Question

segment ad is parallel to segment bc, and segment ab is parallel to segment cd. which statement about the measure of \\( \angle d \\) is not true? \\( m \angle d = m \angle a \\) \\( m \angle b = m \angle d \\) \\( m \angle d = 180 ^ { \circ } - m \angle c \\) \\( m \angle d = 180 ^ { \circ } - m \angle a \\)

Explanation:

Step1: Identify the figure type

Since \(AD\parallel BC\) and \(AB\parallel CD\), the figure is a parallelogram.

Step2: Recall parallelogram properties

In a parallelogram, opposite angles are equal (\(\angle A=\angle C\), \(\angle B = \angle D\)), and consecutive angles are supplementary (\(\angle A+\angle D=180^{\circ}\), \(\angle D+\angle C = 180^{\circ}\)).

• For \(m\angle D=m\angle A\): In a parallelogram, \(\angle A\) and \(\angle D\) are consecutive angles. \(m\angle A + m\angle D=180^{\circ}\) (supplementary). They are equal only if \(m\angle A=m\angle D = 90^{\circ}\) (a special case of a parallelogram - a rectangle). But in a general parallelogram, this is not true.
• For \(m\angle B=m\angle D\): By the property of parallelogram (opposite angles are equal), this is true.
• For \(m\angle D=180^{\circ}-m\angle C\): Since \(\angle D\) and \(\angle C\) are consecutive angles in a parallelogram (supplementary), this is true.
• For \(m\angle D=180^{\circ}-m\angle A\): Since \(\angle D\) and \(\angle A\) are consecutive angles in a parallelogram (supplementary), this is true.

Answer:

\(m\angle D = m\angle A\)