Question

given abcd is a parallelogram. prove m∠a + m∠b + m∠c + m∠d = 360°. by the definition of a parallelogram, ad∥bc and ab∥dc. using, ad as a transversal, ∠a and ∠ are same - side interior angles, so they are. by the definition of supplementary, m∠a + m∠d = 180. using side as a transversal, ∠b and ∠c are same - side interior angles, so they are supplementary. by the definition of supplementary, m∠b + m∠c = 180. so, m∠a + m∠d + m∠b + m∠c = 180 + 180 by the property. simplifying, we have m∠a + m∠b + m∠c + m∠d = 360°

Explanation:

Step1: Identify same - side interior angles

Using \(AD\) as a transversal, \(\angle A\) and \(\angle D\) are same - side interior angles and are supplementary. Using \(BC\) as a transversal, \(\angle B\) and \(\angle C\) are same - side interior angles and are supplementary.

Step2: Write angle - sum equations

We know that \(m\angle A + m\angle D=180^{\circ}\) and \(m\angle B + m\angle C = 180^{\circ}\).

Step3: Combine the equations

By the addition property of equality, \(m\angle A+m\angle D + m\angle B + m\angle C=180 + 180\).

Step4: Simplify

Simplifying gives \(m\angle A + m\angle B + m\angle C + m\angle D=360^{\circ}\).

Answer:

The blanks should be filled as follows: \(\angle D\); supplementary; \(BC\); addition.