Call Now to Set Up Tutoring:

(888) 888-0446

Basic Geometry : Triangles

Study concepts, example questions & explanations for Basic Geometry

Create An Account

All Basic Geometry Resources

9 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Find If Right Triangles Are Congruent

Complete the congruence statement

$\bigtriangleup ABC \cong \square$

Possible Answers:

$\bigtriangleup CED$

$\bigtriangleup DEC$

$\bigtriangleup CDE$

$\bigtriangleup DCE$

$\bigtriangleup EDC$

Correct answer:

$\bigtriangleup DEC$

Explanation:

Since we know that $\overleftarrow{A}\overrightarrow{B}\parallel \overleftarrow{D}\overrightarrow{E}$ , we know that $\angle B$ is also a right angle and is thus congruent to $\angle D$ .

We are given that $\overline{BC}\cong \overline{CE}$ . Furthermore, since $\angle BCA$ and $\angle DCE$ are vertical angles, they are also congruent.

Therefore, we have enough evidence to conclude congruence by Angle-Side-Angle. Vertex matches up with , vertex matches up with , and matches up to . Thus, our congruence statement should look the following

$\bigtriangleup ABC\cong \bigtriangleup DEC$

Report an Error

Example Question #4 : How To Find If Right Triangles Are Congruent

Figures ABC and A'B'C' are triangles.

$\overline{AB}=\overline{A'B'}=\sqrt{3}, \overline{BC}=\overline{B'C'}=1, \overline{AC}=\overline{A'C'}=2$

Triangles_3

Are $\Delta ABC$ and $\Delta A'B'C'$ congruent?

Possible Answers:

There is not enough information given to answer this question.

Yes.

No.

Correct answer:

Yes.

Explanation:

We know that congruent triangles have equal corresponding angles and equal corresponding sides. We are given that the corresponding sides are equal and are in the ratio of $1:2:\sqrt{3}$ . A triangle whose sides are in this ratio is a $30^{\circ}-60^{\circ}-90^{\circ}$ , where the shortest side lies opposite the $30^{\circ}$ angle, the longest side is the hypotenuse and lies opposite the right angle, and the third side lies opposite the $60^{\circ}$ angle. (Remember $\sqrt{3}<2$ .) So we know the corresponding angles are equal. Therefore, the triangles are congruent.

Report an Error

Example Question #491 : Triangles

Figures ABC and A'B'C' are triangles.

$\overline{AB}=\overline{A'B'}=3, \overline{BC}=\overline{B'C'}=\sqrt{3}, \overline{AC}=\overline{A'C'}=2\sqrt{3}$

Triangles_3

Are $\Delta ABC$ and $\Delta A'B'C'$ congruent?

Possible Answers:

There is not enough information given to answer this question.

Yes.

No.

Correct answer:

Yes.

Explanation:

We know that congruent triangles have equal corresponding angles and equal corresponding sides. We are given that the corresponding sides are equal and are in the ratio of $\sqrt{3}:2\sqrt{3}:3$ .

Simplify the ratio by dividing by $\sqrt{3}$

$\frac{\sqrt{3}}{\sqrt{3}}:\frac{2\sqrt{3}}{\sqrt{3}}:\frac{3}{\sqrt{3}}$

$\frac{\sqrt{3}}{\sqrt{3}}=1$

$\frac{2\sqrt{3}}{\sqrt{3}}=2$

$\frac{3}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}=\frac{3\sqrt{3}}{3}=\sqrt{3}$

Thus, the corresponding sides are in the ratio $1:2:\sqrt{3}$ and we know both triangles are $30^{\circ}-60^{\circ}-90^{\circ}$ triangles. Since the corresponding angles and the corresponding sides are equal, the triangles are congruent.

Report an Error

Example Question #6 : How To Find If Right Triangles Are Congruent

Figures ABC and A'B'C' are triangles.

$m\angle A=30^{\circ}, m\angle B=90^{\circ}, m\angle B'=90^{\circ}, m\angle C'=60^{\circ}$

$\overline{AB}=\overline{A'B'}=3, \overline{BC}=\overline{B'C'}=\sqrt{3}, \overline{AC}=\overline{A'C'}=2\sqrt{3}$

Triangles_3

Are $\Delta ABC$ and $\Delta A'B'C'$ congruent?

Possible Answers:

Yes.

There is not enough information given to answer this question.

No.

Correct answer:

Yes.

