. Since rhombus ABCD has \( m\angle A = 114^\circ \), it follows that \( m\angle C = 114^\circ \) as well (since \( \angle A \) and \( \angle C \) are opposite angles).

Now, to find the measures of angles B and D, we can use the fact that the sum of all angles in a quadrilateral is \( 360^\circ \):

\[
m\angle A + m\angle B + m\angle C + m\angle D = 360^\circ
\]

Substituting the known values:

\[
114^\circ + m\angle B + 114^\circ + m\angle D = 360^\circ
\]

This simplifies to:

\[
228^\circ + m\angle B + m\angle D = 360^\circ
\]

Now, simplify further to find:

\[
m\angle B + m\angle D = 360^\circ - 228^\circ
\]
\[
m\angle B + m\angle D = 132^\circ
\]

Since \( m\angle B \) and \( m\angle D \) are also equal (opposite angles in a rhombus):

\[
m\angle B = m\angle D
\]

We can substitute \( m\angle D \) with \( m\angle B \):

\[
m\angle B + m\angle B = 132^\circ
\]
\[
2m\angle B = 132^\circ
\]

Dividing both sides by 2:

\[
m\angle B = 66^\circ
\]

Thus, the measure of angle B in rhombus ABCD is \( \boxed{66^\circ} \).