( 1)

∠∠A =∠F =∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠

∴AC∥DF。 . . (Internal misalignment angles are equal)

BC∨DE

∵∠ 1=∠2

∠DMF=∠ 1 .。 . (Equal vertex angle)

∴∠2=∠DMF

Lines d, m and B***, lines e, n and C***.

∴BD∥CE。 . . (Internal misalignment angles are equal)

∴ quadrilateral BCDE is a parallelogram. . . (Two groups of parallelograms with parallel opposite sides are parallelograms)

(2)

(1) quadrilateral BCDE is a parallelogram.

BC=DE

DBN=∠CNB .。 . (Internal misalignment angles are equal)

∵BN equal division ∠DBC

∴∠NBC=∠DBN(= 1/2∠DBC)

National Broadcasting Corporation =∠CNB

∴△CNB is isosceles

CN=CB(BC)

DE=2

∴CN=2

9. Solution:

( 1)

∫D' is △ADE, which is obtained by folding AE.

∴△ADE≌△AD'E

∠DAE=∠D'AE

∴D'A∥AD

A quadrilateral is a parallelogram.

∴D'E∥ and = A.D.

And point d' is on AB, e is on DC, and quadrilateral ABCD is a parallelogram.

∴D'E∥ and = BC

A quadrilateral is a parallelogram.

(2)

D' is △ADE, which is obtained by folding AE.

∴∠AED=∠AED'

AE equal division ∠DED'

And ∵ divide evenly ∠ABC and e are in DC.

∴∠AEB= 1/2∠CED (flat angle) = 90.

∴△AEB is RT△

∴AB^2=AE^2+BE^2。 . . (Pythagorean theorem)