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.
Please write a two-column proof. Also, there should be ONLY given.
*If you could*, please choose some of these statements and use some of these reasons to prove the statements true. (Those below are the ideas for you to use if you want to.)
Definition of ~ triangles
Definition of median
Definition of midpoint
If 4 quantities are in proportion, then like powers are in proportion.
If 2 angles have their sides perpendicular, right side to right side and left side to left side, the angles are equal.
In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.
Angles inscribed in the same segment or equal segments are equal.
If a line is drawn parallel to the base of a triangle, it cuts off a triangle similar to the given triangle.
Two isosceles triangles are similar if any angle of one equals the corresponding angle of the other.
C.A.S.T.E. - corresponding angles of similar triangles are equal
C.S.S.T.P. - corresponding sides of similar triangles are proportional
Theorem 57- If two triangles have the three angles of one equal respectively to the three angles of the other, then the triangles are similar
Corollary 57-1 If two angles of one triangle are equal respectively to two angles of another, then the triangles are similar. (a.a.)
Corollary 57-2 Two right triangles are similar if an acute angle of one is equal to an acute angle of the other.
Theorem 58-If two triangles have two pairs of sides proportional and the included angles equal respectively, then the two triangles are similar. (s.a.s.)
Corollary 58-1 If the legs of one right triangle are proportional to the legs of another, the triangles are similar. (l.l.)
Theorem 59- If two triangles have their sides respectively proportional, then the triangles are similar. (s.s.s.)
Theorem 60- If two parallels are cut by three or more transversals passing through a common point, then the corresponding segments of the parallels are proportional.
Theorem 61-If in a right triangle the perpendicular is drawn from the vertex of the right angle to the hypotenuse.
Theorem 62-The square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs.
Corollary 62-1 The difference of the square of the hypotenuse and the square of one leg equals the square of the other leg.