How to find if right triangles are similar - ISEE Upper Level Mathematics Achievement

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Question

$\Delta ABC \sim \Delta DEF$

$\angle B$ is a right angle; , .

Which is the greater quantity?

(a) $m \angle A$

(b) $60 ^{\circ }$

Answer

$\Delta ABC \sim \Delta DEF$ . Corresponding angles of similar triangles are congruent, so since $\angle B$ is a right angle, so is $\angle E$ .

The hypotenuse $\overline{DF}$ of $\Delta DEF$ is twice as long as leg $\overline{DE}$ ; by the $30 ^{\circ } - 60^{\circ } -90^{\circ }$ Theorem, $m \angle D = 60 ^{\circ }$ . Again, by similiarity,

$m \angle A = m \angle D = 60 ^{\circ }$ .

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