November 10, 2008

This resource has been contributed by Winpossible, and can also be accessed on their website by clicking here - Congurent and Similar Triangle Theorems.

In this section, you'll learn about the explanation of the theorem of congruency and similarity for triangles and how to use it. It will be done with the help of some examples and practice questions with solution, watching video as well as explanation by the instructor in own hand writing.

In this section, you'll learn about the explanation of the theorem of congruency and similarity for triangles and how to use it. It will be done with the help of some examples and practice questions with solution, watching video as well as explanation by the instructor in own hand writing.

Now we deal with the statement of angle-angle similarity theorem (AA), side-angle-side similarity theorem, side-side-side similarity theorem and how to apply to solve the given problems.

Angle-Angle-Similarity (AA) theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. E.g. the triangle *ABC* has, *A* = 75 and *B* = 60, and triangle *DEF* has, *D* = 75 and *E* = 60. Here you will see that m*A* = m*D* and m*B* = m*E*. Therefore, ?*ABC* ~ ?*DEF*.

Side-Angle-Side Similarity (SAS) states that if one angle of a triangle is congruent to one angle of another triangle and the sides that include those angles are proportional, then the two triangles are similar.E.g. the triangle *ABC* has, sides *AB* = 5, *BC* = 4 and *B* = 150 and triangle *DEF* has, sides *DE* = 10, *EF* = 8 and *E* = 150. Here you will see that m*B* = m*E* and *AB* : *BC* is proportional to *DE*:*EF* i.e. 5:4. Therefore, ?*ABC* ~ ?*DEF*.

Side-Side-Side Similarity (SSS) states that if all pairs of corresponding sides of two triangles are proportional, then the triangles are similar.E.g. the triangle *ABC* has, sides *AB* = 7, *BC* = 10 and *AC* = 8, and triangle *DEF* has, sides *DE* = 10.5, *EF* = 15 and *DF* = 12. Here you will see that *AB* : *DE *is proportional to *BC* : *EF* i.e. 2 : 3 and also *AB* : *DE* is proportional to *AC* : *DF* i.e. 2 : 3. Therefore, *AB* : *DE* = *BC* : *EF* = *AC* : *DF* = 2 : 3. Thus, ?*ABC* ~ ?*DEF*.

Side-Angle-Side congruence (SSS) states that If each side of one triangle is congruent to corresponding side of another triangle, then the triangles are congruent.

Angle-Side-Angle Congruence (ASA) states that if two angles and the side between them in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent.

Angle-Angle-Side Congruence states that if two angles and a side not between them in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent.

Hypotenuse Leg Congruence states that if the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent. For example, *AD* intersects *BE* at point *C*, *B* and *E* are right angles, and *BC* = *CE*. If *AB* = 4*x* + 15 and *DE* = *x* + 30, you will see that ?*ABC* & ?*DEC* are congruent. Therefore, *AB* = *DE* and when you plug the value of *AB* and *DE*, and solve for *x*, it gives *x* = 5. Hence, *AB* = 4(5) + 15 = 35.

This FREE mini-lesson is a part of Winpossible's online course that covers all topics within Geometry. Click on the video below to go through it. If you like it, you can buy our online course in Geometry by clicking here.

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