Question

1. in the figure shown, what is ( mangle a )? explain.

(a) 57; ( \triangle abc ) is an isosceles triangle with base angles ( a ) and ( c ).
( mangle a=mangle c ).
(b) 66; ( \triangle abc ) is an isosceles triangle with base angles ( b ) and ( c ).
( mangle b = mangle c = 57 ), and ( mangle a+mangle b+mangle c = 180 ).
(c) 57; ( \triangle abc ) is an equilateral triangle.
(d) there is not enough information to find ( mangle a ).

Explanation:

Step1: Identify the triangle type

The marks on the sides of \(\triangle ABC\) indicate that \(AB = AC\). So, \(\triangle ABC\) is an isosceles triangle with base angles \(B\) and \(C\). That means \(m\angle B=m\angle C = 57^{\circ}\).

Step2: Use the angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). Let \(m\angle A=x\). Then, by the angle - sum property \(x + m\angle B+m\angle C=180^{\circ}\). Substitute \(m\angle B = 57^{\circ}\) and \(m\angle C = 57^{\circ}\) into the equation: \(x+57 + 57=180\).

Step3: Solve for \(x\) (which is \(m\angle A\))

Simplify the left - hand side of the equation: \(x + 114=180\). Subtract 114 from both sides: \(x=180 - 114\). So, \(x = 66^{\circ}\).

Answer:

B. 66; \(\triangle ABC\) is an isosceles triangle with base angles \(B\) and \(C\). \(m\angle B=m\angle C = 57\), and \(m\angle A+m\angle B+m\angle C = 180\).