Question

question 9 (multiple choice worth 1 points) (03.09 hc) which of the following explains how δaei could be proven similar to δdeh using the aa similarity postulate? diagram of a geometric figure with points a, i, b, f, e, g, c, h, d ∠aei ≅ ∠deh because vertical angles are congruent, reflect δhed across segment fg, then translate point d to point a to confirm ∠iae ≅ ∠hde ∠aei ≅ ∠deh because vertical angles are congruent, rotate δhed 180° around point e, then translate point d to point a to confirm ∠iae ≅ ∠hde ∠aei ≅ ∠deh because vertical angles are congruent, rotate δhed 180° around point e, then dilate δhed to confirm $overline{ed} cong overline{ea}$ ∠aei ≅ ∠deh because vertical angles are congruent, reflect δhed across segment fg, then dilate δhed to confirm $overline{ed} cong overline{ei}$

Explanation:

To prove \(\triangle AEI \sim \triangle DEH\) using AA similarity, we need two pairs of congruent angles. First, \(\angle AEI\) and \(\angle DEH\) are vertical angles, so they are congruent (\(\angle AEI \cong \angle DEH\)). Next, we need to show \(\angle IAE \cong \angle HDE\).

Looking at the options, the correct transformation to confirm \(\angle IAE \cong \angle HDE\) involves rotating \(\triangle HED\) \(180^\circ\) around point \(E\) (which aligns the sides) and then translating point \(D\) to point \(A\) to check the angle congruence. Let's analyze each option:

• Option 1: Reflecting over \(FG\) and translating \(D\) to \(A\) – reflection might not align the angles correctly as needed.
• Option 2: \(\angle AEI \cong \angle DEH\) (vertical angles), rotate \(\triangle HED\) \(180^\circ\) around \(E\) (aligns the triangle's orientation), then translate \(D\) to \(A\) to confirm \(\angle IAE \cong \angle HDE\) – this makes sense for AA similarity (two angles congruent).
• Option 3: Dilating to confirm \(\overline{ED} \cong \overline{EA}\) – AA similarity doesn't require side congruence, it's about angles.
• Option 4: Reflecting and dilating to confirm \(\overline{ED} \cong \overline{EI}\) – again, side congruence is not needed for AA, and the reflection/translation logic is off.

Answer:

B. \(\angle AEI \cong \angle DEH\) because vertical angles are congruent, rotate \(\triangle HED\) \(180^\circ\) around point \(E\), then translate point \(D\) to point \(A\) to confirm \(\angle IAE \cong \angle HDE\)