Question

select the correct answer from each drop-down menu.

points (a), (b), and (c) form a triangle. complete the statements to prove that the sum of the interior angles of (\triangle abc) is (180^{\circ}).

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$$\begin{tabular}{|l|l|} \\hline statement & reason \\\\ \\hline points (a), (b), and (c) form a triangle. & given \\\\ \\hline let $\\overline{de}$ be a line passing through (b) and parallel to $\\overline{ac}$. & definition of parallel lines \\\\ \\hline $\\angle 3 \\cong \\angle 5$ and $\\angle 1 \\cong \\angle 4$ & alternate interior angles theorem \\\\ \\hline (m\\angle 1 = m\\angle 4) and (m\\angle 3 = m\\angle 5) & choose an answer \\\\ \\hline (m\\angle 4 + m\\angle 2 + m\\angle 5 = 180^{\\circ}) & choose an answer \\\\ \\hline (m\\angle 1 + m\\angle 2 + m\\angle 3 = 180^{\\circ}) & substitution property \\\\ \\hline \\end{tabular}$$

Explanation:

⚡ Using what you learned: lines and angles (parallel, transversal, vertical) · properties of triangles

Step 1: Analyze the first dropdown

The statement is:
\[ \angle 3 \cong \angle 5 \text{ and } \angle 1 \cong \angle 4 \]

Since line \( \overline{DE} \) is parallel to line segment \( \overline{AC} \), the transversal lines \( AB \) and \( BC \) create alternate interior angles. Therefore, the reason is:

• Alternate Interior Angles Theorem (already selected in the image)

Step 2: Analyze the second dropdown

The statement is:
\[ m\angle 1 = m\angle 4 \text{ and } m\angle 3 = m\angle 5 \]

This statement converts the geometric congruence of the angles (\( \cong \)) into equal numerical measures (\( = \)). By definition, congruent angles have equal measures. Therefore, the reason is:

• Congruent angles have equal measures.

Step 3: Analyze the third dropdown

The statement is:
\[ m\angle 4 + m\angle 2 + m\angle 5 = 180^\circ \]

These three angles form a straight line along the line \( \overline{DE} \) at vertex \( B \). Therefore, the reason is:

• angles that form a straight line (or definition of a straight angle / straight line)

Step 4: Analyze the fourth dropdown

The statement is:
\[ m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ \]

This is obtained by substituting \( m\angle 1 \) for \( m\angle 4 \) and \( m\angle 3 \) for \( m\angle 5 \) in the previous equation. Therefore, the reason is:

• substitution (or Substitution Property of Equality)

Answer:

• Row 3 Reason: Alternate Interior Angles Theorem
• Row 4 Reason: Congruent angles have equal measures.
• Row 5 Reason: angles that form a straight line
• Row 6 Reason: substitution