Call Now to Set Up Tutoring:

(888) 888-0446

SSAT Upper Level Math : Geometry

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

Create An Account

All SSAT Upper Level Math Resources

6 Diagnostic Tests 311 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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,

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

Report an Error

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

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

Report an Error

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

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

Report an Error

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

Report an Error

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).

Report an Error

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.

Report an Error

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.

Report an Error

Example Question #581 : Geometry

$\bigtriangleup ABC \sim \bigtriangleup DEF$ ; AB = AC = 50, BC = 40 .

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

Possible Answers:

$\bigtriangleup DEF$ is isosceles and obtuse.

$\bigtriangleup DEF$ is scalene and obtuse.

$\bigtriangleup DEF$ is isosceles and acute.

$\bigtriangleup DEF$ is scalene and acute.

None of the other responses is correct.

Correct answer:

$\bigtriangleup DEF$ is isosceles and acute.

Explanation:

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

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

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

DE = DF

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;

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

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

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

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

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.

Report an Error

Example Question #142 : Properties Of Triangles

Which of the following statements would prove that the statement

$\bigtriangleup ABC \sim \bigtriangleup DEF$

is false?

Possible Answers:

$m \angle A + m \angle B \ne m \angle D + m \angle E$

$AB + BC \ne DE+EF$

$\bigtriangleup ABC$ and $\bigtriangleup DEF$ have different perimeters

None of the other statements alone would prove the statement $\bigtriangleup ABC \sim \bigtriangleup DEF$ to be false.

$\bigtriangleup ABC$ and $\bigtriangleup DEF$ have different areas

Correct answer:

$m \angle A + m \angle B \ne m \angle D + m \angle E$

Explanation:

Triangles that are similar need not have congruent sides, so it does not follow that AB + BC = DE+EF , or that their perimeters are equal. Consequently, their areas need not be equal either.

However, if $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then corresponding angles are congruent; specifically, $m \angle A = m \angle D$ and $m \angle B =m \angle E$ . Therefore, $m \angle A + m \angle B = m \angle D + m \angle E$ . Contrapositively, if $m \angle A + m \angle B \ne m \angle D + m \angle E$ , then $\bigtriangleup ABC \nsim \bigtriangleup DEF$ .

Report an Error

Example Question #15 : How To Find If Two Acute / Obtuse Triangles Are Similar

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

AB = 30, DE = 40, DF = 60, EF = 50, GI = 150 .

What is the ratio of the area of $\bigtriangleup GHI$ to that of $\bigtriangleup ABC$ ?

Possible Answers:

36:1

100:9

16:1

225:2

625:36

Correct answer:

100:9

Explanation:

The similarity ratio of two triangles is the ratio of the lengths of their corresponding sides.

The similarity ratio of $\bigtriangleup DEF$ to $\bigtriangleup ABC$ is

$\frac{DE}{AB} = \frac{40}{30} = \frac{4}{3}$ .

The similarity ratio of $\bigtriangleup GHI$ to $\bigtriangleup DEF$ is

$\frac{GI}{DF} = \frac{150}{60} = \frac{5}{2}$

Multipliy these to get the similarity ratio of $\bigtriangleup GHI$ to $\bigtriangleup ABC$ :

$\frac{4}{3} \times \frac{5}{2} = \frac{20}{6}= \frac{10}{3}$

The ratio of the areas of two similar figures is the square of their similarity ratio, so the ratio of the areas of the triangles is

$\left ( \frac{10}{3} \right )^{2} = \frac{100}{9}$ .

The correct choice is 100:9 .

Report an Error

View SSAT- Upper Level Tutors

Meredith
Certified Tutor

University of Arkansas at Little Rock, Bachelor in Arts, English. University of Arkansas at Little Rock, Masters in Education...

View SSAT- Upper Level Tutors

Liban
Certified Tutor

Emory University, Bachelor of Science, Mathematics/Economics.

View SSAT- Upper Level Tutors

Robert
Certified Tutor

University of North Carolina at Chapel Hill, Bachelor in Arts, English. North Carolina Central University, Master of Arts, En...

All SSAT Upper Level Math Resources

6 Diagnostic Tests 311 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Statistics Tutors in New York City, Statistics Tutors in Dallas Fort Worth, ISEE Tutors in Boston, ISEE Tutors in Los Angeles, Biology Tutors in Washington DC, GMAT Tutors in San Diego, Algebra Tutors in Phoenix, LSAT Tutors in Houston, Chemistry Tutors in Boston, MCAT Tutors in Philadelphia

Popular Courses & Classes

GRE Courses & Classes in New York City, GRE Courses & Classes in Atlanta, ACT Courses & Classes in Phoenix, ISEE Courses & Classes in Atlanta, SAT Courses & Classes in New York City, SAT Courses & Classes in Dallas Fort Worth, SAT Courses & Classes in Philadelphia, GMAT Courses & Classes in New York City, LSAT Courses & Classes in Boston, ACT Courses & Classes in New York City

Popular Test Prep

MCAT Test Prep in San Diego, ACT Test Prep in Phoenix, MCAT Test Prep in Atlanta, GMAT Test Prep in Miami, SAT Test Prep in Washington DC, MCAT Test Prep in San Francisco-Bay Area, ACT Test Prep in Houston, LSAT Test Prep in Atlanta, LSAT Test Prep in Denver, GMAT Test Prep in New York City

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

SSAT Upper Level Math : Geometry

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

All SSAT Upper Level Math Resources

Example Questions

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

Example Question #581 : Geometry

Example Question #142 : Properties Of Triangles

Example Question #15 : How To Find If Two Acute / Obtuse Triangles Are Similar

All SSAT Upper Level Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: