If m∠C = m∠Z and AC=XZ, then which of the following conditions is necessary for ΔABC and ΔXYZ to be congruent?

In a quadrilateral ABCD, a line segment BD is a diagonal such that AB=CD and ∠ABD=∠CDB. Are the triangles ABD and CDB congruent? If so, by what rule?

A right-angled triangle has a hypotenuse of 13 cm and one leg of 5 cm. A second right-angled triangle is similar to the first, and its hypotenuse is 39 cm. What is the area of the second triangle?

In a trapezoid ABCD with AB parallel to CD, the diagonals AC and BD intersect at E. What is the ratio of the area of △ABE to the area of △CDE?

Two right-angled triangular blocks, ABC and DEF, have ∠B = ∠E = 90°. If the lengths of the hypotenuses AC and DF are equal, and the sides AB and DE are equal, are the triangles congruent? If so, by what rule?

The ratio of the lengths of two corresponding sides of two similar triangles is 15 : 14.
The ratio of the areas of these two triangles, in the order mentioned, is:

If ∆ ABC ∼ ∆ XYZ, AB = 6 cm, XY = 8 cm, YZ = 12 cm and ZX = 16 cm, then find the perimeter of ∆ ABC.

The ratio of the lengths of two corresponding sides of two similar triangles is 3 : 10. The ratio of the areas of these two triangles, in the order mentioned, is:

In △ABC, DE ∥ AC, where D and E are the points on sides AB and BC, respectively. If BD = 17 cm and AD = 14 cm, then what is the ratio of the area of △BDE to that of the trapezium ADEC?

In ΔABC, DE || AC, where D and E are the points on sides AB and BC, respectively. If BD = 12 cm and AD = 11 cm, then what is the ratio of the area of ΔBDE to that of the trapezium ADEC?

If ΔABC ≅ ΔPQR, such that ∠ABC = 77°, ∠BCA = (x − y)°, AC = 48 cm, ∠PQR = (3x − 4)°, PR = x + 3y, then find the value of ∠QRP.​