Question

2 in parallelogram abcd on the right, given points e and f are placed on diagonal bd, so that be = df. prove that ae = cf as follows. fill the appropriate words in the blanks. proof in △abe and △cdf, since the opposite sides of a parallelogram are equal, (the first property of a parallelogram ) □ = □ ……① from ab || dc, ∠□ = ∠□ ……② given: □ = □ ……③ from ①, ②, and ③, since the two □ and the included □ are equal respectively, △□ ≅ △□. therefore, □ = □.

Explanation:

Step1: Identify equal sides

In parallelogram \(ABCD\), \(AB = DC\) (opposite sides of a parallelogram are equal).

Step2: Identify equal angles

Since \(AB\parallel DC\), \(\angle ABE=\angle CDF\) (alternate - interior angles).

Step3: Use given condition

Given \(BE = DF\).

Step4: Apply SAS congruence

From \(AB = DC\), \(\angle ABE=\angle CDF\), and \(BE = DF\), since the two sides and the included angle are equal respectively, \(\triangle ABE\cong\triangle CDF\) (by SAS congruence criterion).

Step5: Conclude equal sides

Therefore, \(AE = CF\) (corresponding parts of congruent triangles are equal).

Answer:

\(AB = DC\); \(\angle ABE=\angle CDF\); \(BE = DF\); sides, angle; \(\triangle ABE\cong\triangle CDF\); \(AE = CF\)