Congruent Triangles – Review, Short Tutorial and Practice Questions

In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. This means that either object can be re-positioned and reflected (but not resized) so as to coincide precisely with the other object. More about congruence

2 geometrical shapes are congruent if their elements (sides and angles) are equal, but they don’t have to have the same direction. So, here we are only interested in the size and shapes. For example, angles α and β are congruent if they have the same size, but not necessarily the same direction:

We write the congruence like this:

We can also say that 2 triangles are congruent if their appropriate elements are equal:

So, we write:  ABC  A’ B’ C’

Most important congruence rules are concerning triangles, and they are used for congruence of almost all 2D shapes (quadrilaterals, hexagons…). There are 4 of these rules:

 1) Side-Angle-Side (SAS)

If 2 triangles have 2 equal sides and an equal angles that are between those sides, then we can conclude that these 2 triangles are congruent.

2) Side-Side-Side (SSS)

If 2 triangles have all 3 sides that are equal, then these 2 triangles are congruent.

  3) Angle-Side-Angle (ASA)

If 2 triangles have 2 equal angles and an equal side that is between them, then these 2 triangles are congruent.

4) Side-Side-Angle (SSA)

If 2 triangles have 2 equal sides, and an equal angle that is not between those 2 sides, then these 2 triangles are congruent.

GED

1. For triangles ABC and A’B’C’ have:

AB = A’ B’
RA = R A’
RB = R B’

Are these 2 triangles congruent?

Yes
No

2. For triangles ABC and A’B’C’ have:

BC = B’C’
AC = A’C’
RA = RA’

Are these 2 triangles congruent?

3. Rectangles ABCD and A’B’C’D’ have following equal
elements:

CD = C’D’
AC = A’C’

Are these 2 rectangles congruent?

Yes
No

1. Yes
We see that we have a case of ASA:

These 2 triangles are congruent

2. Yes

This is a case of SSA:

3. Yes the rectangles are congruent.
We mark the elements that are the same:
CD = C’D’
AC = A’C’

We mark the elements that are the same: