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

Step2: Triangle Angle - Sum Theorem

For \(\triangle ABC\), by the Triangle Angle - Sum Theorem, \(m\angle CAB + m\angle ACB + m\angle CBA = 180^{\circ}\).

Step3: Subtraction Property of Equality

Since \(m\angle ZAB + m\angle CAB=m\angle CAB + m\angle ACB + m\angle CBA\), we can subtract \(m\angle CAB\) from both sides. By the Subtraction Property of Equality, \(m\angle ZAB=m\angle ACB + m\angle CBA\).

Answer:

The first blank: Linear Pair Postulate; the second blank: Triangle Angle - Sum Theorem; the third blank: Subtraction Property of Equality.