Similar Triangles: Criteria, Theorems, and Ratio Problems Class 10

In geometry, two figures that have identical shapes and identical sizes are called congruent ($\cong$). But what if two geometric figures share the exact same shape, yet differ in physical size—like a miniature blueprint of an architectural stadium, a reduced photographic print, or a magnified image projected onto a cinema screen?

In mathematics, figures having the same shape but not necessarily the same size are called Similar Figures (denoted by the symbol $\sim$).

In CBSE Class 10 Mathematics, Chapter 6 (Triangles) is the largest and most theorem-intensive geometry chapter on the syllabus. Understanding the Three Criteria for Triangle Similarity (AA, SSS, SAS) and applying them to solve ratio, altitude, and shadow problems is essential for securing full marks in Section C and Section D.

What You Will Learn

• Congruence vs. Similarity: The geometric distinction
• The two conditions for polygon similarity
• The Three Similarity Criteria: AA (Angle-Angle), SSS (Side-Side-Side), and SAS (Side-Angle-Side)
• Relationship between corresponding sides, perimeters, altitudes, and medians
• The classic Vertical Pole and Building Shadow Problem
• The Ladder Sliding Down a Wall problem
• Right triangle similarity corollaries and board exam presentation rubrics

1. What Makes Two Triangles Similar?

Two triangles $\Delta ABC$ and $\Delta DEF$ are said to be similar ($\Delta ABC \sim \Delta DEF$) if and only if:

1. Their corresponding angles are equal: $\angle A = \angle D, \quad \angle B = \angle E, \quad \angle C = \angle F$
2. Their corresponding sides are in the same ratio (proportional): $\mathbf{\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}}$

2. The Three Criteria for Triangle Similarity

Just as we don't need to measure all 6 elements to prove triangle congruence, we do not need to check all 6 conditions to prove similarity:

                            Criteria for Similarity of Triangles
                                             |
       +-------------------------------------+-------------------------------------+
       |                                     |                                     |
AAA / AA CRITERION                    SSS CRITERION                         SAS CRITERION
Two angles of one triangle equal     All three corresponding sides         Two sides proportional AND
to two angles of another             are in the same ratio                 included angles are EQUAL
∠A = ∠D,  ∠B = ∠E                    AB/DE = BC/EF = AC/DF                 AB/DE = AC/DF  and  ∠A = ∠D

1. The AA (Angle-Angle) Similarity Criterion

If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

(Why only two angles? Because if two pairs of angles are equal, the third pair must automatically be equal by the Angle Sum Property of triangles: $180^\circ - (\angle A + \angle B)$!)

2. The SSS (Side-Side-Side) Similarity Criterion

If in two triangles, the corresponding sides of one triangle are proportional to the corresponding sides of the other triangle, then their corresponding angles are equal and the triangles are similar.

3. The SAS (Side-Angle-Side) Similarity Criterion

If one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are proportional, then the two triangles are similar.

Important: <u>In the SAS criterion, the equal angle MUST strictly be the INCLUDED angle between the two proportional sides! If the angle is outside the proportional sides, similarity CANNOT be established!</u>

3. High-Yield Solved Board Examination Problems

Problem 1: The Vertical Pole and Shadow Problem (CBSE Classic)

Problem: A vertical pole of length $6\text{ m}$ casts a shadow $4\text{ m}$ long on the ground, and at the same time a tower casts a shadow $28\text{ m}$ long. Find the height of the tower.

            Pole (6 m)                         Tower (h)
                A                                  D
                |\                                 |                | \                                |             6 m |  \                           h   |                  |   \                              |                   +----+                             +----+
                B 4m C                             E 28m F

Solution:

1. Let $AB = 6\text{ m}$ be the pole, and $BC = 4\text{ m}$ be its shadow.
2. Let $DE = h\text{ metres}$ be the tower, and $EF = 28\text{ m}$ be its shadow.
3. Compare $\Delta ABC$ and $\Delta DEF$:
   • Both the pole and the tower are vertical to the ground: $\angle B = \angle E = 90^\circ$
   • At the same time of day, the Sun's rays strike the ground at the same angular elevation: $\angle C = \angle F$
4. By the AA Similarity Criterion: $\mathbf{\Delta ABC \sim \Delta DEF}$
5. Since the triangles are similar, their corresponding sides are proportional: $\frac{AB}{DE} = \frac{BC}{EF}$ $\frac{6}{h} = \frac{4}{28}$ $\frac{6}{h} = \frac{1}{7} \implies h = 6 \times 7 = \mathbf{42\text{ metres}}$
6. Therefore, <u>the height of the tower is $42\text{ metres}$</u>.

Problem 2: Proving Similarity in Trapeziums

Problem: Diagonals $AC$ and $BD$ of a trapezium $ABCD$ with $AB \parallel DC$ intersect each other at point $O$. Using a similarity criterion for two triangles, show that $\frac{OA}{OC} = \frac{OB}{OD}$.

Proof:

1. In $\Delta OAB$ and $\Delta OCD$:
   • Since $AB \parallel DC$, taking transversal $AC$: $\angle OAB = \angle OCD \quad \text{(Alternate interior angles)}$
   • Taking transversal $BD$: $\angle OBA = \angle ODC \quad \text{(Alternate interior angles)}$
   • Vertically opposite angles: $\angle AOB = \angle COD$
2. Therefore, by the AAA (or AA) Similarity Criterion: $\mathbf{\Delta OAB \sim \Delta OCD}$
3. Since corresponding sides of similar triangles are in proportion: $\frac{OA}{OC} = \frac{OB}{OD}$ Hence Proved.

4. Fundamental Ratios of Similar Triangles

If two triangles are similar ($\Delta ABC \sim \Delta DEF$ with ratio of sides $\frac{AB}{DE} = k$):

1. Ratio of Perimeters: Equal to the ratio of their corresponding sides: $\mathbf{\frac{\text{Perimeter}(\Delta ABC)}{\text{Perimeter}(\Delta DEF)} = \frac{AB}{DE} = k}$
2. Ratio of Altitudes: Equal to the ratio of corresponding sides: $\mathbf{\frac{\text{Altitude } h_1}{\text{Altitude } h_2} = \frac{AB}{DE} = k}$
3. Ratio of Medians and Angle Bisectors: Also equal to the ratio of corresponding sides ($k$).

5. Summary and Examination Tips

Exam Tip: In writing similarity statements, letter order matters strictly! Writing $\Delta ABC \sim \Delta DEF$ means $\angle A = \angle D$. If $\angle A = \angle E$, you MUST write $\Delta ABC \sim \Delta EDF$! Misordering vertex letters in the similarity statement invalidates corresponding side ratios and loses marks!

Common Mistake: Confusing congruence with similarity. All congruent triangles are similar, but similar triangles are NOT congruent unless their ratio of sides is $1$!