Geometry: Congurent and Similar Triangle Theorems
 This is a free lesson from our course in Geometry

 In this lesson you’ll explore concepts of Congruent and Similar Triangle, an important part of the geometric applications. First the related and earlier learnt basics will be reviewed, which will follow on the important properties, related theorems/postulates 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 learn to identify and understand why; two triangles are ‘congruent’ or ‘similar’ and how it can be applied to solve home work problems as also use them for real life applications. (More text below video...)
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(Continued from above) Two triangles are said to be congruent if the corresponding angles and sides have the same measurements. Similar triangles have the same shape, but the size may be different. Notice that congruent is represented by ‘()’ and similar to is represented by ‘(~)’. E.g. Two triangles are similar if:
- two pairs of corresponding angles are congruent (thus the third pair of is also congruent).
- three pairs of corresponding sides are proportional.
If you know triangles are similar, you can use the proportion of corresponding sides to help determine an unknown dimension.

Proving Congruence of Triangles:
To prove that triangles are congruent, it is not necessary to prove that all three pairs of corresponding angles and all three pairs of corresponding sides are congruent. The other approach can be: E.g. if two pairs of corresponding angles are congruent, then the third angle pair is also congruent, since triangles interior angles equals 180°. The other approach could be shorter method for determining congruence between triangles without proving the congruence of all the six corresponding parts. These are called SAS, ASA and SSS.
Congruent Triangles
The smallest number of sides that a polygon can have is three. A triangle is a polygon with exactly three sides.
ABC  DEF

The correspondence establishes six facts for these triangles. Three facts about corresponding sides and corresponding angles:
Corresponding parts of congruent triangles are equal in measure.
- A and D are corresponding congruent angles.
opposite A, and is opposite D.
- and are corresponding congruent sides

Equivalence Relation of Congruence
The relation “is congruent to” is an equivalence relation for the set of triangles or the set of polygons with a given number of sides. The related important postulates are given below:
- Reflexive property: ABC ABC i.e. any geometric figure is congruent to itself.
- Symmetric property: If ABC DEF, then DEF ABC i.e. congruence may be expressed in either order.
- Transitive property: If ABC DEF and DEF RST, then ABC RST i.e. the two geometric figures congruent to the same geometric figure, are congruent to each other.

Proving Triangles Congruent Using Side, Angle, Side
Postulate
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. (in short written as, SAS)
 in ABC DEF => ABC DEF
Proving Triangles Congruent Using Angle, Side, Angle
Postulate
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. (in short written as, ASA)
 In ABC and DEF, If => ABC DEF
Proving Triangles Congruent Using Side, Side, Side
Postulate
Two triangles are congruent if the three sides of one triangle are congruent, respectively, to the three sides of the other. (in short written as, SSS)
 In ABC and DEF, If then => ABC DEF
Proving Similarity of Triangles:
There can be three ways to do. It involves the methods; without actually knowing the measure of all the six parts of each triangle, to show that all corresponding angles are congruent and all corresponding sides are proportional.
Proving Triangles Similarity Using Angle, Angle
If two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. Because if two angle pairs are the same, then the third pair shall also be equal. When the three angle pairs are all equal, the three side pairs must also be proportional (Fig: below): You may notice that if any of the A’, B’, C’ moves the other two should also move to form the triangle and of the same shape. Thus any triangle with pair of three congruent angles will be similar. In the case if three vertices of the triangles are at the same distance from one another, then the triangle s will be congruent i.e. congruent triangles are a subset of similar triangles.

Proving Triangles Similarity Using Side, Angle, Side
If two pairs of corresponding sides are in proportion, and the included angle of each pair is equal, then the two triangles are similar. E.g. A’B’C’ ~ ABC. Note that with the three vertices fixed and two of the pairs of sides proportional, the third pair of sides must also be proportional (Figure: below, NTS).
Proving Triangles Congruent Using Side, Side, Side Look at the other method now known as by side, side, and side (in short form SSS,). If the measures of corresponding sides are known, then their proportionality can be calculated. If all three pairs are in proportion, then the triangles are similar (Figure: below, NTS).

Look at some of the illustrations to explain for deeper understanding of some of these situations below:

Example 1: Given- In the Figure (NTS) below, there are two overlapping triangles ANM and BMN. If we know that A =B and BMN = ANM, are these triangles congruent?

 Step 1: Since these triangles share a common side MN, notice that it is AAS situation. Step 2: This can be explained as, two corresponding angles are equal and a third side MN, shared by both triangles, has to be the same in both triangles. Therefore, the other three corresponding parts must also be equal. E.g. sides AN and BM have the same measure. Thus ANM BMN, is the final answer.
Example 2: Given- in the Figure below (NTS) of ABC and PQR, assume that each hypotenuse of these right triangles, AB and PQ, has a measure of 2m and that one of the acute angles in each has a measure of 37º. State what other information can you have about these triangles? Step 1: The hypotenuses and one acute angle of each triangle are equal (given) br /> Step 2: Now, on the basis of known fact- “Two triangles are congruent if their hypotenuses and one of the acute angles have the same measure”, these triangles are congruent and all the six corresponding parts are equal in measure.
SStep 3: For example, both angles A and P must have measure 53º in order for the acute angles to be complementary (the three angles of these triangles sum up to 180º).
Thus, PQR ABC, as the final answer. .
 Example 3/b>: Given- in the figure below (NTS), calculate the similarity ratio. Step 1: Recall the ratio of the lengths of corresponding sides is called the similarity ratio. SStep 2: In the figure on right (NTS)- PQR ~ PST, (Given) - PQR = 45°, PRQ  = 120°, and thus P = 15° & PTS = 120°, and P =15°. Step 4: Also PR = 6, PQ =8 and QR =4 (units), PT = 12, PS = 16 and ST = 8 (units) These two triangles i.e. PQR ~ PST and have a similarity ratio of 1/2, the final answer.
 Example 4/b>:: Given- ABC ~ A’B’C’. What is the perimeter of A’B’C’? The ABC is similar to (~) A’B’C’ (as given). Step 1: The ratio of perimeter of A’B’C’ & ABC i.e. P ( A’B’C’)/ P ( ABC) Step 2: P (A’B’C’)/ P (ABC) = 8/4 = 2. Thus P (A’B’C’) = 2 * P (ABC) Step 3: P (img src="http://www.winpossible.com/App_Themes/default/Images/CourseImages/triangle.JPG" /> A’B’C’) = 2 * (4 + 6 + 8) = 2 * 18 = 36 units The perimeter of A’B’C’ is 36 units, as the final answer.

Example 5
: Given- which of the following conditions are true for similar triangles?
1) A pair of triangles has one equal side as well as one equal angle. br /> 2) A pair of triangles has three equal angles and proportional sides.
3) A pair of triangles has only one proportional side.
4) A pair of triangles has two proportional angles.
The correct choice is 2) i.e. ‘A pair of triangles has three equal angles and proportional sides’, as the final answer.

Example 6:: Given- identify if triangle ABC congruent to triangle PQR? Mention logic for your answer.
Step 1: No. Though, triangle ABC appears congruent to triangle PQR. But the vertices of ABC and PQR don't correspond right for them to be congruent, as the final answer.

Remember: br /> - if limited measures of two geometric figures are known, you may be able to prove their congruence or similarity.
- if the figures are congruent or similar, you can calculate measures of relationship of their corresponding parts that are not known.
The video above will explain more in detail about Congruent and Similar Triangles, and how to apply the concepts in solving real-world 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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