Given:
m∠EAC = 120° and m∠HBI = 84°.
To find:
The measure of ∠BCA.
Solution:
If two lines intersect each other then the vertically opposite angles are equal.
[tex]\angle HBI \cong \angle ABC[/tex] (Vertically opposite angles)
[tex]m\angle HBI = m\angle ABC[/tex]
[tex]84^\circ = m\angle ABC[/tex] ...(i)
If two angles forms a linear pair, then their sum is 180 degrees.
[tex]m\angle BAC+m\angle EAC=180^\circ[/tex] (Linear pair)
[tex]m\angle BAC+120^\circ=180^\circ[/tex]
[tex]m\angle BAC=180^\circ-120^\circ[/tex]
[tex]m\angle BAC=60^\circ[/tex] ...(ii)
According to the angle sum property, the sum of all interior angles of a triangle is 180 degrees.
[tex]m\angle ABC+m\angle BAC+m\angle BCA=180^\circ[/tex] (Angle sum property)
[tex]84^\circ+60^\circ+m\angle BCA=180^\circ[/tex] [Using (i) and (ii)]
[tex]144^\circ+m\angle BCA=180^\circ[/tex]
[tex]m\angle BCA=180^\circ-144^\circ[/tex]
[tex]m\angle BCA=36^\circ[/tex]
Therefore, the correct option is C.
