Question

In the figure $\angle BAC = {75^ \circ }$, $\angle ABC = {35^ \circ }$. Find the measure of \[\angle BAZ\]

Answer 1

Hint: We will find the value of \[\angle ACB\] using the angle sum property of a triangle, which states that sum of all angles of the triangle is ${180^ \circ }$. Then, use the property that the value of an exterior angle is equal to the sum of the opposite interior angles to find the value of \[\angle BAZ\].

Complete step by step answer:

We are given that angles $\angle BAC = {75^ \circ }$ and $\angle ABC = {35^ \circ }$.
Consider the triangle $\vartriangle ABC$, where the sum of all angles of the triangle is ${180^ \circ }$ by the angle sum property of the triangle.
Then, $\angle ABC + \angle BAC + \angle ACB = {180^ \circ }$
On substituting the given values, we will get,
$
  {35^ \circ } + {75^ \circ } + \angle ACB = {180^ \circ } \\
   \Rightarrow {110^ \circ } + \angle ACB = {180^ \circ } \\
   \Rightarrow \angle ACB = {180^ \circ } - {110^ \circ } \\
   \Rightarrow \angle ACB = {70^ \circ } \\
$
As it is known that the value of an exterior angle is equal to the sum of the opposite interior angles.
Here, we have to find the value of the \[\angle BAZ\] which will be equal to the sum of the opposite interior angles, that are, $\angle ABC$ and \[\angle ACB\].
Hence, \[\angle BAZ = \angle ABC + \angle ACB\]
On substituting the values, we will get,
$
  \angle BAZ = {35^ \circ } + {70^ \circ } \\
   \Rightarrow \angle BAZ = {105^ \circ } \\
$
Hence, the value of the \[\angle BAZ\] is ${105^ \circ }$.

Note: \[\angle BAZ\] can also be calculated using the property that the sum of angles on a straight line is ${180^ \circ }$. That is, \[\angle BAZ + \angle BAC = {180^ \circ }\]. We will substitute the value $\angle BAC = {75^ \circ }$ to find the value of \[\angle BAZ\]. Then,
$
  \angle BAZ + {75^ \circ } = {180^ \circ } \\
  \angle BAZ = {110^ \circ } \\
$