List of top Mathematics Questions on Congruence of Triangles

AB is a line segment and P is its mid-point. D and E are points on the same side of AB such that ∠ BAD = ∠ ABE and ∠ EPA = ∠ DPB (see Fig). Show that 

(i) ∆ DAP ≅ ∆ EBP 

(ii) AD = BE

AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that 

(i) AD bisects BC 

(ii) AD bisects ∠A.

∆ ABC and ∆ DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see Fig.). If AD is extended to intersect BC at P, show that

(i) ∆ABD ≅ ∆ACD 

(ii) ∆ABP≅ ∆ACP 

(iii) AP bisects ∠A as well as ∠D. 

(iv) AP is the perpendicular bisector of BC.

In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B (see Fig. 7.23). Show that: 

(i) ∆ AMC ≅ ∆ BMD 

(ii) ∠ DBC is a right angle. 

(iii) ∆ DBC ≅ ∆ ACB (iv) CM = \(\frac{1}{2}\) AB

Line l is the bisector of an angle ∠ A and B is any point on l. BP and BQ are perpendiculars from B to the arms of ∠A (see Fig. 7.20). Show that: 

(i) ∆ APB ≅ ∆ AQB 

(ii) BP = BQ or B is equidistant from the arms of ∠A.

In Fig. 7.21, AC = AE, AB = AD and ∠ BAD = ∠ EAC. Show that BC = DE.

l and m are two parallel lines intersected by another pair of parallel lines p and q (see Fig. 7.19). Show that ∆ ABC ≅ ∆ CDA.

AD and BC are equal perpendiculars to a line segment AB (see Fig). Show that CD bisects AB.

ABCD is a quadrilateral in which AD = BC and ∠ DAB = ∠ CBA (see Fig. 7.17). Prove that 

(i) ∆ ABD ≅ ∆ BAC 

(ii) BD = AC 

(iii) ∠ ABD = ∠ BAC.

In quadrilateral ACBD, AC = AD and AB bisects ∠ A (see Fig. 7.16). Show that ∆ ABC ≅ ∆ ABD. What can you say about BC and BD?