ABC and DEF are two isosceles triangles on bases BC and EF respectively. If AB : BC = DE : EF (proportion), prove that the two triangles are similar.

Question

Please write a two-column proof. Also, there should ONLY be one given.

Answer 1

Proof in step / reason form:

Statement Reason

AB/BC = DE/EF given

for the given isoceles

right triangles

tan ACB = AB/BC and defintion of tangent ratio

tan DFE = DE/EF

tan ACB = tan DFE substitution/transitive

angles ACB and DFE property of right triangles

are acute angles

angle ACB = angle DFE inverse tangent of both sides

angle DEF = angle ABC All right triangles

are congruent 90 degrees

angle BAC = angle EDF Theorem:1 pair of corresponding angles in

a right triangle is congruent, then

the other pair of corresponding angles

must be congruent

Proof in paragraph form:

Then angles DFE and ACB have the same tangent ratio....

tan DFE = tan ACB

So then those two angles MUST be equal, upon taking inverse tangent of both sides,

as they are acute angles per right triangle, forcing them to be in quadrant 1

Right angles are congruent

angle DFE = angle ACB

Therefore, the 3rd pair of corresponding angles must also be congruent...

which prove similarlty