Question

find ( x ) and then ( mangle bac ) and ( mangle bca ).
note: always make sure the value of ( x ) produces positive angles.
( x=)
( mangle bac=)
( mangle bca=)

Explanation:

Step1: Use the property of isosceles triangle

Since \(AB = BC\), \(\angle BAC=\angle BCA=(5x - 1)^{\circ}\).

Step2: Apply the triangle - angle sum theorem

The sum of interior angles of a triangle is \(180^{\circ}\). So \(\angle B+\angle BAC+\angle BCA = 180^{\circ}\).
Substitute the values: \(62+(5x - 1)+(5x - 1)=180\).
Simplify the left - hand side: \(62 + 10x-2=180\), which becomes \(10x+60 = 180\).
Subtract 60 from both sides: \(10x=180 - 60\), so \(10x=120\).
Divide both sides by 10: \(x = 12\).

Step3: Calculate \(\angle BAC\) and \(\angle BCA\)

Substitute \(x = 12\) into \((5x - 1)^{\circ}\).
\(5x-1=5\times12 - 1=60 - 1=59\).

Answer:

\(x = 12\), \(m\angle BAC = 59^{\circ}\), \(m\angle BCA=59^{\circ}\)