Congruence and Similarity in Geometry

Table of Contents

1. Introduction
2. Congruence
3. Similarity
4. Congruence & Similarity in Other Polygons
5. Real-World Applications
6. Quiz

1. Introduction

Congruence and similarity are fundamental concepts in geometry that help us compare and understand relationships between different geometric figures. While these concepts are related, they describe different types of relationships:

Congruence: Two figures are congruent if they have exactly the same size and shape.
Similarity: Two figures are similar if they have the same shape but possibly different sizes.

These concepts are particularly important in establishing relationships between triangles and other polygons, and have numerous applications in mathematics, engineering, architecture, and everyday life.

2. Congruence

2.1 Definition & Properties

Two geometric figures are congruent if they have exactly the same shape and size. This means all corresponding angles and sides are equal.

Key Properties of Congruent Figures:

All corresponding angles are equal.
All corresponding sides or distances are equal.
The perimeters of congruent figures are equal.
The areas of congruent figures are equal.
Congruent figures can be transformed into one another by a sequence of rigid motions (translations, rotations, and reflections).

We denote congruence using the symbol ≅. For example, if triangle ABC is congruent to triangle DEF, we write: △ABC ≅ △DEF.

Figure 1: Congruent triangles △ABC ≅ △DEF

2.2 Congruent Triangles

Triangles are particularly important in congruence. Two triangles are congruent if and only if their corresponding sides and angles are equal:

Three pairs of corresponding sides are equal
Three pairs of corresponding angles are equal

When working with congruent triangles, the order of vertices matters. If △ABC ≅ △DEF, then:

Vertex A corresponds to vertex D
Vertex B corresponds to vertex E
Vertex C corresponds to vertex F

This means that angles A and D are equal, angles B and E are equal, angles C and F are equal, and sides AB and DE, BC and EF, AC and DF are equal.

2.3 Triangle Congruence Criteria

There are several criteria for determining if two triangles are congruent without having to check all six measures (3 sides and 3 angles). These criteria are abbreviated as:

1. SSS (Side-Side-Side): If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

Figure 2: SSS Triangle Congruence

2. SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.

Figure 3: SAS Triangle Congruence

3. ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.

Figure 4: ASA Triangle Congruence

4. AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle, the triangles are congruent.

Figure 5: AAS Triangle Congruence

5. HL (Hypotenuse-Leg): For right triangles, if the hypotenuse and one leg of one triangle are equal to the hypotenuse and the corresponding leg of another triangle, the triangles are congruent.

Figure 6: HL Triangle Congruence (for right triangles only)

Note: The HL criterion only applies to right triangles. Also, AA (Angle-Angle) is NOT a congruence criterion because it does not provide information about the size of the triangles.

2.4 Examples & Applications

Example 1: Proving Triangle Congruence using SSS

Problem: In the figure below, if AB = DE, BC = EF, and AC = DF, prove that △ABC ≅ △DEF.

Solution:

Given:

AB = DE (first pair of corresponding sides)
BC = EF (second pair of corresponding sides)
AC = DF (third pair of corresponding sides)

Since all three pairs of corresponding sides are equal, by the SSS congruence criterion, △ABC ≅ △DEF.

Example 2: Proving Triangle Congruence using SAS

Problem: In the figure below, if AB = QR, ∠B = ∠R, and BC = RS, prove that △ABC ≅ △QRS.

Solution:

Given:

AB = QR (first pair of corresponding sides)
∠B = ∠R (included angle between the two pairs of sides)
BC = RS (second pair of corresponding sides)

Since two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, by the SAS congruence criterion, △ABC ≅ △QRS.

Example 3: Using Triangle Congruence to Find Unknown Values

Problem: In the figure below, if △ABC ≅ △DEF, find the value of x.

Solution:

Given that △ABC ≅ △DEF, all corresponding parts of the triangles are equal.

We can see that:

AB = 7 cm corresponds to DE = 7 cm
BC = 5 cm corresponds to EF = 5 cm
AC = x cm corresponds to DF = 8 cm

Since corresponding parts of congruent triangles are equal, AC = DF.

Therefore, x = 8 cm.

3. Similarity

3.1 Definition & Properties

Two geometric figures are similar if they have exactly the same shape but possibly different sizes. This means all corresponding angles are equal, and all corresponding sides are proportional.

Key Properties of Similar Figures:

All corresponding angles are equal.
All corresponding sides are proportional by the same scale factor.
The ratio of perimeters of similar figures equals the scale factor.
The ratio of areas of similar figures equals the square of the scale factor.
The ratio of volumes of similar 3D figures equals the cube of the scale factor.

We denote similarity using the symbol ~. For example, if triangle ABC is similar to triangle DEF, we write: △ABC ~ △DEF.

Figure 7: Similar triangles △ABC ~ △DEF

3.2 Similar Triangles

Two triangles are similar if and only if their corresponding angles are equal and their corresponding sides are proportional:

Three pairs of corresponding angles are equal
Three pairs of corresponding sides are proportional

If △ABC ~ △DEF, then:

∠A = ∠D, ∠B = ∠E, ∠C = ∠F
AB/DE = BC/EF = AC/DF = k (where k is the scale factor)

Note that the perimeter ratio equals k, and the area ratio equals k².

3.3 Triangle Similarity Criteria

There are several criteria for determining if two triangles are similar:

1. AA (Angle-Angle): If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

Note: Since the sum of angles in a triangle is 180°, if two angles are equal, the third angle must also be equal.

Figure 8: AA Triangle Similarity

2. SAS Similarity: If the ratio of two sides of one triangle equals the ratio of two sides of another triangle, and the included angles are equal, then the triangles are similar.

Figure 9: SAS Triangle Similarity

3. SSS Similarity: If the ratios of all three pairs of corresponding sides of two triangles are equal, then the triangles are similar.

Figure 10: SSS Triangle Similarity

3.4 Scale Factors & Proportions

The scale factor (k) is a key concept in similarity. It represents the ratio of corresponding sides between similar figures.

Scale Factor and Proportions:

If △ABC ~ △DEF with scale factor k = AB/DE, then:

AB/DE = BC/EF = AC/DF = k
Perimeter of △ABC / Perimeter of △DEF = k
Area of △ABC / Area of △DEF = k²

Example: Scale Factor Calculations

Problem: If △ABC ~ △DEF, AB = 8 cm, DE = 4 cm, and the area of △DEF is 12 cm², find the area of △ABC.

Solution:

First, we find the scale factor:

k = AB/DE = 8/4 = 2

Since the triangles are similar, the ratio of their areas equals k²:

Area of △ABC / Area of △DEF = k² = 2² = 4

Therefore:

Area of △ABC = 4 × Area of △DEF = 4 × 12 = 48 cm²

3.5 Examples & Applications

Example 1: Proving Triangle Similarity using AA

Problem: In the figure below, if ∠A = ∠D and ∠B = ∠E, prove that △ABC ~ △DEF.

Solution:

Given:

∠A = ∠D (first pair of corresponding angles)
∠B = ∠E (second pair of corresponding angles)

In a triangle, the sum of all angles is 180°. Therefore:

∠C = 180° - ∠A - ∠B

∠F = 180° - ∠D - ∠E

Since ∠A = ∠D and ∠B = ∠E, we can substitute:

∠F = 180° - ∠A - ∠B = ∠C

Now we have established that all three pairs of corresponding angles are equal:

∠A = ∠D, ∠B = ∠E, and ∠C = ∠F

By the AA similarity criterion, △ABC ~ △DEF.

Example 2: Finding Unknown Sides using Similarity

Problem: In the figure below, △ABC ~ △DEF. If AB = 12 cm, BC = 8 cm, AC = 15 cm, and DE = 8 cm, find EF and DF.

Solution:

First, we find the scale factor using the known corresponding sides:

k = AB/DE = 12/8 = 3/2

Since the triangles are similar, all pairs of corresponding sides are proportional by the same scale factor:

EF = BC/k = 8/(3/2) = 8 · (2/3) = 16/3 ≈ 5.33 cm

DF = AC/k = 15/(3/2) = 15 · (2/3) = 10 cm

Example 3: The Shadow Problem

Problem: A person who is 5 feet tall casts a shadow of 8 feet. At the same time, a nearby building casts a shadow of 80 feet. How tall is the building?

Solution:

This is a classic application of similar triangles. The sun's rays create similar triangles with the person and their shadow, and with the building and its shadow.

Let h be the height of the building. We can set up a proportion based on the similar triangles:

h/5 = 80/8 = 10

Therefore, h = 5 × 10 = 50 feet.

The building is 50 feet tall.

4. Congruence & Similarity in Other Polygons

The concepts of congruence and similarity extend beyond triangles to all polygons.

Congruent Polygons

Two polygons with n sides are congruent if and only if:

They both have n sides.
Their corresponding sides are equal in length.
Their corresponding angles are equal in measure.

Similar Polygons

Two polygons with n sides are similar if and only if:

They both have n sides.
Their corresponding angles are equal in measure.
Their corresponding sides are proportional by the same scale factor.

For Quadrilaterals:

Unlike triangles, having all corresponding angles equal does NOT necessarily make two quadrilaterals similar. The corresponding sides must also be proportional.

Example: Similar Regular Polygons

Problem: A regular hexagon has sides of length 6 cm. A similar regular hexagon has sides of length 9 cm. Find the ratio of their perimeters and the ratio of their areas.

Solution:

First, we find the scale factor:

k = 9/6 = 3/2 = 1.5

The ratio of perimeters equals the scale factor:

Perimeter ratio = k = 1.5

The ratio of areas equals the square of the scale factor:

Area ratio = k² = (1.5)² = 2.25

Therefore, the perimeter of the larger hexagon is 1.5 times the perimeter of the smaller one, and the area is 2.25 times as large.

5. Real-World Applications

Congruence and similarity have numerous practical applications in the real world:

Applications of Congruence

Manufacturing: Ensuring that all products have exactly the same specifications and measurements.
Construction: Verifying that structural elements like beams, columns, and supports are identical.
Art and Design: Creating identical patterns, tiles, or elements in a design.
Measurement: Using congruent triangles to indirectly measure distances that cannot be measured directly.

Applications of Similarity

Scale Models: Creating scaled-down models of buildings, vehicles, or other objects.
Maps: Representing geographical areas at a smaller scale while preserving proportions and angles.
Photography: Enlarging or reducing images while maintaining proportions.
Indirect Measurement: Using similar triangles to measure heights of tall objects like trees or buildings.
Shadow Problems: As demonstrated in Example 3 in the similarity section.
Computer Graphics: Scaling objects up or down in design software.

Real-World Example: Measuring the Height of a Tree

A classic application of similar triangles is measuring the height of a tall object like a tree:

Place a mirror on the ground at point M.
Stand at a location where you can see the top of the tree reflected in the mirror.
Measure the distance from your feet to the mirror (d₁).
Measure the distance from the mirror to the base of the tree (d₂).
Measure your eye height (h₁).