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 )

statement	reason
let ( overleftrightarrow{de} ) be a line passing through b and parallel to ( overline{ac} ).	definition of parallel lines
( angle 3 cong angle 5 ) and ( angle 1 cong angle 4 )
( mangle 1 = mangle 4 ) and ( mangle 3 = mangle 5 )
( mangle 4 + mangle 2 + mangle 5 = 180^circ )	angle addition and definition of a straight line
( mangle 1 + mangle 2 + mangle 3 = 180^circ )	substitution

Explanation:

To solve the problem of proving the sum of the interior angles of \(\triangle ABC\) is \(180^\circ\), we analyze the given statements and reasons:

First Drop - Down (for \(\angle 3 \cong \angle 5\) and \(\angle 1 \cong \angle 4\)):

When a transversal (e.g., \(BC\) for \(\angle 3\) and \(\angle 5\), or \(AB\) for \(\angle 1\) and \(\angle 4\)) intersects two parallel lines (\(DE \parallel AC\)), alternate interior angles are congruent. Thus, the reason for \(\angle 3 \cong \angle 5\) and \(\angle 1 \cong \angle 4\) is the Alternate Interior Angles Theorem.

Second Drop - Down (for \(m\angle 1 = m\angle 4\) and \(m\angle 3 = m\angle 5\)):

If two angles are congruent (e.g., \(\angle 1 \cong \angle 4\) or \(\angle 3 \cong \angle 5\)), their measures are equal by the Definition of Congruent Angles (congruent angles have equal measures).

Final Answer (for the drop - down menus):

• For \(\angle 3 \cong \angle 5\) and \(\angle 1 \cong \angle 4\): \(\boldsymbol{\text{Alternate Interior Angles Theorem}}\)
• For \(m\angle 1 = m\angle 4\) and \(m\angle 3 = m\angle 5\): \(\boldsymbol{\text{Definition of Congruent Angles}}\)

Answer:

To solve the problem of proving the sum of the interior angles of \(\triangle ABC\) is \(180^\circ\), we analyze the given statements and reasons:

First Drop - Down (for \(\angle 3 \cong \angle 5\) and \(\angle 1 \cong \angle 4\)):

When a transversal (e.g., \(BC\) for \(\angle 3\) and \(\angle 5\), or \(AB\) for \(\angle 1\) and \(\angle 4\)) intersects two parallel lines (\(DE \parallel AC\)), alternate interior angles are congruent. Thus, the reason for \(\angle 3 \cong \angle 5\) and \(\angle 1 \cong \angle 4\) is the Alternate Interior Angles Theorem.

Second Drop - Down (for \(m\angle 1 = m\angle 4\) and \(m\angle 3 = m\angle 5\)):

If two angles are congruent (e.g., \(\angle 1 \cong \angle 4\) or \(\angle 3 \cong \angle 5\)), their measures are equal by the Definition of Congruent Angles (congruent angles have equal measures).

Final Answer (for the drop - down menus):

• For \(\angle 3 \cong \angle 5\) and \(\angle 1 \cong \angle 4\): \(\boldsymbol{\text{Alternate Interior Angles Theorem}}\)
• For \(m\angle 1 = m\angle 4\) and \(m\angle 3 = m\angle 5\): \(\boldsymbol{\text{Definition of Congruent Angles}}\)