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

By the Linear Pair Postulate, if two angles form a linear pair, their measures add up to \(180^{\circ}\). Since \(\angle ZAB\) and \(\angle CAB\) are a linear pair, \(m\angle ZAB + m\angle CAB=180^{\circ}\).

Step2: Triangle Angle - Sum Theorem

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

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. Using 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:

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