Given: I and m are two parallel lines intersected by another pair of parallel lines p and q.
To Prove: ∆ABC ≅ ∆CDA.
Proof: ∵ AB || DC
and AD || BC
∴ Quadrilateral ABCD is a parallelogram.
| ∵ A quadrilateral is a parallelogram if both the pairs of opposite sides are parallel
∴ BC = AD ...(1)
| Opposite sides of a ||gm are equal
AB = CD ...(2)
| Opposite sides of a ||gm are equal
and ∠ABC = ∠CDA ...(3)
| Opposite angles of a ||gm are equal
In ∆ABC and ∆CDA,
AB = CD | From (2)
BC = DA | From (1)
∠ABC = ∠CDA | From (3)
∴ ∆ABC ≅ ∆CDA. | SAS Rule
Line I is the bisector of an angle ∠A and B is any point on I. BP and BQ are perpendiculars from B to the arms of ∠A (see figure). Show that:
(i) ∆APB ≅ ∆AQB
(ii) BP = BQ or B is equidistant from the arms of ∠A.
(i) ∆ABD ≅ ∆BAC
(ii) BD = AC
(iii) ∠ABD = ∠BAC.
