Question

given: δabc 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 dropdown. we also know m∠cab + m∠acb + m∠cba = 180° because dropdown. 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 dropdown (options: substitution property, reflexive property, subtraction property).

Explanation:

First Dropdown (m∠ZAB + m∠CAB = 180° reason)

Step1: Recall linear pair postulate

A linear pair of angles is supplementary, meaning their measures sum to \(180^\circ\). Since ∠ZAB and ∠CAB are a linear pair, \(m\angle ZAB + m\angle CAB = 180^\circ\) by the Linear Pair Postulate (or definition of supplementary angles for a linear pair).

Second Dropdown (m∠CAB + m∠ACB + m∠CBA = 180° reason)

Step1: Recall triangle angle sum theorem

The sum of the interior angles of a triangle is \(180^\circ\). For \(\triangle ABC\), the angles are ∠CAB, ∠ACB, and ∠CBA, so \(m\angle CAB + m\angle ACB + m\angle CBA = 180^\circ\) by the Triangle Angle Sum Theorem.

Third Dropdown (Conclusion step property)

Step1: Analyze the equation manipulation

We have \(m\angle ZAB + m\angle CAB = m\angle CAB + m\angle ACB + m\angle CBA\). To isolate \(m\angle ZAB\), we subtract \(m\angle CAB\) from both sides. This is done using the Subtraction Property of Equality (if \(a + b = a + c\), then \(b = c\) by subtracting \(a\) from both sides).

Final Answers for Each Dropdown:

1. First dropdown: Linear Pair Postulate (or "supplementary angles (linear pair)" concept)
2. Second dropdown: Triangle Angle Sum Theorem
3. Third dropdown: subtraction property

(Note: For the dropdowns, the first should be filled with the reason for linear pair supplementary (e.g., "Linear Pair Postulate"), the second with "Triangle Angle Sum Theorem", and the third with "subtraction property" as we subtract \(m\angle CAB\) from both sides of the equation.)

Answer:

First Dropdown (m∠ZAB + m∠CAB = 180° reason)

Step1: Recall linear pair postulate

A linear pair of angles is supplementary, meaning their measures sum to \(180^\circ\). Since ∠ZAB and ∠CAB are a linear pair, \(m\angle ZAB + m\angle CAB = 180^\circ\) by the Linear Pair Postulate (or definition of supplementary angles for a linear pair).

Second Dropdown (m∠CAB + m∠ACB + m∠CBA = 180° reason)

Step1: Recall triangle angle sum theorem

The sum of the interior angles of a triangle is \(180^\circ\). For \(\triangle ABC\), the angles are ∠CAB, ∠ACB, and ∠CBA, so \(m\angle CAB + m\angle ACB + m\angle CBA = 180^\circ\) by the Triangle Angle Sum Theorem.

Third Dropdown (Conclusion step property)

Step1: Analyze the equation manipulation

We have \(m\angle ZAB + m\angle CAB = m\angle CAB + m\angle ACB + m\angle CBA\). To isolate \(m\angle ZAB\), we subtract \(m\angle CAB\) from both sides. This is done using the Subtraction Property of Equality (if \(a + b = a + c\), then \(b = c\) by subtracting \(a\) from both sides).

Final Answers for Each Dropdown:

1. First dropdown: Linear Pair Postulate (or "supplementary angles (linear pair)" concept)
2. Second dropdown: Triangle Angle Sum Theorem
3. Third dropdown: subtraction property

(Note: For the dropdowns, the first should be filled with the reason for linear pair supplementary (e.g., "Linear Pair Postulate"), the second with "Triangle Angle Sum Theorem", and the third with "subtraction property" as we subtract \(m\angle CAB\) from both sides of the equation.)