What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer. Hint: Draw a diagonal and check for congruent triangles.
A quadrilateral with opposite sides equal is a parallelogram.
Proof using congruent triangles:
Step 1: Consider quadrilateral ABCD with AB = CD and AD = BC.
Step 2: Draw diagonal BD.
Step 3: In triangles ABD and CDB: AB = CD (given) AD = CB (given) BD = BD (common)
Step 4: By SSS congruence, triangle ABD ≅ triangle CDB.
Step 5: Therefore, ∠ABD = ∠CDB and ∠ADB = ∠CBD (corresponding angles of congruent triangles).
Step 6: These are alternate angles, so AB || CD and AD || BC.
Step 7: Hence, ABCD is a parallelogram.
Explanation
The chapter's hint suggests drawing a diagonal to prove this. In quadrilateral ABCD with AB = CD and AD = BC, drawing diagonal BD creates two triangles ABD and CDB. Since AB = CD, AD = CB, and BD is common, triangles ABD and CDB are congruent by SSS. Therefore, ∠ABD = ∠CDB and ∠ADB = ∠CBD (corresponding angles). These are alternate angles, so AB || CD and AD || BC. Thus, opposite sides are parallel, making it a parallelogram.
Solution Steps
Step 1: Consider quadrilateral ABCD with AB = CD and AD = BC.
Step 2: Draw diagonal BD.
Step 3: In triangles ABD and CDB: AB = CD (given), AD = CB (given), BD = BD (common).
Step 4: By SSS congruence, triangle ABD ≅ triangle CDB.
Step 5: Therefore, ∠ABD = ∠CDB and ∠ADB = ∠CBD (corresponding angles of congruent triangles).
Step 6: These are alternate angles, so AB || CD and AD || BC.
Step 7: Hence, ABCD is a parallelogram.