Architects and engineers often need to deal with various shapes and, more importantly, maintain a relative proportion with another shape. Similarity and congruence are two standard methods used to compare two figures. Let’s find out how these two concepts are used and what are the differences between them.

What is congruence?

When two shapes are congruent, they are the exact copies of one another. For example, two coins of the same value are congruent because they have the same radius. Congruent figures can be superimposed on one another. The shapes don’t need to be in alignment; one can rotate the figures to establish congruence.

One of the real-life uses of congruence is seen in comparing triangles, which helps engineers make rigid and robust frameworks.

Generally, we compare the corresponding sides (denoted by S) and angles (denoted by A, shown in red) to check for congruence while ≅ or ≡ are the signs that denote congruence:

The three sides of the triangle are equal (SSS)

Two pairs of corresponding sides and the angle between them are equal (SAS)

Two pairs of corresponding angles and the corresponding side between them are equal (ASA)

Two pairs of corresponding angles are equal, and the corresponding side NOT between the two angles is equal (AAS)

In a right-angled triangle, if the corresponding hypotenuse and one other side are equal, then the two triangles are congruent

What are similar figures?

Similar shapes are of the same shape but are different in size. Take the example of a magnifying glass. When we see something through a magnifying glass, the object appears larger than it actually is, but the shape of the object remains the same. Similar figures, especially triangles, are extensively used to indirectly scale a building or tower without climbing the structure. Similarity between two triangles is denoted by ∼. We label two triangles as similar if:

If two corresponding angles are equal (AA)

If all three corresponding sides of the two triangles are in the same ratio: AB/PQ = BC/QR=CA/RP=k

If the two corresponding sides of the two triangles are proportional and the corresponding angle between them is equal.

Congruent vs. Similar

Congruent

Similar

Two congruent figures can be superimposed on one another despite their alignment in space.

Two similar figures are not superimposable on one another.

Congruent shapes are a subtype of similar shapes.

Similar shapes can be congruent if their sizes are the same.

Congruent shapes need to be of the same size.

Similar shapes may or may not be of the same size.

Application of congruent and similar figures

Scale drawing:
Similar figures help us make “pilot” or small-scale models of ships, skyscrapers, etc., which can be tested in labs. The lab mimics the environment the actual skyscraper or the ship would face. This technique effectively finds faults in the design or measurements without the risk of a heavy investment. On the artistic end, scale drawings are used to perceive depth. For example, an artist draws a portrait on the canvas. However, he can’t fit the actual dimensions of the model he is drawing. Instead, the artist estimates the distance between the two eyes or the length of the limbs and shrinks the measurements to fit the canvas size.

Shadows:
There’s a fun way to see how scales work with the help of shadows
Step 1: Cut out a triangle (or any other shape you like).
Step 2: Light a flashlight in the direction of the triangle.
Step 3: Move the flashlight back and forth, and notice how the shadow changes. Measure the shadow and try to find the proportion between the corresponding sides.
You’ll notice that a larger shadow forms when the flashlight is nearer, and the shadow shrinks while moving the flashlight away from the triangle.

Cameras and depth perception; Photography involves sizing down a bigger material to a smaller one. Yet, we often estimate the size of the object. This is because we can perceive depth, or in simple terms, our eyes can roughly estimate how far or near the thing is just by looking at the surroundings. The farther the object, the smaller the size, and vice versa.

Our eyes unite us with geometry through similar shapes and congruency. We can’t always bring a ruler or test an actual ship on the unruly waves. The success of humans in geometry would have been impossible without one man: Euclid. Congruency and similarity are the brainchild of him. You can learn more about him here.

Copyright @smorescience. All rights reserved. Do not copy, cite, publish, or distribute this content without permission.

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