Question

find x and then m∠bac and m∠bca.
note: always make sure the value of x produces positive angles.
x=
m∠bac=
m∠bca=

Explanation:

Step1: Use the property of an equilateral triangle

In an equilateral triangle, the base angles are equal. So \(m\angle BAC=m\angle BCA=(5x + 1)^{\circ}\).

Step2: Use the angle - sum property of a triangle

The sum of angles in a triangle is \(180^{\circ}\). So \(m\angle BAC+m\angle BCA+m\angle ABC = 180^{\circ}\).
Substitute the values: \((5x + 1)+(5x + 1)+78=180\).
Simplify the left - hand side: \(10x+80 = 180\).
Subtract 80 from both sides: \(10x=180 - 80\), so \(10x=100\).
Divide both sides by 10: \(x = 10\).

Step3: Find \(m\angle BAC\) and \(m\angle BCA\)

Substitute \(x = 10\) into \(5x+1\).
\(m\angle BAC=5x + 1=5\times10+1=51^{\circ}\).
Since \(m\angle BAC=m\angle BCA\), \(m\angle BCA = 51^{\circ}\).

Answer:

\(x = 10\), \(m\angle BAC=51^{\circ}\), \(m\angle BCA=51^{\circ}\)