2016 · Q55
Let the triangles ABC and DEF be such that ∠ABC = ∠DEF, ∠ACB = ∠DFE and ∠BAC = ∠EDF. Let L be the midpoint of BC and M be the midpoint of EF. Consider the following statements :
Statement ITriangles ABL and DEM are similar.
Statement IITriangle ALC is congruent to triangle DMF even if AC ≠ DF.
Which one of the following is correct in respect of the above statements?
Elementary MathematicsGeometry: Lines, Angles and Triangles
- ABoth Statement I and Statement II are true and Statement II is the correct explanation of Statement I
- BBoth Statement I and Statement II are true but Statement II is not the correct explanation of Statement I
- CStatement I is true but Statement II is false
- DStatement I is false but Statement II is true
Explanation
The angle equalities give ΔABC ∼ ΔDEF (AAA) with AB/DE = BC/EF; since L, M bisect BC, EF, also AB/DE = BL/EM with included angles ∠ABL = ∠DEM, so ΔABL ∼ ΔDEM by SAS — Statement I true. Statement II claims congruence with AC ≠ DF, impossible since congruent triangles have all corresponding sides equal — false. Hence option C.