Criteria for Similarity of Triangles

Similarity in geometry deals with figures that have the same shape but not necessarily the same size. Two figures are considered similar if their corresponding angles are equal and their corresponding sides are proportional. There are several criteria (or theorems) that allow us to determine if two triangles are similar without having to check all angles and sides. These criteria significantly simplify the process of proving similarity.

Formulae

The key concepts involve the relationships between the sides and angles of triangles. While there aren’t specific “formulae” in the traditional sense, the following relationships are crucial:

• AAA/AA Similarity: If all three angles of one triangle are equal to the corresponding angles of another triangle (AAA), the triangles are similar. If two angles of one triangle are equal to two angles of another triangle (AA), then the triangles are similar (since the third angles must also be equal).
• SSS Similarity: If the three sides of one triangle are proportional to the three corresponding sides of another triangle, the triangles are similar.
• SAS Similarity: If two sides of one triangle are proportional to the corresponding two sides of another triangle, and the included angles (the angles between those two sides) are equal, then the triangles are similar.

Examples

Here are a few examples illustrating the application of these similarity criteria:

Example-1 (AA Similarity): Consider two triangles, $\triangle ABC$ and $\triangle DEF$. Suppose we know that $\angle A = 50^\circ$, $\angle B = 70^\circ$, $\angle D = 50^\circ$, and $\angle E = 70^\circ$. Since $\angle A = \angle D$ and $\angle B = \angle E$, by the AA similarity criterion, $\triangle ABC \sim \triangle DEF$.

Example-2 (SSS Similarity): Suppose we have two triangles, $\triangle PQR$ and $\triangle STU$. We are given that $PQ = 6$, $QR = 8$, and $RP = 10$. Also, $ST = 3$, $TU = 4$, and $US = 5$. We see that $\frac{PQ}{ST} = \frac{6}{3} = 2$, $\frac{QR}{TU} = \frac{8}{4} = 2$, and $\frac{RP}{US} = \frac{10}{5} = 2$. Since the sides are proportional (with a ratio of 2), by the SSS similarity criterion, $\triangle PQR \sim \triangle STU$.

Theorem with Proof

Theorem: AA Similarity Theorem If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

Proof:

1. Given: $\triangle ABC$ and $\triangle DEF$ with $\angle A \cong \angle D$ and $\angle B \cong \angle E$.
2. To Prove: $\triangle ABC \sim \triangle DEF$.
3. Proof:
   • Since the sum of the angles in a triangle is $180^\circ$, we have:
     • $\angle A + \angle B + \angle C = 180^\circ$
     • $\angle D + \angle E + \angle F = 180^\circ$
   • Since $\angle A \cong \angle D$ and $\angle B \cong \angle E$, we can substitute: $\angle D + \angle E + \angle C = 180^\circ$ and $\angle D + \angle E + \angle F = 180^\circ$.
   • Therefore, $\angle C = \angle F$.
   • Since $\angle A \cong \angle D$, $\angle B \cong \angle E$, and $\angle C \cong \angle F$, the three angles of $\triangle ABC$ are congruent to the three corresponding angles of $\triangle DEF$.
   • By definition of similarity, $\triangle ABC \sim \triangle DEF$.

Common mistakes by students

• Confusing Similarity with Congruence: Students often confuse similar triangles (same shape, different size) with congruent triangles (same shape, same size). Make sure to understand the difference.
• Incorrect Side Proportions (SSS and SAS): When using SSS or SAS, students sometimes set up the side ratios incorrectly. Ensure you match the corresponding sides correctly (e.g., the longest side of one triangle to the longest side of the other, etc.). For SAS, make sure the angle is *included* between the two proportional sides.
• Not Recognizing AA: Students sometimes struggle to recognize AA similarity if the given angles are not directly provided. For instance, they may not see AA when given one angle and then finding a second angle through vertical angles or linear pairs.
• Ignoring the order of vertices: When stating similarity, the order of the vertices matters. For example, if $\triangle ABC \sim \triangle DEF$, that means $\angle A$ corresponds to $\angle D$, $\angle B$ to $\angle E$, and $\angle C$ to $\angle F$.

Real Life Application

Similarity has many real-world applications:

• Scale Drawings and Maps: Maps and blueprints use scale factors to represent real-world objects or areas. Understanding similarity is crucial for interpreting these representations correctly.
• Architecture and Engineering: Architects and engineers use similar triangles to design buildings and structures, ensuring that different parts of a design are proportional.
• Photography and Videography: Camera lenses use the principles of similarity to focus and zoom in on subjects. The size of the image on the sensor is related to the distance to the object using similar triangles.
• Shadow Calculations: Using the shadows cast by objects, we can calculate the height of trees, buildings, or other inaccessible objects using similar triangles.

Fun Fact

The concept of similarity was instrumental in the development of trigonometry! Trigonometric ratios (sine, cosine, tangent) are based on the relationships between the sides of similar right triangles. Understanding similarity unlocks many doors in higher-level mathematics and its applications.

Recommended YouTube Videos for Deeper Understanding

Q.1 Which of the following conditions guarantees that two triangles are similar?

Check Solution

Ans: A

AAA (or AA) similarity, SSS similarity, and SAS similarity are the three main criteria for similarity. Option A describes AAA similarity.

Q.2 In $\triangle ABC$, $AB = 6$, $BC = 8$, and $\angle B = 60^\circ$. In $\triangle DEF$, $DE = 9$, $EF = 12$, and $\angle E = 60^\circ$. Are $\triangle ABC$ and $\triangle DEF$ similar? If so, by what criterion?

Check Solution

Ans: B

We have two sides and an included angle. Check the proportionality of the sides: $AB/DE = 6/9 = 2/3$ and $BC/EF = 8/12 = 2/3$. Since the ratios are equal, and the included angles are equal, the triangles are similar by SAS similarity.

Q.3 Two triangles have angles of $40^\circ$ and $60^\circ$. What must be the measures of the angles in the other triangle to guarantee the triangles are similar?

Check Solution

Ans: C

For AAA similarity, we need two angles to be the same to have the triangles similar.

Q.4 $\triangle ABC$ has sides of length 3, 4, and 5. $\triangle XYZ$ has sides of length 6, 8, and 10. Are the triangles similar? If so, by what criterion?

Check Solution

Ans: B

We can use the SSS criterion by checking if all three ratios of corresponding sides are equal. $AB/XY=3/6=1/2, BC/YZ=4/8=1/2, AC/XZ=5/10=1/2$.

Q.5 If two triangles are similar by SAS similarity, which of the following must be true?

Check Solution

Ans: C

By definition, SAS similarity states that two sides are proportional and the included angle is congruent.

Next Topic: Basic Proportionality Theorem (Thales Theorem) & its Converse