CONGRUENT TRIANGLES In this tutorial we will learn

CONGRUENT TRIANGLES In this tutorial, we will learn about congruent triangles, principles of congruent triangles and their applications.

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At the and of this lesson you will be able to : define the concept of congruent shape. n explain the congruence of two triangles. n n determine the sufficient conditions for congruence of two triangles. n use the cogruence of triangles for solving problems.

TABLE OF CONTENTS The topicis divided into the following subsections: n Introduction n Congruence of Triangles n Rules of Congurent Triangles n n n n The Side-Side (SSS) Congruence Rule The Side-Angle-Side(SAS)Congruence Rule The Angle-Side-Angle (ASA) Congruence Rule The Angle-Side(AAS) Congruence Rule The Right Angle-Hypotenuse-Side (RHS) Congruence Rule Summary References

Introduction Table of contents n If two figures have the same size and shape, then they are said to be congruent figures. n Square ABCD is congruent to square EFGH as their corresponding sides and angles are equal. n Note: Congruent figures are exact duplicates of each other. One could be fitted over the other so that their corresponding parts coincide. The concept of congruence applies to figures of any type.

Congruence of Triangles Table of contents n Two triangles are said to be congruent if all the sides and the angles of one triangle are respectively equal to corresponding sides and angles of other triangle. n The equal sides and angles may not be in the same position (if there is a turn or a flip), but they will be there.

Table of contents △ABC and △DEF have the same size and shape. In fact, if you could “pick up” triangle ABC you fit it exactly over triangle DEF so that the vertices and sides coincide. We say △ABC is congruent to △DEF. In symbolic form: △ABC ≅ △DEF

Table of contents n The order of the points in the names of the triangles is important. △ABC and △DEF says that the triangles will coincide when A is placed on D, B on E, and C on F. ( We would not say that △ABC ≅ △EFD. ) n The congruent triangles ABC and DEF have equal corresponding sides (AB=DE, BC=EF, CA=FD) and equal corresponding angles ( <A=<D, <B=<E, <C=<F)

Rules of Congruent Triangles Table of contents 1) The Side-Side (SSS) Congruence Rule 2) The Side-Angle-Side(SAS)Congruence Rule 3) The Angle-Side-Angle (ASA) Congruence Rule 4) The Angle-Side(AAS) Congruence Rule 5) The Right Angle-Hypotenuse-Side (RHS) Congruence Rule

Table of contents 1)The Side-Side (SSS) Congruence Rule : If three sides of one triangle are equal to three sides of the other triangle, then the two triangles are congruent. △ABC ≅ △DEF

Table of contents Question : Find the value of each of pronumerals in the given pair of triangles. △ABC ≅ △DEF (SSS) Click here to see the solution x = 89, y = 58, z = 33 (corresponding angles of congurent triangles)

Table of contents 2) The Side-Angle-Side (SAS) Congruence Rule : If two sides and the included angle of one triangle are congurent to the corresponding parts of another triangle, the triangles are congruent. △ABC ≅ △DEF Click here to review the included angle

Question: (Click the space to see the solution. ) Table of contents We should use Vertical Angles Theorem. The teorem states that vertical angles are congruent, so we know that <ACB=<DCE. Now we have two pairs of corresponding, congruent sides and congruent included angles. (SAS congruence)

Table of contents 3)The Angle-Side-Angle (ASA) Congruence Rule: If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. △ABC ≅ △DEF

Table of contents Question: Use the data in the diagram to prove that △ABD ≅ △CDB In △ABD and △CDB, a=b (Alternate Angles) BD=DB (Common Side) Click here to see the solution x=y (Alternate Angles) So, △ABD ≅ △CDB (ASA).

Table of contents 4)The Angle-Side(AAS) Congruence Rule: If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent. △ ACB ≅ △A’C’B’

Compare AAS with AAS : Compare AAS with ASA : Compare AAS with AAS : Table of contents For the ASA rule the given side must be included and for AAS rule the side given must not be included. The trick is we must use the same rule for both the triangles that we are comparing

Table of contents Question: Which of the following conditions would be sufficent for the above triangles to be congruent? A a=e , x=u, c=f B x=u, y=t, z=s C a=e, y=s, z=t D a=f, y=t, z=s Solution for c) Step 1: a = e gives the S y = s gives the A z = t gives the A Step 2: a and e are non-included sides. Click here to see the solution! Follows the AAS rule. Answer: a = e, y = s, z = t is sufficient show that the above are congruent triangles.

Table of contents 5)The Right Angle-Hypotenuse-Side (RHS) Congruence Rule: If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent. △BCA ≅ △EFD

Table of contents Question : According to given triangles, decide following statements are true/false. • △ABC ≅ △DEF TRUE FALSE • x=50 y=40 TRUE FALSE △ABC ≅ △DEF (RHS congruent rule) Click for the explanation X=40, y=50 (Corresponding angles of congruent triangles)

There is a useful simulation about Congruence Rules: n http: //www. mathopenref. com/congruentsss. ht ml

summary Table of contents In this lesson you learned that: n n Congruent figures have the same size and shape. Congruent triangles have the same size and the same shape. The corresponding sides and the corresponding angles of congruent triangles are equal. There are five types of congruence rules. (SSS, SAS, ASA, AAS, RHS). How to use them by solving questions.

references The following resources have been used in this tutorial: n n n The Math Curriculum of The Ministry of Education http: //www. mathsteacher. com. au/year 9/ch 13_geometry/ 07_congruent/triangles. htm http: //www. onlinemathlearning. com/congruenttriangles. html#aas http: //www. excellup. com/classnine/mathnine/triangletheo rem. aspx http: //www. mathopenref. com/congruentsss. html