how do we prove that opposite angles and sides of a parallelogram are equal in measure?
See http://www.algebra.com/algebra/homework/Parallelograms/Opposite-sides-of-a-parallelogram-are-equal.lesson
To prove that opposite angles and sides of a parallelogram are equal in measure, you can use the properties of parallel lines and theorems related to angles and sides. Here are the steps to prove this:
1. Draw a parallelogram with sides labeled as AB, BC, CD, and DA, and diagonals AC and BD.
2. Start by proving that opposite sides of a parallelogram are parallel. Draw a line segment EF that is parallel to side AB, and a line segment GH that is parallel to side CD, forming transversals AC and BD.
3. Use the property of alternate interior angles to show that angle A and angle C are congruent, and angle B and angle D are congruent, since they are formed by parallel lines and a transversal.
4. Next, prove that alternate exterior angles are congruent. Use the property of alternate exterior angles to show that angle E is congruent to angle G, and angle F is congruent to angle H, since they are formed by parallel lines and a transversal.
5. Now, observe that corresponding angles of the parallelogram are congruent. Use the property of corresponding angles to show that angle A is congruent to angle D, and angle B is congruent to angle C, since they are formed by parallel lines and a transversal.
6. To prove that the opposite sides of a parallelogram are congruent, use the property of congruent triangles. By drawing diagonal AC, you can prove that triangle ABC is congruent to triangle CDA using the side-angle-side (SAS) congruence criterion. Therefore, side AB is congruent to side CD.
7. Similarly, draw diagonal BD to prove that triangle BCD is congruent to triangle DAB, using the side-angle-side (SAS) congruence criterion. Therefore, side BC is congruent to side DA.
By using these steps, you have proven that opposite sides and angles of a parallelogram are congruent.