(Q17)In △ABC and △XYZ,if ∠A and ∠X are acute angles such that cos A=cos X then show that ∠A=∠X

BCA
YZX

Solution:

Given,

for △ABC and △XYZ

∠A and ∠X are acute triangle

Where cos A=cos X

Also given,to show that ∠A=∠X

cos=
length of adjacent side
length of hypotenuse
AB
AC
 =
XY
XZ
Let
AB
AC
 =
XY
XZ
=k(where k is constant)
so we get,
AB
XY
 =
AC
XZ
=k----------eq(1)

taking then separately we get,

AB
XY
=k

AB=

AC
XZ
=k

AC=

now, let us consider both the triangles opposite sides with their distance

d2=X22-X21

(BC)2
(ZY)2
=
(AC)2-(AB)2
(XZ)2-(XY)2
(from we get equation(1))

by substitute the values we get

(BC)2
(ZY)2
=2
BC
ZY
=

[let us now substitute k in eq(1)]

AB
XY
=
AC
XZ
=
BC
ZY

by similarly we get △ABC ~△XYZ

By property of similarly ∠A =∠X