Question

the figure is not to scale.
which of the following additional statements would allow us to prove that the sum of the interior angle measures in △abc is 180°?
choose 1 answer:
a ∠1 ≅ ∠2 only
b line ℓ is parallel to (overleftrightarrow{ab}) only
c either statement is sufficient.
d we need both statements.
e even with both statements, we still could not prove that the sum of the interior angle measures in △abc is 180°.

Explanation:

Step1: Analyze Option A

$\angle 1 \cong \angle 2$ implies $\triangle ABC$ is isosceles, but this alone doesn't help prove the angle - sum property. It just gives information about the equality of two interior angles, not related to the straight - line (180°) concept needed for angle - sum.

Step2: Analyze Option B

If line $\ell$ is parallel to $\overleftrightarrow{AB}$, then by the alternate interior angles theorem, $\angle 1=\angle 4$ and $\angle 2 = \angle 5$. Since $\angle 3+\angle 4+\angle 5 = 180^{\circ}$ (linear pair), substituting $\angle 4$ with $\angle 1$ and $\angle 5$ with $\angle 2$ gives $\angle 1+\angle 2+\angle 3=180^{\circ}$, which proves the sum of interior angles of $\triangle ABC$ is $180^{\circ}$.

Step3: Analyze Other Options

Option C is wrong because Option A's statement is not sufficient. Option D is wrong as we don't need Option A's statement. Option E is wrong as Option B's statement is sufficient.

Answer:

B. Line $\boldsymbol{\ell}$ is parallel to $\overleftrightarrow{AB}$ only