Given: In quadrilateral ABCD, the diagonals AC and BD intersect at point E.
Additionally: AE = BE and CE = DE
a
Prove: AD = BC
Choose the correct congruence theorem:
We use triangles ΔAED and ΔBEC
Correct! The triangles are congruent by SAS
Explanation:
• AE = BE (given)
• DE = CE (given)
• ∠AED = ∠BEC (vertical angles)
• Therefore ΔAED ≅ ΔBEC by SAS
• From the congruence: AD = BC
b
Prove: ∠ABD = ∠BAC
What type of triangle is ΔABE?
Correct! An isosceles triangle
Explanation:
• Since AE = BE, triangle ΔABE is isosceles
• In an isosceles triangle, the base angles are equal: ∠ABE = ∠BAE
• ∠ABD = ∠ABE (same angle, since B-E-D are collinear)
• ∠BAC = ∠BAE (same angle, since A-E-C are collinear)
• Therefore ∠ABD = ∠BAC
c
Let F be the midpoint of CD. Prove: AF = BF
What is the key idea in the proof?
Correct! The symmetry through E is the key
Explanation:
• AE = BE means that point E lies on the perpendicular bisector of AB
• CE = DE means that point E lies on the perpendicular bisector of CD
• F is the midpoint of CD, so line EF is the perpendicular bisector of CD
• By the symmetry of the figure, F also lies on the perpendicular bisector of AB
• Any point on the perpendicular bisector of AB is equidistant from A and B
• Therefore AF = BF
Well done! You completed Question 42!
You understood congruence and symmetry proofs in quadrilaterals!