Question

given \\( \triangle abc \\) prove \\( m \angle zab = m \angle acb + m \angle cba \\) we start with triangle abc and see that angle zab is an exterior angle created by the extension of side ac angles zab and cab are a linear pair by definition we know that \\( m \angle zab + m \angle cab = 180 ^ { \circ } \\) by the we also know \\( m \angle cab + m \angle acb + m \angle cba = 180 ^ { \circ } \\) because using substitution, we have \\( m \angle zab + m \angle cab = m \angle cab + m \angle acb + m \angle cba \\) therefore, we conclude \\( m \angle zab = m \angle acb + m \angle cba \\) using the

Explanation:

Step1: Linear Pair Postulate

Angles \(ZAB\) and \(CAB\) are a linear pair. By the Linear Pair Postulate, if two angles form a linear pair, then the sum of their measures is \(180^{\circ}\). So, \(m\angle ZAB + m\angle CAB=180^{\circ}\)

Step2: Triangle Angle - Sum Theorem

For \(\triangle ABC\), by the Triangle Angle - Sum Theorem, the sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle CAB + m\angle ACB + m\angle CBA = 180^{\circ}\)

Step3: Subtraction Property of Equality

We have \(m\angle ZAB + m\angle CAB=m\angle CAB + m\angle ACB + m\angle CBA\). Subtract \(m\angle CAB\) from both sides of the equation. By the Subtraction Property of Equality (\(a + b=a + c\Rightarrow b = c\) when we subtract \(a\) from both sides), we get \(m\angle ZAB=m\angle ACB + m\angle CBA\)

Answer:

1. Linear Pair Postulate
2. Triangle Angle - Sum Theorem
3. Subtraction Property of Equality