Question

if ( mangle abc = 35^{circ}), ( mangle bac = 29^{circ}), and (angle acbcongangle deb), what is ( mangle deb)?

Explanation:

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). For \(\triangle ABC\), we know that \(m\angle ABC + m\angle BAC+m\angle ACB=180^{\circ}\).
Given \(m\angle ABC = 35^{\circ}\) and \(m\angle BAC = 29^{\circ}\), we substitute these values into the formula: \(35^{\circ}+29^{\circ}+m\angle ACB = 180^{\circ}\).

Step2: Solve for \(m\angle ACB\)

First, simplify the left - hand side of the equation: \(64^{\circ}+m\angle ACB=180^{\circ}\).
Then, subtract \(64^{\circ}\) from both sides: \(m\angle ACB=180^{\circ}- 64^{\circ}\).
So, \(m\angle ACB = 116^{\circ}\).

Step3: Use the congruent angle property

Since \(\angle ACB\cong\angle DEB\), by the definition of congruent angles (congruent angles have equal measures), \(m\angle DEB=m\angle ACB\).

Answer:

\(116^{\circ}\)