Question

based on the flowchart, which of the following theorems is established?
a linear pair theorem
b vertical angles theorem
c common angle theorem
d supplementary theorem

Explanation:

Step1: Analyze the Linear Pair Theorem

The Linear Pair Theorem states that if two angles form a linear pair, then they are supplementary (their measures add up to \(180^{\circ}\)). In the flowchart, we see applications of the Linear Pair Theorem in establishing angle - sum relationships (\(m\angle DFC + m\angle DFA=180^{\circ}\) and \(m\angle AFB + m\angle DFA = 180^{\circ}\)).

Step2: Use the Subtraction Property

From \(m\angle DFC + m\angle DFA=180^{\circ}\) and \(m\angle AFB + m\angle DFA = 180^{\circ}\), by the Subtraction Property (if \(a + c=b + c\), then \(a = b\)), we get \(m\angle DFC=m\angle AFB\). This is the Vertical Angles Theorem (vertical angles are congruent).

Answer:

B. Vertical angles theorem