Question

Given: ∠BCD is right; BC ≅ DC; DF ≅ BF; FA ≅ FE Which relationships in the diagram are true? Check all that apply.

△ACF ≅ △ECF by HL
△CBF ≅ △CDF by SSS
△BFA ≅ △DFE by SAS
△CFD ≅ △EFD by SSS
△CBE ≅ △CDA by HL

Answer 1

Triangle CBE =~ Triangle CDA by HL b/c hypotenuse, leg and right angle =~

Triangle BFA =~ Triangle DFE by SAS b/c side, angle, side are =~

Triangle CBF =~ Triangle CDF by SSS b/c side, side, side are =~

Plus, I got it correct on the quiz :)

Explanation:

Answer 2

Given that B C ≅ D C and D F ≅ B F. Also, C F ≅ C F by refrexive property.

Therefore, △ C B F ≅ △ C D F by S S S


Given that F A ≅ F E
i.e. A F ≅ E F
Also, given that D F ≅ B F
i.e. F D ≅ F B

This means that A F + F D ≅ E F + F B which means that A D ≅ E B

Now, given that ∠ B C D is right, this means that △ C B E is a right triangle with the right angle at C and E B as its hypothenus. Also △ C D A is a right triangle with the right angle at C and A D as its hypothenus.

Thus, given that A D ≅ E B and the legs B C ≅ D C, therefore, △ C B E ≅ △ C D A by H L.


Therefore, the true statements are:
△CBF ≅ △CDF by SSS, and

△CBE ≅ △CDA by HL