7. If ∠EAC measures 120oand ∠HBI measures 84o, find the measure of ∠BCA. A.m∠BCA = 60 B.m∠BCA = 84 C.m∠BCA = 36 D.m∠BCA = 54

Question

asked Nov 3, 2022 199k views

1 Answer

Caleb Keller · Answer 1 · 2022-11-09T08:24:50+0000

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.

$\angle HBI \cong \angle ABC$ (Vertically opposite angles)

$m\angle HBI = m\angle ABC$

$84^\circ = m\angle ABC$ ...(i)

If two angles forms a linear pair, then their sum is 180 degrees.

$m\angle BAC+m\angle EAC=180^\circ$ (Linear pair)

$m\angle BAC+120^\circ=180^\circ$

$m\angle BAC=180^\circ-120^\circ$

$m\angle BAC=60^\circ$ ...(ii)

According to the angle sum property, the sum of all interior angles of a triangle is 180 degrees.

$m\angle ABC+m\angle BAC+m\angle BCA=180^\circ$ (Angle sum property)

$84^\circ+60^\circ+m\angle BCA=180^\circ$ [Using (i) and (ii)]

$144^\circ+m\angle BCA=180^\circ$

$m\angle BCA=180^\circ-144^\circ$

$m\angle BCA=36^\circ$

Therefore, the correct option is C.