question_answer

ABC is a triangle, X is a point outside the\[\Delta ABC\]such that CD=CX, where D is the point of intersection of BC and AX" and\[\angle BAX=\angle XAC.\] Which one of the following is correct?

A)  \[\Delta ABD\] and \[\Delta ACX\]are similar

B)  \[\angle BAD<\angle ACD\]

C)  \[AC=CX\]

D)  \[\angle ADB>\angle DXC\]

Correct Answer: A

Solution :

In \[\Delta DCX\] CD = CX                      (Given) \[\angle 3=\angle 4\] (opposite angle of same sides) But       \[\angle 3=\angle 5\] So,       \[\angle 4=\angle 5\] In \[\Delta ABD\] and \[\Delta ACX,\] \[\angle 1=\angle 2\]                   (Given) \[\angle 4=\angle 5\] \[\angle B=\angle ACX\]              (rest angle) \[\therefore \] \[\Delta ABD\tilde{\ }\Delta ACX\]