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

Since angles \( \angle ZAB\) and \( \angle CAB\) are a linear pair, \(m\angle ZAB + m\angle CAB=180^{\circ}\) by the Linear Pair Postulate.

Step2: Triangle Angle - Sum Theorem

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

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. By the Subtraction Property of Equality, \(m\angle ZAB=m\angle ACB + m\angle CBA\).

Answer:

First blank: Linear Pair Postulate; Second blank: Triangle Angle - Sum Theorem; Third blank: Subtraction Property of Equality.