If c and z are acute angles and CosC = CosZ, then prove that angleC = angleZ.
Answers
Answer:
Step-by-step explanation:
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
According to the question:-
Let cosA=
Hypotenuse
sideadjacentA
AB
AC
Similarly,
cosB=
Hypotenuse
sideadjacentB
=
AB
BC
Given that
cosA=cosB
AB
AC
=
AB
BC
AC=BC
In triangle,
angles opposite equal side are equal
∠B=∠A