<1=<2, <3=<4, АЕ=CD . докажите что FE=HD

To solve this problem, we can make use of the property of alternate interior angles.

Given: <1 = <2 (1) <3 = <4 (2) AE = CD (3)

To prove: FE = HD

To prove that FE = HD, we need to show that triangles FAE and HCD are congruent.

Using properties of alternate interior angles, we can derive the following:

<1 = <4 (from (1) and (2)) <1 = <3 (from (1) and (2))

Using the transitive property, we can combine (3) and the above statements to obtain:

AE = CD = FE (4)

Since we have shown that AE = CD and AE = FE, we can conclude that CD = FE.

Similarly, using the properties of alternate interior angles, we can show that triangle HCD is congruent to triangle FAE:

<4 = <2 (from (1) and (2)) <4 = <1 (from (1) and (2))

Again, using the transitive property and combining with (3), we have:

AE = CD = HD (5)

Therefore, we have shown that CD = FE (from (4)) and CD = HD (from (5)), which implies FE = HD. QED.