Question

1. opposite sides of a parallelogram are parallel. prove that opposite angles of a parallelogram are congruent. see example 4 given: abcd is a parallelogram prove: ∠a≅∠c, ∠b≅∠d

Explanation:

Step1: Use property of parallel lines

Since $AB\parallel CD$ and $AD\parallel BC$ in parallelogram $ABCD$, $\angle A+\angle D = 180^{\circ}$ (same - side interior angles of parallel lines are supplementary) and $\angle B+\angle C=180^{\circ}$ (same - side interior angles of parallel lines are supplementary). Also, $\angle A+\angle B = 180^{\circ}$ (same - side interior angles of parallel lines are supplementary).

Step2: Manipulate angle - sum equations

From $\angle A+\angle D = 180^{\circ}$ and $\angle A+\angle B = 180^{\circ}$, we can get $\angle A+\angle D=\angle A+\angle B$. Subtracting $\angle A$ from both sides of the equation $\angle A+\angle D=\angle A+\angle B$, we have $\angle B=\angle D$.

Step3: Prove the other pair of angles congruent

Similarly, from $\angle A+\angle D = 180^{\circ}$ and $\angle B+\angle C = 180^{\circ}$, and since $\angle B=\angle D$, we can substitute $\angle B$ with $\angle D$ in $\angle B+\angle C = 180^{\circ}$ and get $\angle D+\angle C = 180^{\circ}$. Comparing with $\angle A+\angle D = 180^{\circ}$, subtracting $\angle D$ from both sides, we get $\angle A=\angle C$.

Answer:

We have proved that in parallelogram $ABCD$, $\angle A\cong\angle C$ and $\angle B\cong\angle D$.