Question

If C and Z are acute angles and that cos C = cos Z prove that angle C = angle Z​

Answer 1

Given C and Z are acute angles and cos C = cos Z

cos C = BCAC and cos Z = YZXZ

cos C = cos Z =>BCAC= YZXZ

Let BCYZ= ACXZ = k..... (1)

BC = k(yz) and AC = k(xz)

In right triangle ABC, using Pythagoras theorem AB = AC2−BC2−−−−−−−−−−√

AB = (k(xz)2−(k(yz)2)−−−−−−−−−−−−−−√= k(xz)2−(yz)2−−−−−−−−−−√

In right triangle xyz, using Pythagoras theorem xy = (xz)2−(yz)2−−−−−−−−−−√

ABXY = K×(XZ)2−(YZ)2√(XZ)2−(YZ)2√

Or ABXY = k ..... (1)

From 1 and 2, BCYZ = ACXZ = ABXY = k

By SSS similarity criterion, we can conclude ΔABC is similar to ΔXYZ

∴ Corresponding angles of two similar triangles will be equal.

∴ ∠C = ∠Z