Introduction to Congruence
What is Congruence?
Two shapes or objects are congruent if they have the same shape and size. The position and orientation can be different as long as they have the same angles and side lengths in the same relative position.
To identify if two shapes are congruent, we will need to remind ourselves how we identify equivalent angles and lengths of polygons.
Angles
Angles that have the same size are marked on diagrams with equivalent markings. These markings could be values, variables (letters or symbols), colours or lines.
Here are some examples
Example 1
In these two triangles the angles are marked as having the same size using corresponding colours. Hence $\angle BCA=\angle EFD$∠BCA=∠EFD, $\angle ABC=\angle DEF$∠ABC=∠DEF and $\angle BAC=\angle EDF$∠BAC=∠EDF.
example 2
In these two triangles the angles are marked as having the same size using the values of the angles.
Example 3
Here, we can see that angle(MAB)=angle(MAC) because the same symbol has been used.
Notice that a pair of angles that are equal and corresponding are named by their vertices in the same corresponding order. That is to say, if $\angle ABC=\angle DEF$∠ABC=∠DEF:
- vertex A corresponds to vertex D
- vertex B corresponds to vertex E
- vertex C corresponds to vertex F
Sides
Sides that have the same size are marked on diagrams with equivalent markings. These markings could be values, variables (letters or symbols), or lines.
Here are some examples
example 1
In these two triangles the sides are marked with tally marks. Corresponding tally marked sides are equal in length.
length AB = length DE because both use two tally marks.
length BC = length EF because both use one tally mark
length AC = length DF because both use three tally marks
Example 2
In these two quadrilaterals the lengths are marked as having the same size using the values of the lengths.
Examples
Question 1
Write triangle $ABC$ABC is congruent to triangle $DEF$DEF using mathematical notation.
Question 2
Which pairs of the following triangles are congruent?
Question 3
Given that the two triangles are congruent, which angle matches $\angle BCA$∠BCA in the top triangle?