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SSAT Upper Level Math : SSAT Upper Level Quantitative (Math)

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Example Questions

Example Question #801 : Ssat Upper Level Quantitative (Math)

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 $\bigtriangleup ABC \sim \bigtriangleup DEF$ ?

Possible Answers:

The given information is enough to prove the triangles similar.

$\angle B \cong \angle E$

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

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

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

Correct answer:

The given information is enough to prove the triangles similar.

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.

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Example Question #4 : How To Find If Two Acute / Obtuse Triangles Are Similar

$\bigtriangleup ABC \sim \bigtriangleup DEF$ . $\frac{AB}{DE} = \frac{25}{36}$ . Which of the following is the ratio of the area of $\bigtriangleup DEF$ to that of $\bigtriangleup ABC$ ?

Possible Answers:

$\frac{6}{5}$

$\frac{1,296}{625}$

$\frac{36}{25}$

$\frac{625}{1,296}$

$\frac{5}{6}$

Correct answer:

$\frac{1,296}{625}$

Explanation:

The similarity ratio of $\bigtriangleup ABC$ to $\bigtriangleup DEF$ is equal to the ratio of two corresponding sidelengths, which is given as $\frac{25}{36}$ ; the similarity ratio of $\bigtriangleup DEF$ to $\bigtriangleup ABC$ is the reciprocal of this, or $\frac{36}{25}$ .

The ratio of the area of a figure to that of one to which it is similar is the square of the similarity ratio, so the ratio of the area of $\bigtriangleup DEF$ to that of $\bigtriangleup ABC$ is

$\left (\frac{36}{25} \right ) ^{2}= \frac{1,296}{625}$

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Example Question #5 : How To Find If Two Acute / Obtuse Triangles Are Similar

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

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

Possible Answers:

$\bigtriangleup ABC$ is scalene and acute.

None of the other responses is correct.

$\bigtriangleup ABC$ is isosceles and acute.

$\bigtriangleup ABC$ is scalene and obtuse.

$\bigtriangleup ABC$ is isosceles and obtuse.

Correct answer:

$\bigtriangleup ABC$ is scalene and obtuse.

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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Example Question #6 : How To Find If Two Acute / Obtuse Triangles Are Similar

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

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

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,

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

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Example Question #1 : How To Find If Two Acute / Obtuse Triangles Are Similar

$\bigtriangleup ABC \sim \bigtriangleup DEF$ ; AB = 50, BC = DE= 75, DF = 100 .

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Corresponding sides of similar triangles are in proportion; since $\bigtriangleup ABC \sim \bigtriangleup DEF$ ,

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

$\frac{AC}{100} = \frac{50}{75}$

$\frac{AC}{100} \cdot 100 = \frac{50}{75} \cdot 100$

$AC = 66\frac{2}{3}$

Therefore, AB < AC < BC .

The angle opposite the longest (shortest) side of a triangle is the angle of greatest (least) measure, so

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

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Example Question #1 : How To Find If Two Acute / Obtuse Triangles Are Similar

$\bigtriangleup ABC \sim \bigtriangleup DEF \sim \bigtriangleup GHI$ .

AB = DF=HI = 24

AC = 32

GH = 40

Order the triangles by perimeter, least to greatest.

Possible Answers:

$\bigtriangleup ABC, \bigtriangleup GHI,\bigtriangleup DEF$

$\bigtriangleup DEF,\bigtriangleup GHI, \bigtriangleup ABC$

$\bigtriangleup ABC,\bigtriangleup DEF, \bigtriangleup GHI$

$\bigtriangleup GHI,\bigtriangleup DEF,\bigtriangleup ABC$

$\bigtriangleup DEF, \bigtriangleup ABC,\bigtriangleup GHI$

Correct answer:

$\bigtriangleup DEF, \bigtriangleup ABC,\bigtriangleup GHI$

Explanation:

$\overline{DF}$ and $\overline{AC}$ are corresponding sides of their respective triangles, and DF < AC , so it easily follows from proportionality that each side of $\bigtriangleup DEF$ is shorter than its corresponding side in $\bigtriangleup ABC$ . Therefore, $\bigtriangleup DEF$ is of lesser perimeter than $\bigtriangleup ABC$ . By the same reasoning, since AB < GH , $\bigtriangleup ABC$ is of lesser perimeter than $\bigtriangleup GHI$ .

The correct response is

$\bigtriangleup DEF, \bigtriangleup ABC,\bigtriangleup GHI$

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Example Question #11 : How To Find If Two Acute / Obtuse Triangles Are Similar

$\bigtriangleup ABC \sim \bigtriangleup DEF \sim \bigtriangleup GHI$ .

AB = 20, AC = 30, BC = 25,DE = 50, HI = DF .

Evaluate .

Possible Answers:

100

Correct answer:

Explanation:

Corresponding sides of similar triangles are in proportion, so

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

$\frac{DF}{30} = \frac{50}{20}$

$\frac{DF}{30} \cdot 30 = \frac{50}{20}\cdot 30$

DF = 75

Therefore, HI = 75 as well.

Again, by similarity,

$\frac{GH}{AB} = \frac{HI}{BC}$

$\frac{GH}{20} = \frac{75}{25}$

$\frac{GH}{20}\cdot 20 = \frac{75}{25} \cdot 20$

GH = 60

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Example Question #141 : Properties Of Triangles

$\bigtriangleup ABC \sim \bigtriangleup DEF \sim \bigtriangleup GHI$ .

$\angle A \cong \angle I$ .

Which of the following scenarios is possible ?

I) $\bigtriangleup DEF$ is an acute triangle.

II) $\bigtriangleup DEF$ is an obtuse triangle with $\angle D$ the obtuse angle.

III) $\bigtriangleup DEF$ is an obtuse triangle with $\angle E$ the obtuse angle.

IV) $\bigtriangleup DEF$ is an obtuse triangle with $\angle F$ the obtuse angle.

Possible Answers:

II, III, and IV only

II and IV only

I and III only

I, II, III, and IV

I only

Correct answer:

I and III only

Explanation:

Corresponding angles of similar triangles are congruent, so $\angle A \cong \angle D$ and $\angle F \cong \angle I$ . Since $\angle A \cong \angle I$ , it follows that $\angle D \cong \angle F$ . Since a triangle cannot have two angles that measure more than $90^{\circ }$ , both $\angle D$ and $\angle F$ are acute. No information is given about $\angle E$ , so $\angle E$ can be acute, right, or obtuse. Therefore, scenarios (I) and (III) are possible, but not (II) or (IV).

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Example Question #13 : How To Find If Two Acute / Obtuse Triangles Are Similar

$\bigtriangleup ABC \sim \bigtriangleup DEF$ ; AB = 20, BC=DE=40, DF=80 .

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

Possible Answers:

None of the other responses is correct.

$\bigtriangleup ABC$ is scalene and obtuse.

$\bigtriangleup ABC$ is scalene and acute.

$\bigtriangleup ABC$ is isosceles and acute.

$\bigtriangleup ABC$ is isosceles and obtuse.

Correct answer:

$\bigtriangleup ABC$ is isosceles and acute.

Explanation:

Corresponding sides of similar triangles are in proportion, so

$\frac{AC}{DF} = \frac{AB }{DE}$ and $\frac{AB }{DE}= \frac{BC}{EF}$ .

Substituting, we have from the first statement

$\frac{AC}{80} = \frac{20}{40}$

$\frac{AC}{80} \cdot 80 = \frac{20}{40} \cdot 80$

AC = 40

Since AC =BC = 40 , $\bigtriangleup ABC$ is isosceles.

We can compare the sum of the squares of the lesser two sides to that of the greatest.

$20^{2}+40^{2}\; \square \; 40^{2}$

$400+ 1,600 \; \square \; 1,600$

$2,000 \; \square \; 1,600$

2,000 > 1,600

The sum of the squares of the lesser two sides is greater than the square of the third, so $\bigtriangleup ABC$ is acute.

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Example Question #51 : Acute / Obtuse Triangles

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

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

Possible Answers:

$\bigtriangleup DEF$ is scalene and acute.

None of the other responses is correct.

$\bigtriangleup DEF$ is scalene and obtuse.

$\bigtriangleup DEF$ is isosceles and obtuse.

$\bigtriangleup DEF$ is isosceles and acute.

Correct answer:

$\bigtriangleup DEF$ is isosceles and acute.

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.

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SSAT Upper Level Math : SSAT Upper Level Quantitative (Math)

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #801 : Ssat Upper Level Quantitative (Math)

Example Question #4 : How To Find If Two Acute / Obtuse Triangles Are Similar

Example Question #5 : How To Find If Two Acute / Obtuse Triangles Are Similar

Example Question #6 : How To Find If Two Acute / Obtuse Triangles Are Similar

Example Question #1 : How To Find If Two Acute / Obtuse Triangles Are Similar

Example Question #1 : How To Find If Two Acute / Obtuse Triangles Are Similar

Example Question #11 : How To Find If Two Acute / Obtuse Triangles Are Similar

Example Question #141 : Properties Of Triangles

Example Question #13 : How To Find If Two Acute / Obtuse Triangles Are Similar

Example Question #51 : Acute / Obtuse Triangles

All SSAT Upper Level Math Resources

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