Congruence and similarity Knowledge-Based Study

Study notes

1. What is Congruence?

   Two geometric figures are said to be congruent if they have the exact same shape and the exact same size. In formal geometry, two figures are congruent if and only if one can be mapped onto the other through a sequence of rigid transformations (isometries): translations, rotations, and reflections.

   When two polygons are congruent:

   • All pairs of corresponding interior angles are equal in measure.
   • All pairs of corresponding sides are equal in length.

   The symbol for congruence is $\cong$. For example, writing $\triangle ABC \cong \triangle DEF$ indicates that vertex $A$ corresponds to $D$, $B$ corresponds to $E$, and $C$ corresponds to $F$. This yields:

   • $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$
   • $AB = DE$, $BC = EF$, $AC = DF$

   Establishing Triangle Congruence

   To prove that two triangles are congruent, it is not necessary to verify all six pairs of corresponding parts. Specific minimal conditions, known as congruence criteria, guarantee that the triangles are completely congruent:

   • Side-Side-Side (SSS): All three pairs of corresponding sides are equal in length.
   • Side-Angle-Side (SAS): Two pairs of corresponding sides and the included angle between them are equal.
   • Angle-Side-Angle (ASA): Two pairs of corresponding angles and the included side between them are equal.
   • Angle-Angle-Side (AAS): Two pairs of corresponding angles and a non-included side are equal.
   • Right angle-Hypotenuse-Side (RHS or HL): In right-angled triangles, the hypotenuse and one leg are equal to the corresponding hypotenuse and leg.

   Note: The combinations Side-Side-Angle (SSA) and Angle-Angle-Angle (AAA) do not guarantee congruence.

   What is Similarity?

   Two geometric figures are similar if they have the same shape, but not necessarily the same size. In transformational geometry, two figures are similar if one can be transformed into the other via a sequence of rigid motions combined with a dilation (rescaling).

   Two polygons are similar if and only if:

   1. All corresponding angles are equal in measure.
   2. All corresponding side lengths are proportional.

   The symbol for similarity is $\sim$. If $\triangle ABC \sim \triangle PQR$, then:

   $$\angle A = \angle P, \quad \angle B = \angle Q, \quad \angle C = \angle R$$

   and the ratios of corresponding sides equal a constant scale factor $k$:

   $$\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR} = k$$

   Every pair of congruent figures is similar (with scale factor $k = 1$), but similar figures are not necessarily congruent.

   Determining Triangle Similarity

   Similar to congruence, triangles require fewer than all six measurements to establish similarity. The three core criteria are:

   • Angle-Angle (AA): If two angles of one triangle are equal to two angles of another triangle, the third angles must also be equal, making the triangles similar.
   • Side-Angle-Side Similarity (SAS~): If two pairs of corresponding sides are proportional and their included angles are equal, the triangles are similar: $$\frac{AB}{DE} = \frac{AC}{DF} \quad \text{and} \quad \angle A = \angle D$$
   • Side-Side-Side Similarity (SSS~): If all three pairs of corresponding sides are in the same ratio: $$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$$

   Effects of the Scale Factor

   When two two-dimensional figures are similar with a linear scale factor of $k$, their linear measurements and area measurements relate in specific ways:

   • Linear Dimensions: Any corresponding linear lengths, including perimeters, altitudes, medians, and circumferences, have the ratio: $$\frac{\text{Perimeter}_1}{\text{Perimeter}_2} = k$$
   • Area Dimensions: The ratio of their areas is the square of the linear scale factor: $$\frac{\text{Area}_1}{\text{Area}_2} = k^2$$

   Worked Example

   Consider two similar triangles, $\triangle T_1$ and $\triangle T_2$. If a side of $\triangle T_1$ is $6\text{ cm}$ and the corresponding side of $\triangle T_2$ is $15\text{ cm}$:

   • The linear scale factor is $k = \frac{15}{6} = 2.5$.
   • The ratio of their perimeters is $2.5$.
   • The ratio of their areas is $k^2 = (2.5)^2 = 6.25$.