Question

triangle angle theorems proving the exterior angle theorem given △abc prove: m∠zab = m∠acb + m∠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∠zab + m∠cab = 180° by the we also know m∠cab + m∠acb + m∠cba = 180° because using substitution, we have m∠zab + m∠cab = m∠cab + m∠acb + m∠cba therefore, we conclude m∠zab = m∠acb + m∠cba using the substitution property relative property subtraction property

Explanation:

Step1: Recall angle - sum property of triangle

In \(\triangle ABC\), \(m\angle CAB + m\angle ACB+m\angle CBA = 180^{\circ}\) (angle - sum property of a triangle).

Step2: Recall linear - pair property

\(\angle ZAB\) and \(\angle CAB\) are a linear pair, so \(m\angle ZAB + m\angle CAB=180^{\circ}\).

Step3: Substitute and simplify

Since \(m\angle ZAB + m\angle CAB = 180^{\circ}\) and \(m\angle CAB + m\angle ACB+m\angle CBA = 180^{\circ}\), we can substitute \(180^{\circ}\) in the first equation with \(m\angle CAB + m\angle ACB+m\angle CBA\). So \(m\angle ZAB + m\angle CAB=m\angle CAB + m\angle ACB+m\angle CBA\). Using the subtraction property of equality (subtract \(m\angle CAB\) from both sides), we get \(m\angle ZAB=m\angle ACB + m\angle CBA\).

Answer:

The exterior - angle theorem \(m\angle ZAB=m\angle ACB + m\angle CBA\) is proved.