Back To Course

Math 102: College Mathematics15 chapters | 122 lessons | 13 flashcard sets

Are you a student or a teacher?

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 70,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*DaQuita Hester*

DaQuita has taught high school mathematics for six years and has a master's degree in secondary mathematics education.

Similar triangles have the same characteristics as similar figures but can be identified much more easily. Learn the shortcuts for identifying similar triangles here and test your ability with a quiz.

**Similar figures** are figures that have the same shape but are different sizes. They have congruent corresponding angles and proportional corresponding sides. **Similar triangles** are a type of similar figure, and determining their similarity is much easier thanks to the triangle similarity theorems. These theorems, which are Angle - Angle (AA), Side - Angle - Side (SAS) and Side - Side - Side (SSS), make it possible to determine triangle similarity with minimal calculations. Before we go any further, let's review key terms that will help these theorems make sense.

When parts of a figure are **corresponding**, this means that they are in the same location in each figure. Sides are **proportional** when the ratios between the corresponding sides are congruent. So, if you create ratios or fractions comparing all of the corresponding sides, each will have the same value and reduce to the same number.

When discussing congruent or similar figures, the **included angle** is the angle formed by the congruent or proportional sides. In the triangle below, since side AB is congruent to side DE and side BC is congruent to side EF, then angles B and E are the included angles.

Okay. Now that we're refreshed on vocabulary, let's further examine each of the similarity theorems.

We'll begin with Angle - Angle (AA). For two triangles to be similar by Angle - Angle (AA), two angles of one triangle are congruent to two angles of another triangle. Look at triangle JKL below. Angle J is 52 degrees and angle K is 60 degrees. If we subtract 52 and 60 from 180 (the total number of degrees that all angles in a triangle must add up to), we will see that angle L is 68 degrees.

Now, look at triangle MNO below. If angle M is congruent to angle J, and angle N is congruent to angle K, what can we say about angles L and O? We can say that they are congruent as well. With three pairs of congruent angles, the triangles are the same shape but different sizes, meaning that their sides are proportional and the triangles are similar.

For Side - Angle - Side (SAS), an angle of one triangle must be congruent to the corresponding angle of another triangle, and the lengths of the sides including these angles are in proportion. With congruent included angles, the proportional sides can't fluctuate, and the third side in both triangles must be a specific length. So, with two sides already proportional, the lengths of the third sides must also be proportional, proving triangle similarity.

The last theorem is Side - Side - Side (SSS), which means that the three sets of corresponding sides of two triangles are in proportion. If the ratios of all corresponding sides are equal, then the sides are similar and so are the triangles.

When determining which theorem proves similarity, don't overthink it; just look at the letters in each theorem. For triangles to be similar by Angle - Angle (AA), the measures of two angles in each triangle will be provided. If similar by Side - Angle - Side (SAS), then you will have the measures of two sides and the included angles of both triangles. For Side - Side - Side (SSS), you will have all three side lengths for both triangles. Let's practice.

Is triangle ABC similar to triangle DEF?

Let's examine the given information. We have the lengths for two sides in both triangles and the measures of the included angles. This sounds like Side - Angle - Side (SAS). But, before concluding similarity by this theorem, we must check for congruent angles and proportional sides.

Angle B and angle E both measure 63 degrees, so the included angles are congruent. To set up the side proportions, always compare the two smallest sides together and the two largest sides together, going in the same order between triangles. For this example, our ratios are 3/6 and 5/8. If we convert to decimals, 3/6 = .5 and 5/8 = .625. Since these ratios are not equal, triangle ABC is not similar to triangle DEF.

For our next example, determine if triangle RST is similar to triangle WXY.

Since we were given the lengths of all three sides in both triangles, Side - Side - Side (SSS) is the only theorem that can prove similarity. Let's set up our proportions. The two smaller sides have a ratio of 6/3, the largest sides have a ratio of 10/5 and the remaining sides have a ratio of 8/4. From simplifying, all of the ratios equal two. Therefore, triangle RST is similar to triangle WXY by the Side - Side - Side (SSS) similarity theorem.

Let's do one more. Is triangle CRE similar to triangle PHB?

Based on the given information, these triangles can only be congruent by Angle - Angle (AA). But, it appears that only one pair of angles are congruent. So, you may think the triangles aren't similar. Before coming to a conclusion, let's calculate the measure of angle E. Subtracting 180 - 40 - 95, we find that angle E measures 45 degrees. With this information, we now see that angle R and angle H are congruent, as well as angle E and angle B. Therefore, triangle CRE is similar to triangle PHB by the Angle - Angle (AA) similarity theorem.

Similar triangles possess the same characteristics as other similar figures: congruent corresponding angles and proportional corresponding sides. The triangle similarity theorems, which are Angle - Angle (AA), Side - Angle - Side (SAS) and Side - Side - Side (SSS), serve as shortcuts for identifying similar triangles. When assessing similarity, always begin by examining the provided information, which will help you figure out which theorem to use when determining if triangles are similar or not.

After completing this lesson, you'll be able to:

- Define similar figures
- Describe the characteristics required for triangles to be similar
- Explain the three triangle similarity theorems
- Determine whether two triangles are similar using Angle - Angle, Side - Angle - Side or Side - Side - Side theorems

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackDid you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
7 in chapter 14 of the course:

Back To Course

Math 102: College Mathematics15 chapters | 122 lessons | 13 flashcard sets

- Go to Logic

- Go to Sets

- Properties of Shapes: Rectangles, Squares and Rhombuses 5:46
- Properties of Shapes: Triangles 5:09
- Perimeter of Triangles and Rectangles 8:54
- Area of Triangles and Rectangles 5:43
- Circles: Area and Circumference 8:21
- The Pythagorean Theorem: Practice and Application 7:33
- How to Identify Similar Triangles 7:23
- Parallel, Perpendicular and Transverse Lines 6:06
- Types of Angles: Vertical, Corresponding, Alternate Interior & Others 10:28
- Angles and Triangles: Practice Problems 7:43
- Properties of Shapes: Circles 4:45
- Go to Geometry

- Understanding & Influencing Consumer Behavior
- DSST Ethics in Technology
- DSST Introduction to Geology: Practice & Study Guide
- Chemistry 304: Organic Chemistry II
- ILTS Information Guide
- Overview of the Vietnam War During the Nixon Years
- Psychosocial Aspects of Consumer Behavior
- Principles of Genetics
- Overview of the Solar System & Universe
- Marketing Research & Assessing Consumer Behavior
- Texas Teacher Certification Test Limit Waiver
- AFOQT Cost
- What Does the HESI A2 Nursing Exam Consist of?
- How to Learn Pharmacology for NCLEX
- What Are Considered Higher-Level Questions on the NCLEX?
- How to Study for NCLEx in 2 Weeks
- How Hard Is the ASVAB

- Promoting Physical Activity in School & the Community
- Why is System Analysis Important?
- Types of Fitness Testing in Schools
- Brain Contusion: Treatment, Recovery & Long-Term Effects
- Practical Application for Python: Using Print and Input
- What is a Network Scanner? - Definition & Use
- Internet of Things Platform: Definition & Forms
- Practical Application: Addressing Customer Service Complaints
- Quiz & Worksheet - Equitable Learning & Physical Education
- Quiz & Worksheet - The Three Factors of Movement
- Quiz & Worksheet - Teaching Materials for PE Programs
- Quiz & Worksheet - 2nd-Degree Burn Care
- Quiz & Worksheet - Social Awareness Overview
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies

- Writing Skills for Business
- Praxis Business Education - Content Knowledge (5101): Practice & Study Guide
- UExcel Principles of Management: Study Guide & Test Prep
- NES Biology (305): Practice & Study Guide
- Algebra for Teachers: Professional Development
- Using Source Materials - 11th Grade English: Tutoring Solution
- Physical Science - The Universe: Homework Help
- Quiz & Worksheet - Characteristics of Alcoholism
- Quiz & Worksheet - Attenuated Psychosis Syndrome
- Quiz & Worksheet - Fabliau
- Quiz & Worksheet - Transference in Psychotherapy
- Quiz & Worksheet - Assessing Body Fat Content

- What Is Mass Media? - Definition, Types, Influence & Examples
- What Is Methadone? - Treatment, Effects & Withdrawal Symptoms
- California Education Technology K-12 Voucher Program for Professional Development
- Election of 1860 Lesson Plan
- Missouri Alternative Teacher Certification
- Narrative Writing Lesson Plan
- Resources for Teachers of English Language Learners
- Middle School Reading List
- Fun Writing Prompts
- NATA Certification Requirements
- How to Pass the NAPLEX
- Common Core State Standards in Oregon

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject