**Similar triangles proof** is a captivating facet of **geometry**, offering a rich playground for mathematical exploration and proof, embodying the **harmony** and **beauty** in the world of shapes and forms. These **congruent** figures, despite their varying scales, uphold the fundamental principle of **similarity**: maintaining **identical angles** while exhibiting **proportional sides**.

Through this article, we will delve into the intriguing world of **similar triangles**, traversing diverse paths of **proof** that reveal their inherent and mesmerizing properties.

These mathematical deductions will illuminate how these **identical** yet **diverse** geometric figures elegantly uphold the fascinating and expansive world of **geometry**, providing a platform for understanding more **complex shapes** and **theorems**.

## Defining Similar Triangles Proof

**Similar triangles** are **geometric figures** that maintain the same shape but can have different sizes. The formal definition of **similar triangles** is as follows:

Two triangles are **similar** if their corresponding angles are **congruent** (equal in measure) and their corresponding sides are **proportional** in length.

When we talk about “**similar triangle proofs**,” we’re referring to mathematical **proofs** that demonstrate the similarity of two triangles. These proofs are based on one of three **postulates**:

**Angle-Angle (AA) Postulate**

If two angles of one t**riangle** are **congruent** to two angles of another, the two triangles are similar.

**Side-Angle-Side (SAS) Postulate**

If an angle of one **triangle** is **congruent** to an angle of a second triangle, and the sides, including these angles, are **proportional**, then the triangles are similar.

**Side-Side-Side (SSS) Postulate**

If the **corresponding sides** of two triangles are **proportional**, then the** triangles** are similar.

**Example**

Given a right triangle **ABC** with **∠ACB = 90°** and **AD** being the altitude to the hypotenuse **(BC)**, prove that **△ABC ~ △BAD ~ △ACD**.

Figure-1.

**Proof**

Firstly, we need to identify the angles in our triangles.

In **△ABC**, **∠BAC** is **∠A** and **∠ABC** is **∠B**.

In **△BAD**, **∠BAD** is **∠A** and **∠ABD** is equal to **∠C** (because **∠ABD** and** ∠ACB** are complementary and **∠ACB = 90°**).

In **△ACD**, **∠ACD** is **∠B** and **∠ADC** is **∠A** (because **∠ADC** and **∠ADB** are complementary and** ∠ADB = 90°**).

Now we can prove the similarity:

For **△ABC** and **△BAD**: They have **∠A** in common and** ∠ABC = ∠ABD = ∠C**. So by the **AA (Angle-Angle)** criterion, **△ABC ~ △BAD**.

For **△ABC** and **△ACD**: They have **∠B** in common and **∠BAC = ∠CAD = ∠A**. So by the **AA (Angle-Angle)** criterion, **△ABC ~ △ACD**.

Thus, **△ABC ~ △BAD ~ △ACD**.

The **AA** **postulate** states that if two **triangles** have two **corresponding angles** that are** congruent**, then the **triangles** are similar. This proof illustrates the use of the **AA postulate**, one of the key methods of proving **triangle similarity**.

**Properties**

Following are the properties associated with **similar triangles** that are often leveraged in proofs:

**Corresponding Angles are Congruent**

If two triangles are similar, then the measure of their corresponding angles is the same. This is a **direct consequence** of the **Angle-Angle (AA) criterion** of triangle similarity.

**Corresponding Sides are Proportional**

The **lengths of corresponding sides** in similar triangles maintain the same **ratio**. For instance, if we have two similar triangles ABC and DEF, the **ratio of side AB to side DE** is the same as the **ratio of side BC to side EF**, and the same as the **ratio of side AC to side DF**.

**Proportional Altitudes, Medians, and Angle Bisectors**

In two **similar triangles**, the ratios of **corresponding altitudes**, **medians,** or **angle bisectors** are the same as the ratio of **corresponding sides.**

**Parallel lines and Transversals**

If a line parallel to one side of a triangle intersects the other two sides, it **divides** those sides **proportionally**, creating **similar triangles**.

**Area Ratio**

The ratio of the **areas** of two similar triangles is equal to the **square of the ratio of their corresponding sides**.

**Pythagorean Theorem**

In a right triangle, the **square of the length of the hypotenuse** (the side opposite the right angle) is equal to the **sum of the squares of the lengths of the other two sides**. This can often be used in conjunction with **similarity properties** to prove further similarities or relationships between figures.

Remember, these properties come from the definitions and **postulates** of **similar triangles**. Any proof involving **similar triangles** will most likely make use of one or more of these properties.

**Applications**

**Similar triangles** have a wide range of applications across numerous fields. Let’s explore a few:

**Engineering and Architecture**

Similar triangles are used in the **scaling** of models. When engineers or architects design buildings or machinery, they often build a **scale model** first. The real object and its model are similar “**triangles**” in three dimensions – they have the same shape but different sizes.

**Astronomy and Physics**

In **astronomy**, similar triangles are used to calculate distances that would be otherwise difficult to measure. For instance, astronomers use **parallax**, which involves creating similar triangles, to calculate the distance to nearby stars. In **physics**, similar triangles can be used to calculate the height of physical objects from their shadows and the angle of the sun.

