Question

Write a proof to show that opposite angles of a parallelogram are congruent. Be sure to create and name the appropriate geometric figures. This figure does not need to be submitted

Answer 1

1. A parallelogram is formed by 2 pair of parallel line segments intersecting each other.

2. Check Parallelogram ABCD in the atached figure.

3. Since AD is parallel to BC, angles BAC and ACD are alternate interior angles (angles Z), and their measure is equal, say alpha degrees, as shown in the figure.

Similarly m(DAC)=m(BCA)=beta

4. m(A)=m(C)= alpha + beta

5. Conclusion: the measures of 2 opposite angles of a parallelogram are equal.

so "opposite angles of a parallelogram are congruent. "

Answer 2

Answer:

Let's say that parallelogram ABCD was shown where segment AB is parallel to segment DC and segment BC is parallel to segment AD.

According to the given information, segment AB and segment DC are parallel and same with segment BC and segment AD. Construct diagonal AC with a straightedge. It is congruent to itself by the Reflexive Property of Equality. Angles BAC and DCA are congruent by the Alternate Interior Angles Theorem. Angles BCA and DAC are congruent also by the same theorem. Triangles BCA and DAC are congruent according to the Angle-Side-Angle (ASA) Theorem. Then, by CPCTC, opposite sides AB and CD, as well as sides BC and DA, are congruent.

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