The problem presents a partial proof related to a parallelogram. We need to fill in the missing justifications in the proof and identify the property of parallelograms that the proof demonstrates. The given information is the diagram of a parallelogram with a diagonal, and the angle relationships $a = d$ and $b = c$.

1. Problem Description

The problem presents a partial proof related to a parallelogram. We need to fill in the missing justifications in the proof and identify the property of parallelograms that the proof demonstrates. The given information is the diagram of a parallelogram with a diagonal, and the angle relationships $a = d$ and $b = c$.

2. Solution Steps

The two lines of the parallelogram are parallel.

The diagonal is a transversal intersecting the two parallel lines.

The angles $a$ and $d$ are alternate angles.

The angles $b$ and $c$ are also alternate angles.

Alternate angles are equal.

Therefore:

$a = d$ because alternate angles are equal.

$b = c$ because alternate angles are equal.

Since $a = d$ and $b = c$, then $a + b = d + c$.

The sum of two adjacent angles in a parallelogram is $180^\circ$. Since $a+b = d+c$, the sum of angles on the same side of the parallelogram is the same. Therefore $a+b$ must be equal to $c+d$.

The question then becomes what property is implied by $a+b = c+d$.

The sum of all 4 angles in any quadrilateral is 360 degrees. Therefore $a+b+c+d=360$.

Since $a+b=c+d$, we can rewrite that as $2(a+b)=360$.

Dividing by two, we get $a+b=180$.

3. Final Answer

a)

$a = d$ because alternate angles are equal.

$b = c$ because alternate angles are equal.

b)

The proof shows that adjacent angles in a parallelogram are supplementary (sum to 180 degrees).