Question

select the correct statement in the table.
michelle knows $\triangle abc \sim \triangle def \sim \triangle ghi$. in those triangles, $ab = 36$ cm, $bc = 60$ cm, $de = 60$ cm, $df = 120$ cm, $hi = 90$ cm, and $gi = 108$ cm.
which statement is true?

$ac = 42$ cm	$ef = 100$ cm	$gh = 180$ cm

Explanation:

Step1: Recall Similar Triangles Properties

Similar triangles have proportional sides and equal corresponding angles. So, for \(\triangle ABC \sim \triangle DEF \sim \triangle GHI\), \(\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}\) (for \(\triangle ABC\) and \(\triangle DEF\)) and corresponding angles are equal (e.g., \(\angle A = \angle D = \angle G\), \(\angle B = \angle E = \angle H\), \(\angle C = \angle F = \angle I\)).

Step2: Check Side Proportions for \(\triangle ABC\) and \(\triangle DEF\)

First, find the ratio of similarity between \(\triangle ABC\) and \(\triangle DEF\). The ratio of \(AB\) to \(DE\) is \(\frac{AB}{DE}=\frac{36}{60}=\frac{3}{5}\). Now, check \(BC\) and \(EF\): if \(\frac{BC}{EF}=\frac{3}{5}\), then \(EF = \frac{BC\times5}{3}=\frac{60\times5}{3}=100\) cm. Let's verify other sides to be sure. For \(AC\) and \(DF\): \(\frac{AC}{DF}=\frac{3}{5}\), so \(AC=\frac{3}{5}\times120 = 72\) cm, so \(AC = 42\) cm is false. For \(\triangle DEF\) and \(\triangle GHI\), ratio of \(DF\) to \(GI\) is \(\frac{120}{108}=\frac{10}{9}\), ratio of \(HI\) to \(BC\) is \(\frac{90}{60}=\frac{3}{2}\), wait, maybe better to check angles.

Step3: Check Angle Properties

Corresponding angles in similar triangles are equal. So \(\angle A = \angle D\), \(\angle B = \angle E = \angle H\), \(\angle C = \angle F = \angle I\).

• For the angle statement: "if \(m\angle A = 30^\circ\), then \(m\angle D = 50^\circ\)" – false, since \(\angle A=\angle D\).
• "if \(m\angle A = 45^\circ\), then \(m\angle F = 45^\circ\)" – \(\angle A\) corresponds to \(\angle D\), \(\angle F\) corresponds to \(\angle C\), so unless \(\angle A=\angle C\) (which isn't given), this is false.
• "if \(m\angle B = 124^\circ\), then \(m\angle H = 134^\circ\)" – false, since \(\angle B=\angle H\). Wait, but earlier side check for \(EF\) gave \(EF = 100\) cm which matched the ratio. Wait, maybe I made a mistake in angle correspondence. Wait, \(\triangle ABC \sim \triangle DEF\), so \(\angle B\) corresponds to \(\angle E\), and \(\triangle DEF \sim \triangle GHI\), so \(\angle E\) corresponds to \(\angle H\), so \(\angle B=\angle E=\angle H\), so if \(m\angle B = 124^\circ\), \(m\angle H\) should be \(124^\circ\), not \(134^\circ\). Wait, but the side \(EF = 100\) cm: let's recheck the ratio. \(AB = 36\), \(DE = 60\), ratio \(3:5\). \(BC = 60\), so \(EF\) should be \(60\times\frac{5}{3}=100\), which is correct. \(AC\): \(DF = 120\), so \(AC = 120\times\frac{3}{5}=72\), so \(AC = 42\) is wrong. \(GH\): let's see, ratio between \(\triangle ABC\) and \(\triangle GHI\): \(AB = 36\), \(GI = 108\), ratio \(36:108 = 1:3\), \(BC = 60\), so \(HI = 90\), ratio \(60:90 = 2:3\), wait, maybe \(\triangle DEF\) and \(\triangle GHI\): \(DE = 60\), \(GI = 108\), \(DF = 120\), \(HI = 90\). Ratio of \(DE\) to \(GH\)? Wait, maybe I messed up the correspondence. Wait, the side \(EF = 100\) cm: from \(\triangle ABC\) and \(\triangle DEF\), ratio \(AB/DE = 36/60 = 3/5\), so \(BC/EF = 3/5\) => \(EF = (60*5)/3 = 100\), which is correct. So \(EF = 100\) cm is true. Let's check other angle statements again. The angle statement with \(EF = 100\) cm: the side \(EF = 100\) cm calculation is correct via similarity ratio. The angle statements:

• \(\angle A = \angle D\), so first angle statement is wrong.
• \(\angle A\) and \(\angle F\): \(\angle F\) is corresponding to \(\angle C\), so unless \(\angle A = \angle C\), which isn't given, so second angle statement is wrong.
• \(\angle B = \angle H\), so third angle statement is wrong (124 vs 134). So the correct statement is \(EF = 100\) cm.

Answer:

\(EF = 100\) cm (the statement "EF = 100 cm" is true)