**Geography and Surveying**

S**imilar triangles** are used in **cartography** to create **scale maps**. In **surveying**, they are used to determine distances and heights that can’t be measured directly.

**Computer Science and Graphics**

In **computer graphics** and **gaming**, similar triangles are often used to **scale images** and determine **perspective**, enabling objects to appear smaller the further they are from the viewer to create an illusion of depth.

**Trigonometry and Calculus**

The foundations of **trigonometry** are laid on the concept of similar triangles. For instance, **trigonometric ratios** remain constant for a given angle in similar triangles. In **calculus**, similar triangles are used in the derivation of derivative rules for trigonometric functions.

**Medicine and Biology**

In **medical imaging**, similar triangles can be used to **calculate the sizes or distances** of objects in an image. In **biology**, they are used in various models and analyses.

The **ubiquitous nature of similar triangles** in diverse fields stems from their **fundamental properties** and the relative ease of working with them mathematically.

**Exercise **

**Example 1**

**Proving Similarity Using AA Criterion**

Given two triangles **ABC** and **DEF** such that **∠A = ∠D** and **∠B = ∠E**. Prove that **△ABC ~ △DEF**.

Figure-2.

**Solution**

Since **∠A = ∠D** and **∠B = ∠E**, by the **AA (Angle-Angle)** similarity **postulate**, **△ABC ~ △DEF**.

**Example 2**

**Proving Similarity Using SAS Criterion**

Given two triangles **ABC** and** DEF**, if **AB/DE = BC/EF** and** ∠B = ∠E**, prove that **△ABC ~ △DEF**.

**Solution**

Given the** ratio** of two sides and their included angles are equal, by the **SAS (Side-Angle-Side)** similarity **postulate**, **△ABC ~ △DEF**.

**Example 3**

**Proving Similarity Using SSS Criterion**

Given two triangles **ABC** and **DEF**, if **AB/DE = BC/EF = AC/DF**, prove that **△ABC ~ △DEF**.

### Solution

Since the **ratios** of the lengths of corresponding sides of the **triangles** are equal, by the **SSS (Side-Side-Side)** similarity** postulate**, **△ABC ~ △DEF**.

**Example 4**

**Finding Unknowns Using Similarity**

In similar triangles **△ABC** and **△DEF**, if **AB = 3 cm**, **BC = 4 cm**, **AC = 5 cm** and **EF = 12 cm**. Find **DE** and **DF**.

Figure-3.

### Solution

Since **△ABC ~ △DEF**, we have **DE/AB = EF/BC = DF/AC**.

Therefore:

DE = (AB/BC) * EF

DE = (3/4) * 12

DE = 9 cm

Also:

DF = (AC/BC) * EF

DF = (5/4) * 12

DF = 15 cm

**Example 5**

**Using Similar Triangles in Circles**

In a **circle** with center **O**, chord **AB** is parallel to a tangent and the radius **OC** **perpendicular** to the **tangent** meets the chord at **G**. Prove that **OG** is half the length of **OC**.

### Solution

Draw the radius **OA** to point **A**. Then **△OGA** and **△OCA** are similar (angle-angle criterion: **∠OGA = ∠OCA and ∠OAG = ∠OAC**). Therefore, the ratio of corresponding sides is the same:

OG/OA = OA/OC

which simplifies to:

OG = $OA^2$/OC

But since **OA = OC** (both are radii of the circle), we have **OG = OC/2**, as required.

**Example 6**

**Using Similarity in Right Triangles**

In **right triangle** **ABC**, **∠ACB = 90°**, **AD** is an altitude. Prove that **△ABC ~ △BAD ~ △ACD**.

### Solution

Firstly, since **∠BAD = ∠B and ∠ABD = ∠ABC**, by the **AA criterion**, **△ABC ~ △BAD**.

Secondly, since **∠ACD = ∠B and ∠ADC = ∠ABC**, by the **AA criterion, △ABC ~ △ACD**.

Therefore, **△ABC ~ △BAD ~ △ACD**.

**Example 7**

**Shadow Problems**

A **6ft** man is standing near a **lamp post** when he notices his shadow is **4ft** long. At the same time, the lamp post’s shadow is 1**2ft long**. How tall is the lamp post?

### Solution

The triangles formed by the man and his shadow and the lamp post and its shadow are similar (by AA, as the angles of the sunlight are equal and each has a right angle). The ratio of the **lamp post’s** **height (h)** to the man’s height is the same as the **ratio** of the lamp post’s **shadow** to the man’s shadow. Therefore:

h/6 = 12/4

h = 18ft

**Example 8**

**Using Similar Triangles in Trigonometry**

In a right triangle **ABC** with **∠ACB = 90°**, let **∠ABC = θ**. If we take the side opposite **∠ABC** as the opposite **(O)**, side **BC** as the **hypotenuse (H)**, and side **AC** as the adjacent **(A)**, prove that **sin(θ) = O/H** and **cos(θ) = A/H**.

### Solution

In right triangle **ABC**, the ratios **O/H** and **A/H** are **invariant** with the size of the triangle due to similarity (as all right triangles with angle θ are similar).

These ratios are defined as **sin(θ)** and **cos(θ)**, respectively. Hence, the proof is a definition based on the properties of similar triangles.

*All images were created with GeoGebra.*