Explanation:

We know that congruent triangles have equal corresponding angles and equal corresponding sides. We are given that the corresponding sides are equal, and the measures of two angles. We know that $\angle C=60^{\circ}$ and $\angle A'=30^{\circ}$ because the sum of the angles of a triangle must equal $180^{\circ}$ . So the corresponding angles are also equal. Therefore, the triangles are congruent.

Report an Error

Example Question #7 : How To Find If Right Triangles Are Congruent

Figures ABC and A'B'C' are triangles.

$m\angle A=45^\circ, m\angle B=90^\circ, m\angle B'=90^\circ, m\angle C'=45^\circ$

$\overline{AB}=\overline{BC}=\overline{A'B'}=\overline{B'C'}=1, \overline{AC}=\overline{A'C'}=\sqrt{2}$

Triangles_4

Are $\Delta ABC$ and $\Delta A'B'C'$ congruent?

Possible Answers:

Yes.

There is not enough information given to answer this question.

No.

Correct answer:

Yes.

Explanation:

We know that congruent triangles have equal corresponding angles and equal corresponding sides. We are given that the corresponding sides are equal, and the measures of two angles. We know that $\angle C=45^{\circ}$ and $\angle A'=45^{\circ}$ because the sum of the angles of a triangle must equal $180^{\circ}$ . So the corresponding angles are also equal. Therefore, the triangles are congruent.

Report an Error

Example Question #4 : How To Find If Right Triangles Are Congruent

Figures ABC and A'B'C' are triangles.

$\overline{AB}=\overline{A'B'}=\overline{BC}=\overline{B'C'}=1, \overline{AC}=\overline{A'C'}=\sqrt{2}$

Triangles_4

Are $\Delta ABC$ and $\Delta A'B'C'$ congruent?

Possible Answers:

Yes.

There is not enough information given to answer this question.

No.

Correct answer:

Yes.

Explanation:

We know that congruent triangles have equal corresponding angles and equal corresponding sides. We are given that the corresponding sides are equal and are in the ratio of $1:1:\sqrt{2}$ . A triangle whose sides are in this ratio is a $45^{\circ}-45^{\circ}-90^{\circ}$ , where the shorter sides lies opposite the $45^{\circ}$ angles, and the longer side is the hypotenuse and lies opposite the right angle. So we know the corresponding angles are equal. Therefore, the triangles are congruent.

Report an Error

Example Question #2 : How To Find If Right Triangles Are Congruent

Figures ABC and A'B'C' are triangles.

$\overline{AB}=\overline{A'B'}=\overline{BC}=\overline{B'C'}=\sqrt{2}, \overline{AC}=\overline{A'C'}=2$

Triangles_4

Are $\Delta ABC$ and $\Delta A'B'C'$ congruent?

Possible Answers:

Yes.

There is not enough information given to answer this question.

No.

Correct answer:

Yes.

Explanation:

We know that congruent triangles have equal corresponding angles and equal corresponding sides. We are given that the corresponding sides are equal and are in the ratio of $\sqrt{2}:\sqrt{2}:2$ .

Simplify the ratio by dividing by $\sqrt{2}$

$\frac{\sqrt{2}}{\sqrt{2}}:\frac{\sqrt{2}}{\sqrt{2}}:\frac{2}{\sqrt{2}}$

$\frac{\sqrt{2}}{\sqrt{2}}=1$

$\frac{2}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}=\frac{2\sqrt{2}}{2}=\sqrt{2}$

Thus, the corresponding sides are in the ratio $1:1:\sqrt{2}$ and we know both triangles are $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Since the corresponding angles and the corresponding sides are equal, the triangles are congruent.

Report an Error

Example Question #1481 : Plane Geometry

Are these right triangles congruent?

Congruent right triangles

Possible Answers:

Cannot be determined - we need at least one pair of angles, or all three sides

No - at least one pair of corresponding sides is not congruent

No - the angles are different

Yes - all three pairs of sides must be congruent by Pythagorean Theorem

Yes - by the angle-angle-side theorem

Correct answer:

Yes - all three pairs of sides must be congruent by Pythagorean Theorem

Explanation:

Right now we can't directly compare these triangles because we do not know all three side lengths. However, we can use Pythagorean Theorem to determine both missing sides. The left triangle is missing the hypotenuse:

$\small 48^2 + 55^2 = c^2$

$\small 2,304 + 3,025 = c^2$

$\small 5,329 = c$

$\small c = \sqrt{5,329}= 73$

The right triangle is missing one of the legs:

$\small x^2 + 48^2 = 78^2$

$\small x^2 + 2,304 = 5,329$ subtract 2,304 from both sides

$\small x^2 = 3,025$

$\small x = \sqrt{3,025} = 55$

This means that the two triangles both have side lengths 48, 55, 73, so they must be congruent.

