List of top Mathematics Questions on Some Properties of a Triangle

∆ABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see Fig. 7.34). Show that ∠ BCD is a right angle.

ABC and DBC are two isosceles triangles on the same base BC (see Fig). Show that ∠ABD = ∠ACD.

In ∆ ABC, AD is the perpendicular bisector of BC (see Fig. 7.30). Show that ∆ ABC is an isosceles triangle in which AB = AC.

ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see Fig). Show that 

(i) ∆ ABE ≅ ∆ ACF 

(ii) AB = AC, i.e., ABC is an isosceles triangle.

ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see Fig. 7.31). Show that these altitudes are equal.

In an isosceles triangle ABC, with AB = AC, the bisectors of ∠ B and ∠ C intersect each other at O. Join A to O. Show that : 

(i) OB = OC 

(ii) AO bisects ∠ A