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If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Did you know that there are five ways you can prove triangle congruency?  Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher) It’s true! In today’s geometry lesson, we’re going to tackle two of them, the Side-Side-Side and Side-Angle-Side postulates. You’ll quickly learn how to prove triangles are congruent using these methods. In addition, you’ll see how to write the associated two column proof. Let’s jump in! So we already know, two triangles are congruent if they have the same size and shape. This means that the pair of triangles have the same three sides and the same three angles (i.e., a total of six corresponding congruent parts). Thankfully we don’t need to prove all six corresponding parts are congruent… we just need three! Why?
But there is a warning; we must be careful about identifying the accurate side and angle relationships! As Math is Fun accurately states, there only five different congruence postulates that will work for proving triangles congruent. So we need to learn how to identify congruent corresponding parts correctly and how to use them to prove two triangles congruent. Triangle Congruence PostulatesThe first two postulates, Side-Angle-Side (SAS) and the Side-Side-Side (SSS), focus predominately on the side aspects, whereas the next lesson discusses two additional postulates which focus more on the angles. Those are the Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS) postulates. Side-Angle-SideIf we can show that two sides and the included angle of one triangle are congruent to two sides and the included angle in a second triangle, then the two triangles are congruent. This is called the Side Angle Side Postulate or SAS. And as seen in the image, we prove triangle ABC is congruent to triangle EDC by the Side-Angle-Side Postulate
SAS Postulate Example Side-Side-SideOr, if we can determine that the three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. We refer to this as the Side Side Side Postulate or SSS. And as seen in the image to the right, we show that trianlge ABC is congruent to triangle CDA by the Side-Side-Side Postulate.
SSS Postulate Example I’m confident that after watching this lesson you will agree with me that proving triangles congruent is fun and straightforward. Triangle Congruency – Lesson & Examples (Video)38 min
Get access to all the courses and over 450 HD videos with your subscription Monthly and Yearly Plans Available Get My Subscription Now You are probably familiar with what it means to be included: You are a part of the group; you belong. Angles and line segments aren't much different. Line segments can include an angle, and angles can include a line segment. The two sides of a triangle that form an angle are said to include that angle of the triangle. Similarly, any two angles of a triangle must have a common side, and these two angles are said to include that side. For example, Figure 12.3 shows a picture of ABC. ¯AC and ¯CB include C, and A and B include ¯AB.
Figure 12.3In ABC, ¯AC and ¯CB include C , and A and B include ¯AB. You can use included angles and line segments to prove that two triangles are congruent.
The order of the letters in the name SAS Postulate will help you remember that the two sides that are named actually form the angle.
Figure 12.4¯PN ¯MQ and ¯MN ~= ¯NQ.
You will see several theorems about isosceles triangles. Most of these will be proven using the SAS postulate. For example, if ABC is an isosceles triangle with ¯AB ~= ¯BC , you can show that ABC ~= CBA by SAS. Thus A ~= C by CPOCTAC. These are the angles opposite the congruent sides in ABC. This is the first of many theoremsabout isosceles triangles.
Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc. To order this book direct from the publisher, visit the Penguin USA website or call 1-800-253-6476. You can also purchase this book at Amazon.com and Barnes & Noble.
Congruent triangles are triangles with identical sides and angles. The three sides of one are exactly equal in measure to the three sides of another. The three angles of one are each the same angle as the other. Triangle Congruence PostulatesFive ways are available for finding two triangles congruent:
Included PartsAn included angle lies between two named sides. In △CAT below, included ∠A is between sides t and c: An included side lies between two named angles of the triangle. Side Side Side PostulateA postulate is a statement taken to be true without proof. The SSS Postulate tells us,
If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. Congruence of sides is shown with little hatch marks, like this: ∥. For two triangles, sides may be marked with one, two, and three hatch marks. If △ACE has sides identical in measure to the three sides of △HUM, then the two triangles are congruent by SSS: Side Angle Side PostulateThe SAS Postulate tells us,
If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. △HUG and △LAB each have one angle measuring exactly 63°. Corresponding sides g and b are congruent. Sides h and l are congruent. A side, an included angle, and a side on △HUG and on △LAB are congruent. So, by SAS, the two triangles are congruent. Angle Side Angle PostulateThis postulate says,
If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. We have △MAC and △CHZ, with side m congruent to side c. ∠A is congruent to ∠H, while ∠C is congruent to ∠Z. By the ASA Postulate these two triangles are congruent.
Angle Angle Side TheoremWe are given two angles and the non-included side, the side opposite one of the angles. The Angle Angle Side Theorem says,
If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. Here are congruent △POT and △LID, with two measured angles of 56° and 52°, and a non-included side of 13 centimeters: [construct as described] By the AAS Theorem, these two triangles are congruent. HL PostulateExclusively for right triangles, the HL Postulate tells us,
Two right triangles that have a congruent hypotenuse and a corresponding congruent leg are congruent. The hypotenuse of a right triangle is the longest side. The other two sides are legs. Either leg can be congruent between the two triangles. Here are right triangles △COW and △PIG, with hypotenuses of sides w and i congruent. Legs o and g are also congruent: [insert congruent right triangles left-facing △COW and right facing △PIG] So, by the HL Postulate, these two triangles are congruent, even if they are facing in different directions. Proof Using CongruenceGiven: △MAG and △ICG MC ≅ AI AG ≅ GI Prove: △MAG ≅ △ICG Statement Reason MC ≅ AI Given AG ≅ GI ∠MGA ≅ ∠ IGC Vertical Angles are Congruent △MAG ≅ △ICG Side Angle Side If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. Next Lesson:Triangle Congruence Theorems
Instructor: Malcolm M. |
What other information must be known to prove congruence of the two triangles by SAS postulate?
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