How to find if two acute / obtuse triangles are similar

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SSAT Upper Level Quantitative › How to find if two acute / obtuse triangles are similar

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$\bigtriangleup ABC \sim \bigtriangleup DEF$ ; ; $\bigtriangleup DEF$ has perimeter 400.

Which of the following is equal to $DF \div DE$ ?

CORRECT

$0.\overline{6}$

$0.8\overline{3}$

Explanation

The perimeter of $\bigtriangleup DEF$ is actually irrelevant to this problem. Corresponding sides of similar triangles are in proportion, so use this to calculate $DF \div DE$ , or $\frac{DF}{DE}$ :

$\frac{DF}{DE} = \frac{AC}{AB} = \frac{36}{24} = 1.5$

$\bigtriangleup ABC \sim \bigtriangleup DEF$ ; ; ; $\bigtriangleup DEF$ has perimeter 90.

Give the perimeter of $\bigtriangleup ABC$ .

CORRECT

$53\frac{1}{3}$

Explanation

The ratio of the perimeters of two similar triangles is the same as the ratio of the lengths of a pair of corresponding sides. Therefore,

$\frac{P_{\bigtriangleup ABC }}{P_{\bigtriangleup DEF}} = \frac{AB}{DE}$

$\frac{P_{\bigtriangleup ABC }}{90} = \frac{12}{40}$

$\frac{P_{\bigtriangleup ABC }}{90}\cdot 90 = \frac{12}{40} \cdot 90$

$P_{\bigtriangleup ABC} = 27$

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ ; $\angle A \cong \angle D$ and $\angle C \cong \angle F$ .

Which of the following statements would not be enough, along with what is given, to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ ?

The given information is enough to prove the triangles similar.

CORRECT

$\frac{AB}{DE} = \frac{AC}{DF}$

$\frac{AB}{DE} = \frac{BC}{EF}$

$\frac{AC}{DF}= \frac{BC}{EF}$

$\angle B \cong \angle E$

Explanation

Two pairs of corresponding angles are stated to be congruent in the main body of the problem; it follows from the Angle-Angle Similarity Postulate that the triangles are similar. No further information is needed.

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ ; $\frac{AB}{DE} = \frac{AC}{DF}$ .

Which of the following statements would not be enough, along with what is given, to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ ?

$\angle C \cong \angle F$

CORRECT

$\frac{AB}{DE} = \frac{BC}{EF}$

$\angle A \cong \angle D$

$\frac{AC}{DF}= \frac{BC}{EF}$

The given information is enough to prove the triangles similar.

Explanation

From both the given proportion statement and either $\frac{AB}{DE} = \frac{BC}{EF}$ or $\frac{AC}{DF}= \frac{BC}{EF}$ , it follows that $\frac{AB}{DE} =\frac{AC}{DF}= \frac{BC}{EF}$ —all three pairs of corresponding sides are in proportion; by the Side-Side-Side Similarity Theorem, $\bigtriangleup ABC \sim \bigtriangleup DEF$ . From the given proportion statement and $\angle A \cong \angle D$ , since these are the included angles of the sides that are in proportion, then by the Side-Angle-Side Similarity Theorem, $\bigtriangleup ABC \sim \bigtriangleup DEF$ . From the given proportion statement and $\angle C \cong \angle F$ , since these are nonincluded angles of the sides that are in proportion, no similarity can be deduced.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ .

$m\angle A + m\angle D = 94^{\circ }$

$m\angle B + m\angle F = 94^{\circ }$

Evaluate $m \angle C + m \angle E$ .

$172^{\circ }$

CORRECT

$188^{\circ }$

$94^{\circ }$

$86^{\circ }$

These triangles cannot exist.

Explanation

The similarity of the triangles is actually extraneous information here. The sum of the measures of a triangle is $180 ^{\circ }$ , so:

$m \angle A + m \angle B + m \angle C = 180^{\circ }$

$m \angle D + m \angle E + m \angle F = 180^{\circ }$

$m \angle A + m \angle B + m \angle C +m \angle D + m \angle E + m \angle F = 180^{\circ }+ 180^{\circ }$

$m \angle C + m \angle E + (m \angle A +m \angle D) + (m \angle B + m \angle F )= 360^{\circ }$

$m \angle C + m \angle E +94 ^{\circ } +94 ^{\circ }= 360^{\circ }$

$m \angle C + m \angle E +188 ^{\circ }= 360^{\circ }$

$m \angle C + m \angle E +188 ^{\circ }- 188^{\circ } = 360^{\circ } - 188^{\circ }$

$m \angle C + m \angle E = 172^{\circ }$

$\bigtriangleup ABC \sim \bigtriangleup DEF$ ; ; $\bigtriangleup DEF$ has perimeter 300.

Evaluate .

CORRECT

Insufficient information is given to answer the problem.

