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Similar Triangles and Angles, Side Relationship
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Similar Triangles and Angles, Side Relationship

Similar triangles are an important topic in geometry, in terms of 'real life' applications. Similarity is the concept that forms the basis of scale drawing in architecture and engineering, used in building scale models from toy model airplanes to scale models in industry and architecture. The concepts are also used for measuring the heights of inaccessible objects e.g. finding the heights of inaccessible buildings, mountains, and distances in navigation etc.

In the professional world, there are many other practical applications that can be used reliably on triangle similarity. E.g. one such application is in medical science, where similar triangles are used to calculate the position of radiation treatment for cancer patients. The properties can be used to calculate the position of radiation beams so as to avoid their overlap on the spinal cord. Can you think of any other applications of triangle similarity?
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Similar Triangles: While understanding the concept of similar triangles, you may recall the earlier learning about 'Transformations'. Two triangles are said to be similar if they have the same shape, but not the same size necessarily. Further, the triangles can be oriented in different ways, or even could be a mirror reflection of each other. Say in case of congruent triangles, corresponding angles must have equal measures; however, the corresponding sides are not equal but proportional.

Congruent Triangles. Two triangles are called congruent if they have the same shape and size. Notice that the triangles need not be oriented the same way to be congruent. Two triangles are congruent if one can be lined up with the other by performing one of the transformations like: rotation or reflection, slide or any other combination of the transformations. Remember: corresponding sides of congruent triangles have the same length, and the measure of corresponding angles is also the same.

Now you'll look at angles that appear in some simple geometric pictures, such as two intersecting lines or a triangle. In such cases, it is less important to emphasize which side is the initial or which is the terminal. The concern is to be on the absolute value of measure of the angle.

You need to understand some of the following related terms:

Vertical Angles. Take case of an angle and extend its sides to form another angle. The angles thus obtained are pair of vertical angles. Notice that two intersecting lines always make two pairs of vertical angles and vertical angles have equal measures.

Parallel and Transversal Lines. Two lines are parallel if they lie in the same plane and do not intersect. A (third) line that intersects two parallel lines is called a transversal. The intersection of parallel lines and a transversal will form eight angles, some of which are with equal measures. To identify this, we make use of the above property of vertical angles, and the property that corresponding angles have equal measures.

With all the learning so far, you can actually prove a very important property of any triangle i.e. the sum of the measures of the angles of a triangle is 180 . It is known as 'Angle Sum of a Triangle' property.

Triangle Types:

The triangles can be classified according to angles and sides:

Angles:
Acute triangle: all angles are acute; Right triangle: one right angle; Obtuse triangle: one obtuse angle.

Sides:
Equilateral triangle: all he sides are equal; Isosceles triangle: two sides are equal; Scalene triangle: no sides equal.

Example: On a sunny day, Peter and Amy noticed that their shadows were appearing of different lengths. Amy measured Peter's shadow and found that it was 96 inches long. Peter then measured Amy's shadow and found that it was 102 inches long. If Peter is 5 feet 4 inches tall, how tall is Amy?
Solution:
Step 1: You may recall the properties of similar triangles and apply.
Step 2: Convert Peter's height into inches, gives 64 inches. Now set up the proportion, and you would have:
64 / x = 96 / 102.
When simplified, you'll get, x = 68
Convert 68 inches the measure in feet, it gives 5 ft and 8 inches.
Thus Amy is 5 feet 8 inches tall, as the final answer.
 
   
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