In this lesson you’ll learn the basics and concepts of Congruence and Similarity,
an important part of the geometric applications. First the related and earlier learnt basics will be reviewed, which will follow on the important properties and relationship of congruence and similarity that help to apply them to solve real world application problems. The presentation covering such content will be done by the instructor in own handwriting, using video and with the help of several examples and solution. This will help you understand important trigonometric relationships to solve home work problems and also use them for real life applications.
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Understanding the expressions and used Terms:
Congruent Polygons  Polygons, whose corresponding sides and interior angles are all congruent. Similarly Congruent Triangles are those
whose corresponding angles and sides are all congruent.
Some of the common terms used while explaining are given below:
AA  In a triangle, if two angles are congruent to their corresponding
parts in another triangle, then the triangles are similar.
AAS  if two angles and a side not included by those angles are
congruent to their corresponding parts in another triangle, then the triangles are
congruent.
ASA  if two angles and their included side are congruent to their
corresponding parts in another triangle, then the triangles are congruent.
SAS  if two sides are congruent or proportional and their corresponding parts e.g. included angle is congruent to the parts of another triangle, then the triangles are congruent or similar, respectively.
SSS  The triangles are congruent, if the three sides of a triangle are proportional to their corresponding parts in another triangle, then the triangles are similar. Hypotenuse Leg  A method for proving congruence of right triangles: if one leg and the hypotenuse are congruent to their corresponding parts in another right triangle, the right triangles are congruent.
Similar Triangles  are the triangles, whose corresponding angles are congruent and whose corresponding sides are proportional. Notice that congruence is a subset of similarity.
Now on the basic concepts of Congruence and Similarity are explained with example:
Congruent shapes are of the same size with corresponding lengths and angles equal i.e. they are exactly the same size and shape. If you know the size and shape of one, the size and shape of the others is also known. E.g.
Notice that each of the above shapes is congruent to each other. The only difference is in their orientation, or the way they are rotated. If you lay them over each other, they will fit exactly over each other.
Congruent Polygons and corresponding parts:
Congruent polygons are polygons that have the same size and shape.
The congruent polygons are named in such a way that each vertex of ABCD corresponds to exactly one vertex of EFGH and each vertex of
EFGH corresponds to exactly one vertex of ABCD. This relationship is called a onetoone correspondence.
Two polygons are congruent if and only if there is a onetoone correspondence between their vertices such that corresponding angles are congruent and corresponding sides are congruent.
Corresponding Parts of Congruent Polygons Corresponding Angles
In congruent polygons ABCD and EFGH, vertex A corresponds to vertex E. Angles A and E are called corresponding angles.
AE BF CG
DH
Corresponding Sides
In congruent polygons ABCD and EFGH, since A corresponds to E and B corresponds to F,
corresponding sides.
Corresponding parts of congruent polygons are congruent.
Triangles
Two triangles are congruent, if they have the same shape and size and have congruent corresponding parts. The expression ‘corresponding parts’ broad sense mean the matching angles and sides of congruent triangles. Also matching vertices are corresponding vertices and are in the same order (Figure below):
In case of triangles, it may be noted that you don’t have to check every pair of sides and angles to determine whether two triangles are congruent. You found that SSS, SAS, ASA, and SAA are congruence shortcuts. Also here you will find that the similarity shortcuts also i.e. given that two pairs of corresponding sides are proportional.
In some cases, where even though the angles may correspond in size and position, however, the sides do not correspond. It is not case of being congruent (Fig: A below).
Similarity Similar shapes are like congruent shapes in the sense they have the same shape, but not in size i.e. the size is different. Their corresponding angles are congruent, but the corresponding sides are in proportion (Fig: B, above).
Similarity and proportion: you may notice that enlargement preserves the angles of a figure but multiplies each of its lengths by the same constant (the scale factor) i.e. mathematically, any original shape and its enlarged image are similar. E.g. look at the two similar triangles (see the figure below, NTS) marked on them, the measurements:
In the given measurements, find out the ratio of the lengths in each pair of sides i.e. the scale factor of enlargement. In this case, you may check it is 1.5.
Look at some of the illustrations below, to help understand better:
Example 1: Given Fig: 1, Fig: 2, Fig: 3 below. Find out which of them appear to be congruent.
Step 1: Recall the conditions to be met for any polygon to be congruent. It is polygons whose corresponding sides and interior angles are all congruent.
Step 2: Now look at the figures carefully.
Notice that figure 1 & 3 i.e. hexagonal polygons meet the condition.
Therefore, Fig: 1, and Fig: 3 are congruent, is the final answer.
Example 2: Given Amy got a job at Ford car center, where she was asked to enlarge a photo to create a poster. The photo was rectangle in shape 6 in high X 10 in wide. The poster paper size is 32 in height. Find out the scale factor and width of the poster paper.
Step 1: Scale factor = 6/32 = 5.33
Step 2: To find out the width of the paper:
32/6 = (width)/10
Width = (32/6)) 10
= 53.3 in say 54 in (rounded),
Scale factor – 5.33 and width of paper 54 in, as the final answer.
Remember:
• Any geometric figure is congruent to itself. (Reflexive Property)
• Congruence may be expressed in either order. (Symmetric Property)
• Two geometric figures congruent to the same geometric figure are congruent to each other. (Transitive Property)
• Two triangles are congruent if two sides and the included angle of one triangle are congruent, respectively, to two sides and the included angle of the other. (SAS).
• Two triangles are congruent if two angles and the included side of one triangle are congruent, respectively, to two angles and the included side of the other. (ASA)
• Two triangles are congruent if the three sides of one triangle are congruent, respectively, to the three sides of the other. (SSS)
The video above will explain more in detail about Congruence and Similarity,
and how to apply the concepts in solving realworld problems. This is explained with the help of several examples and done watching video. This helps you to deal with solving problems and help doing the Geometry home work.
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