Explanation

The ratio of the perimeters of two similar triangles is equal to the ratio of the lengths of a pair of corresponding sides. Therefore,

$\frac{DE}{AB}= \frac{P_{\bigtriangleup DEF}}{P_{\bigtriangleup ABC }}$ and $\frac{EF}{BC}= \frac{P_{\bigtriangleup DEF}}{P_{\bigtriangleup ABC }}$ , or

$\frac{DE}{AB}=\frac{EF}{BC} = \frac{P_{\bigtriangleup DEF}}{P_{\bigtriangleup ABC }}$

By one of the properties of proportions, it follows that

$\frac{DE+EF}{AB+BC} = \frac{P_{\bigtriangleup DEF}}{P_{\bigtriangleup ABC }}$

The perimeter of $\bigtriangleup ABC$ is

, so

$\frac{DE+EF}{24+20} = \frac{300}{80}$

$\frac{DE+EF}{44} = \frac{300}{80}$

$\frac{DE+EF}{44} \cdot 44 = \frac{300}{80} \cdot 44$

$\bigtriangleup ABC \sim \bigtriangleup DEF$ ; $m \angle A = 74 ^{\circ }, m \angle E = 32^{\circ }$

Which of the following is true about $\bigtriangleup DEF$ ?

$\bigtriangleup DEF$ is isosceles and acute.

CORRECT

$\bigtriangleup DEF$ is scalene and obtuse.

$\bigtriangleup DEF$ is isosceles and obtuse.

$\bigtriangleup DEF$ is scalene and acute.

None of the other responses is correct.

Explanation

Corresponding angles of similar triangles are congruent, so the measures of the angles of $\bigtriangleup ABC$ are equal to those of $\bigtriangleup DEF$ .

$\angle D \cong \angle A$ , so $m \angle D = m \angle A = 74 ^{\circ }$ . Also $m \angle E = 32^{\circ }$ , so

$m \angle F = 180^{\circ } - (m \angle D + m \angle E )$ .

$= 180^{\circ } - (74^{\circ }+ 32^{\circ } )$

$= 180^{\circ } - 106^{\circ }$

$= 74 ^{\circ }$

All three angles have measure less than $90^{\circ }$ , so $\bigtriangleup DEF$ is acute. Also, two of the angles are congruent, so by the Converse of the Isosceles Triangle Theorem, $\bigtriangleup DEF$ is isosceles.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ ; .

Which of the following is true about $\bigtriangleup DEF$ ?

$\bigtriangleup DEF$ is isosceles and acute.

CORRECT

$\bigtriangleup DEF$ is isosceles and obtuse.

$\bigtriangleup DEF$ is scalene and acute.

$\bigtriangleup DEF$ is scalene and obtuse.

None of the other responses is correct.

Explanation

$\bigtriangleup ABC \sim \bigtriangleup DEF$ , so corresponding sides are in proportion; it follows that

$\frac{DE}{AB} = \frac{DF}{AC}$

$\frac{DE}{50} = \frac{DF}{50 }$

Therefore, $\bigtriangleup DEF$ is isosceles.

Also, corresponding angles are congruent, so if $\bigtriangleup ABC$ acute (or obtuse), so is $\bigtriangleup DEF$ . We can compare the sum of the squares of the lesser two sides to that of the greatest;

$40^{2}+50^{2}; \square ; 50^{2}$

$1,600+2,500; \square ; 2,500$

$4,100; \square ; 2,500$

The sum of the squares of the lesser sides is greater than the square of the greatest side, so $\bigtriangleup ABC$ is acute - and so is $\bigtriangleup DEF$ . The correct response is that $\bigtriangleup DEF$ is isosceles and acute.

$\bigtriangleup ABC \sim \bigtriangleup DEF$ ; $m \angle A = 66^{\circ }$ ; $m \angle F = 63^{\circ }$ .

Which of the following correctly gives the relationship of the angles of $\bigtriangleup ABC$ ?

$m \angle B < m \angle C <m \angle A$

CORRECT

$m \angle C <m \angle B < m \angle A$

$m \angle B = m \angle C <m \angle A$

$m \angle C =m \angle B < m \angle A$

$m \angle C <m \angle B = m \angle A$

Explanation

$m \angle A = 66^{\circ }$

Corresponding angles of similar triangles are congruent, so $m \angle C =m \angle F = 63^{\circ }$ .

Consequently,

$m \angle B = 180^{\circ } - (m \angle A +m \angle C)$

$m \angle B = 180^{\circ } - ( 66^{\circ }+ 63^{\circ })$

$m \angle B = 180^{\circ } - 129^{\circ } = 51^{\circ }$

Therefore,

$m \angle B < m \angle C <m \angle A$ .

$\bigtriangleup ABC \sim \bigtriangleup DEF$ ; $m \angle D = 44 ^{\circ }, m \angle E = 26^{\circ }$

Which of the following is true about $\bigtriangleup ABC$ ?

$\bigtriangleup ABC$ is scalene and obtuse.

CORRECT

$\bigtriangleup ABC$ is scalene and acute.

$\bigtriangleup ABC$ is isosceles and obtuse.

$\bigtriangleup ABC$ is isosceles and acute.

None of the other responses is correct.

Explanation

Corresponding angles of similar triangles are congruent, so the measures of the angles of $\bigtriangleup ABC$ are equal to those of $\bigtriangleup DEF$ .

Two of the angles of $\bigtriangleup DEF$ have measures $44 ^{\circ }$ and $26^{\circ }$ ; its third angle measures

$180^{\circ } - (44^{\circ }+ 26^{\circ } ) = 180^{\circ } - 70 ^{\circ } = 110^{\circ }$ .

One of the angles having measure greater than $90^{\circ }$ makes $\bigtriangleup DEF$ - and, consequently, $\bigtriangleup ABC$ - an obtuse triangle. Also, the three angles have different measures, so the sides do as well, making $\bigtriangleup ABC$ scalene.

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