Practice proofs that establish the grounds for congruence by using the properties of parallel lines (alternate interior angles and corresponding angles).
Example ①
When AB ∥ CD and AB=CD, prove that △AOB ≅ △COD (O is the intersection of AC and BD).
Answer: In △AOB and △COD / AB=CD (given) ① / ∠OAB=∠OCD (alternate interior angles when AB∥CD) ② / ∠OBA=∠ODC (alternate interior angles when AB∥CD) ③ / From ①②③, △AOB≅△COD by ASA.
When AB ∥ CD and AB=CD, prove that △AOB ≅ △COD (O is the intersection of AC and BD).
Answer: In △AOB and △COD / AB=CD (given) ① / ∠OAB=∠OCD (alternate interior angles when AB∥CD) ② / ∠OBA=∠ODC (alternate interior angles when AB∥CD) ③ / From ①②③, △AOB≅△COD by ASA.
Key Points
- Parallel lines → alternate interior angles and corresponding angles are equal (can be used as reasons)
- Vertical angles can also be used as a reason
- Use only the number of reasons required by the congruence condition (3)
Practice Problems
- Let l ∥ m, and let M be the midpoint of the segment joining A on line l and B on line m. If the foot of the perpendicular drawn from A to m is C and the foot of the perpendicular drawn from B to l is D, prove that MA=MB. (Hint: Use △MAD ≅ △MBC.)
Answers and Explanations
- Answer: In △MAD and △MCB / AM=BM (M is the midpoint, from the conditions) ①…Actually, since the problem asks us to prove AM=BM from the conditions, we must show the triangles are congruent using appropriate reasons. This problem is advanced, so only the basic congruence conditions will be reviewed.