Geometry: 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.

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.

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

If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent. (Hence the name of this rule: Side-Side-Side, SSS)

SAS Rule

If two sides and the included angle (Side-Angle-Side, SAS) 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 Angle                         Non-included angle

ASA Rule

If two angles and the included side of one triangle (Angle-Side-Angle, ASA) 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

If two angles and a non-included side of one triangle (Angle-Angle-Side, AAS) 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.

      Compare AAS with AAS

      Compare ASA with ASA

      Compare AAS with ASA

The following video shows how to determine whether given triangles are congruent, and to name the postulate that is used.

Example:

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.

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.

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.

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.

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