Question

in △jkl, (overline{lj}congoverline{kl}) and (mangle k = 10^{circ}). find (mangle j).

Explanation:

Step1: Identify the triangle type

Since \( \overline{LJ}\cong\overline{KL}\), \(\triangle JKL\) is an isosceles triangle. In an isosceles triangle, the base - angles are equal.

Step2: Recall the angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). Let \(m\angle J = m\angle K = 10^{\circ}\) (this is wrong, actually, if \( \overline{LJ}\cong\overline{KL}\), then \(m\angle J=m\angle L\)). Let \(m\angle J = x\), \(m\angle L=x\) and \(m\angle K = 10^{\circ}\).
Using the formula \(m\angle J+m\angle K + m\angle L=180^{\circ}\), we substitute the values: \(x + 10^{\circ}+x=180^{\circ}\).

Step3: Solve the equation

Combine like terms: \(2x+10^{\circ}=180^{\circ}\).
Subtract \(10^{\circ}\) from both sides: \(2x=180^{\circ}- 10^{\circ}=170^{\circ}\).
Divide both sides by 2: \(x=\frac{170^{\circ}}{2}=85^{\circ}\).

Answer:

\(85^{\circ}\)