We prove the properties of parallelograms by using congruence. The key is to draw a diagonal and divide the parallelogram into two triangles.
Example ① (Proof that Opposite Sides Are Equal)
In parallelogram ABCD, prove that AB=DC.
Answer: Draw diagonal BD. In △ABD and △CDB / Since AB∥DC, ∠ABD=∠CDB (alternate interior angles) ① / Since AD∥BC, ∠ADB=∠CBD (alternate interior angles) ② / BD=DB (common side) ③ / By ASA, △ABD≅△CDB. Therefore, AB=DC.
In parallelogram ABCD, prove that AB=DC.
Answer: Draw diagonal BD. In △ABD and △CDB / Since AB∥DC, ∠ABD=∠CDB (alternate interior angles) ① / Since AD∥BC, ∠ADB=∠CBD (alternate interior angles) ② / BD=DB (common side) ③ / By ASA, △ABD≅△CDB. Therefore, AB=DC.
Key Points
- Draw a diagonal as an auxiliary line
- Parallel lines → alternate interior angles are equal (use this as a reason)
Practice Problem
- In parallelogram ABCD, prove that ∠A=∠C (use diagonal BD).
Answers and Explanations
- Answer: From △ABD≅△CDB (the same procedure as in Example ①), ∠A=∠C (corresponding angles).