Report an Error

Example Question #1481 : Basic Geometry

The hypotenuse and acute angle are given for several triangles. Which if any are congruent? Triangle A- Hypotenuse=15; acute angle=56 degrees. Triangle B- Hypotenuse=18; acute angle=56 degrees. Triangle C-Hypotenuse=18; acute angle= 45 degrees.

Possible Answers:

A & C

All three.

B & C

None of these

A & B

Correct answer:

None of these

Explanation:

The correct answer is none of these. There are several pairs of angles and sides or sides and angles that must be the same in order for two triangles to be congruent.

In our case, we need the acute angle and the hypotenuse to both be equal. No two triangles above have this relationship and therefore no two are congruent.

Report an Error

Example Question #1483 : Basic Geometry

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$\angle B$ and $\angle E$ are both right angles.

$\overline{AB } \cong \overline{DE}$

$\overline{AC} \cong \overline{DF}$

True or false: From the above information, it follows that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Possible Answers:

True

False

Correct answer:

True

Explanation:

If we seek to prove that $\bigtriangleup ABC \cong \bigtriangleup DEF$ , then , , and correspond to , , and , respectively.

By the Hypotenuse-Leg Theorem (HL), if the hypotenuse and one leg of a triangle are congruent to those of another, the triangles are congruent.

$\angle B$ and $\angle E$ are both right angles, so $\bigtriangleup ABC$ and $\bigtriangleup DEF$ are both right triangles. $\overline{AB }$ and $\overline{DE}$ are congruent corresponding sides, and moreover, since, each includes the right-angle vertex as an endpoint, they are congruent corresponding legs. $\overline{AC}$ and $\overline{DF}$ are opposite the right angles, making them congruent corresponding hypotenuses.

The conditions of HL are satisfied, so $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Report an Error

View Tutors

Johnathan
Certified Tutor

Central Connecticut State University, Bachelor of Science, Geology. Montana Tech of the University of Montana, Master of Scie...

View Tutors

Cara
Certified Tutor

Winston-Salem State University, Bachelor of Science, Computer and Information Sciences, General. Walden University, Masters i...

View Tutors

Pankaj
Certified Tutor

University of Delhi, Electrical Engineer, Electrical Engineering. Columbia University in the City of New York, Master of Arts...

All Basic Geometry Resources

9 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Physics Tutors in Chicago, SSAT Tutors in Los Angeles, SSAT Tutors in Washington DC, Chemistry Tutors in Los Angeles, Biology Tutors in San Diego, GRE Tutors in Denver, Biology Tutors in Boston, Math Tutors in San Diego, GRE Tutors in Washington DC, GRE Tutors in Houston

Popular Courses & Classes

Spanish Courses & Classes in Los Angeles, SAT Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Philadelphia, GRE Courses & Classes in Los Angeles, Spanish Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in New York City, ISEE Courses & Classes in Seattle, MCAT Courses & Classes in San Diego, GMAT Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Atlanta

Popular Test Prep

SAT Test Prep in Miami, ISEE Test Prep in Dallas Fort Worth, SAT Test Prep in Dallas Fort Worth, GMAT Test Prep in Dallas Fort Worth, SSAT Test Prep in Chicago, SAT Test Prep in Los Angeles, SAT Test Prep in Houston, GRE Test Prep in Los Angeles, SSAT Test Prep in Phoenix, MCAT Test Prep in Los Angeles

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

Basic Geometry : Triangles

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #1 : How To Find If Right Triangles Are Congruent

Example Question #4 : How To Find If Right Triangles Are Congruent

Example Question #491 : Triangles

Example Question #6 : How To Find If Right Triangles Are Congruent

Example Question #7 : How To Find If Right Triangles Are Congruent

Example Question #4 : How To Find If Right Triangles Are Congruent

Example Question #2 : How To Find If Right Triangles Are Congruent

Example Question #1481 : Plane Geometry

Example Question #1481 : Basic Geometry

Example Question #1483 : Basic Geometry

All Basic Geometry Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: