Congruent Figures and Congruence

Key Concepts

• Understanding congruency
• Identifying congruent figures

Understanding congruency 

Two figures or geometrical shapes are said to be congruent if they are of the same size and shape. 

Two figures are said to be congruent if the first image has rigid transformations to form the second image. 

or 

We can describe congruency through rigid transformations like translation, reflection, and rotation. 

Understanding Congruent figures 

1. The following figures are congruent figures. Identify the transformation undergone in these figures. 

2. The first figure has undergone the transformation translation, the second figure has undergone the transformation reflection, and the third has undergone the transformation rotation. 

In figure 1, triangle PWZ and triangle SQV are congruent because ∠WPZ = ∠QVS, ∠ZWP= ∠SQV, ∠WZP= ∠QSV and WP=QV, WZ=QS, PZ=SV. In figure 2, EF=HG, EH=FG, HF=HF (common side). 

∠FEH= ∠ HGF=90° also since EF=HG and EF is parallel to HG, ∠EFH= ∠GHF and ∠EHF= ∠HFG, since the alternate angles are equal. 

The following figure is a congruent figure with coordinates  

G(–8, –8), H(–2, -8), E(–2, –2), and F(–8, –2) has undergone a transformation to 

S(2, 2), R(2, 8), U(8, 8), T(8, 2). 

G(–8, –8), H(–2, –8), E(–2, –2), F(–8, –2) undergoes 90° counterclockwise rotation and is followed by the translation (x, y) to (x, y+10) to have coordinates S(2, 2), R(2, 8), U(8, 8),T(8, 2).  

Since GF = SR and FE = RU and UT = EH and GH = ST, we can say that 

GHEF and UTSR are congruent figures. Also, since rotation and translation are rigid transformations, there is no change in shape and size. 

The triangle PQV is congruent to triangle P’Q’V’ in a coordinate plane since it has undergone reflection with respect to the y-axis. 

The triangle MNP is congruent to triangle M’N’P’ in a coordinate plane that has undergone reflection with respect to the x-axis. 

The triangle GHF is congruent to triangle G’H’F’ in a coordinate plane that has undergone translation (x, y) ® (x + 8, y), or it moves 8 points right. 

Describe a congruence transformation that maps ▱ABCD to ▱EFGH. 

A congruence transformation that ▱ABCD to ▱EFGH is a reflection in the y-axis followed by a translation of 4 units down. 

Check your knowledge. 

How can we identify congruent figures? 

If the figures are of the same size and shape, then those figures are congruent. 

How can we identify congruent figures in a coordinate plane? 

If the figures undergo a rigid transformation, then those figures are congruent figures. 

Also, by finding the distance between the coordinate points, we can find the length of each side of geometric figures, and if it is equal to the length of the corresponding side of the other figure, we can also say that they are congruent figures.  

Name the rigid transformations. 

Translation, rotation, and reflection are rigid transformations. 

Exercise:

1. Explain the congruency between the triangles in the coordinate plane of the following figure: Triangle DEF is congruent to triangle D’E’F’.

2. By using the distance formula for any two points in the coordinate plane, check the congruency for triangle ABC and triangle FGH.

3. Write the congruency statement for the following figure: Triangle BAD congruent to ………………….?

4. If a triangle ABC is congruent to triangle DEF, then name the congruent angles and segments of the triangles.

5. Identify the rigid transformation for the following images:

6. Identify the transformation, and check whether the image is congruent; if so, name the corresponding congruent parts.

7. Rotate the figure 180° counter-clockwise and write the coordinates and name the congruent segments and angles of the preimage to the image.

8. Plot the points A(0,0), B(8,1), and C(5,5) and undergo the transformation reflection with respect to the x-axis. Rotate clockwise 180°. Write the coordinates after the transformation and check whether the image is congruent; if so, write the congruent segment and angles.

Concept Map: 

What we have learned:

• Understanding Congruency
• Understanding Congruency in a co-ordinate plane
• Identify congruent figures