Geometry: Congruent Triangles



In this lesson, we will learn

  • congruent triangles
  • how to tell if triangles are congruent using SSS, SAS, ASA and AAS rules
  • why AAA and SSA does not work as congruence shortcuts
  • the Hypotenuse Leg Rule for right triangles
  • how to use CPCTC (corresponding parts of congruent triangles are congruent)

Congruent Triangles

Congruent triangles are triangles that have the same size and shape. This means that the corresponding sides are equal and the corresponding angles are equal.

comparing triangles

In the above diagrams, the corresponding sides are a and d; b and e ; c and f.

The corresponding angles are x and s; y and t; z and u.


How to tell if triangles are congruent?

We can tell whether two triangles are congruent without testing all the sides and all the angles of the two triangles. There are four rules to check for congruent triangles. They are called the SSS rule, SAS rule, ASA rule and AAS rule. There is also another rule for right triangles called the Hypotenuse Leg rule. As long as one of the rules is true, it is sufficient to prove that the two triangles are congruent.

SSS Rule

The Side-Side-Side (SSS) rule states that

If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

SAS Rule

The Side-Angle-Side (SAS) rule states that

If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent.

An included angle is the angle formed by the two given sides.

included and non-included angle
    Included Angle                         Non-included angle

ASA Rule

The Angle-Side-Angle (ASA) Rule states that

If two angles and the included side of one triangle are equal to two angles and included side of another triangle, then the triangles are congruent.

An included side is the side between the two given angles.

AAS Rule

The Angle-Angle-Side (AAS) Rule states that

If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent.

(This rule may sometimes be refered to as SAA).

For the ASA rule the given side must be included and for AAS rule the side given must not be included. The trick is we must use the same rule for both the triangles that we are comparing.

AAS
      Compare AAS with AAS

ASA
      Compare ASA with ASA

AAS with ASA
      Compare AAS with ASA


The following video will explain the rules: SSS, SAS, ASA, AAS. It will also present two column proofs using the rules.



The following video shows how to determine whether given triangles are congruent, and to name the postulate that is used.
If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent (Side-Side-Side or SSS).
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent (Side-Angle-Side or SAS).
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent (Angle-Side-Angle or ASA). 







Example:

congruent triangles

Which of the following conditions would be sufficient for the above triangles to be congruent?

a) a = e, x = u, c = f

b) a = e, y = s, z = t

c) x = u, y = t, z = s

d) a = f, y = t, z = s

Solution for a):

Step 1:   a = e gives the S

x = u gives the A

c = f gives the S

Step 2:  Beware! x and u are not the included angles. This is not SAS but ASS which is not one of the rules. Note that you cannot compare donkeys with triangles!

Answer: a = e, x = u, c = f is not sufficient for the above triangles to be congruent.

congruent triangles  

Solution for b):

Step 1: a = e gives the S

y = s gives the A

z = t gives the A

Step 2:  a and e are non-included sides. Follows the AAS rule.

Answer: a = e, y = s, z = t is sufficient show that the above are congruent triangles.

congruent triangles 

Solution for c):

Step 1: x = u gives the A

y = t gives the A

z = s gives the A

Step 2:  AAA is not one of the rules.

Answer: x = u, y = t, z = s is not sufficient for the above triangles to be congruent.

congruent triangles 

Solution for d):

Step 1: a, y, z follows AAS (non-included side)

f ,t, s follows the ASA (included side)

Step 2:  Comparing AAS with ASA is not allowed

Answer: a = f, y = t, z = s is not sufficient to show that the above are congruent triangles



Why SSA and AAA Don't Work as Congruence Shortcuts

AAA Does not Work

Triangles with all three corresponding angles equal may not be congruent. These triangles will have the same shape but not necessarily the same size. They are called similar triangles.

SSA Does not Work

Triangles with two corrsponding sides and one non-included angle equal may not be congruent.


SSA Can't Be Used to Prove Triangles are Congruent
This video explains why there isn't an SSA Triangle Congruence Postulate or Theorem.



Non Congruence Theorems AAA SSA
Now that we know 4 ways to prove two triangles are congruent, lets look at two non-congruence theorems. Students often use these to prove triangles are congruent which is incorrect.







Hypotenuse Leg Rule

The Hypotenuse-Leg (HL) Rule states that

If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent.

In the right triangles ΔABC and ΔPQR , if AB = PR, AC = QR then ΔABC ≡ ΔRPQ .



How to use the Hypotenuse - Leg Congruence Theorem







CPCTC

CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent.

CPCTC states that

If two or more triangles are proven congruent by: ASA, AAS, SSS, HL, or SAS, then all of their corresponding parts are congruent as well. This can be used to prove various geometrical problems and theorems.


CPCTC Triangle Congruence The following video shows how to use the principle that corresponding parts of congruent triangles are congruent, or CPCTC.

How do we use congruent triangles to prove line segments or angles congruent?
Proving triangles are congruent using their corresponding parts.

 

This video will teach you how to apply the CPCTC theorem when working with congruent triangles.





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