CONGRUENCE AND SIMILARITY - Amazon S3 .CONGRUENCE AND SIMILARITY ... Two triangles are congruent

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  • CONGRUENCE AND SIMILARITY

    The figures that have the same size and the same shape, i.e. one

    shape fits exactly onto other is called Congruent figures .

    CONGRUENT TRIANGLES:

    1. Two triangles are congruent if they have the same size and the same shape.

    2. If two triangles are congruent, then

    a) Their corresponding angles are equal and

    b) Their corresponding sides are equal.

    3. If ABC = PQR, then

    A P

    B C Q R

    1.1CONGRUENT FIGURES

    A = P AB = PQ

    B = Q BC = QR

    C = R AC = PR

  • TIPS FOR STUDENTS: 1. The symbol means is congruent to 2. The matching diagram shows how the corresponding vertices match . ABC PQR, The corresponding vertices must be named in the correct order. e.g. BCA QRP and ACB PRQ

    TEST OF CONGRUENCY BETWEEN TWO TRIANGLES:

    TEST

    DESCRIPTION

    SSS

    Two triangles are congruent if all Three Corresponding Sides are equal .

    A P B C Q R This is known as the SSS rule. ( side, side, side )

    If AB = PQ

    BC = QR

    AC = PR

    ABC = PQR .

  • TEST

    DESCRIPTION

    SAS

    Two triangles are congruent if Two Corresponding Sides and the Included angle are equal .

    A P B C Q R This is known as the SAS rule. ( side, angle, side )

    TEST

    DESCRIPTION

    AAS

    Two triangles are congruent if Two Angles and a Corresponding Sides are equal . A P

    B C Q R This is known as the AAS rule . ( angle, angle, side )

    If AB = PQ

    AC = PR and

    A = P then

    ABC = PQR .

    If A = P and

    B = Q then

    BC = QR

    ABC = PQR .

  • TEST

    DESCRIPTION

    ASA

    Two triangles are congruent if Two angles and the Included Sides are equal .

    A P B C Q R This is known as the ASA rule. ( angle, side, angle )

    If A = P and

    B = Q then

    AB = PQ

    ABC = PQR .

  • EXAMPLE1:

    In the ABE CDE . BAE = 900, AED = 600, BC = 3 cm and DE = 18 cm.

    FIND A

    a) The length of AE ,

    C

    b) CDE . B E

    3 600

    18

    D

    TEST DESCRIPTION

    RHS

    Two triangles are congruent if both triangles have a Right Angle, equal Hypotenuse and another Side which is equal. A P B C Q R This is known as the RHS rule. ( right angle, hypotenuse, side )

    If C = R

    AB = PQ and

    AC = PR then

    ABC = PQR .

  • SOLUTION :

    A

    a) ABE = CDE ( Given )

    BE = DE = 18 cm C 600

    B E

    3 15 300

    CE = 18 - 3 cm = 15 cm

    AE = CE = 15 cm 18

    D

    b) DCE = BAE = 900

    CED = 600 2

    = 300

    CED = 1800 - 900 300 ( sum of )

    EXAMPLE2:

    In the quadrilateral ABCD, BE = CE and AE = DE .

    Corresponding sides are equal .

    Corresponding sides are equal .

    Corresponding sides are equal .

    AEB = CED sinceCorresponding sides are equal .

  • a) Prove that triangle AEB is congruent to B C

    triangle DEC .

    b) Name a triangle that is congruent to

    triangle ABD .

    c) Name a triangle that is congruent to

    triangle ABC . A D

    SOLUTION:

    a) BE = CE

    Given

    AE = DE

    AEB = DEC ( vert. opp. s )

    AEB = DEC ( SAS rule )

    b) DEC

    E

    BD = CA

    BDA = CAD ( Base s of isos . AEB )

    AD ( common sides )

    AEB = DEC ( SAS rule )

  • c) DCB

    1. Under a transformation, an object is formed onto its image .

    2. A figure and its image are congruent under a translation, a rotation and a reflection .

    Translation Rotation Reflection

    Two figures are similar if

    AC = DB

    ACB = DBC ( Base s of isos . EBC )

    BC ( common sides )

    ABC = DCB ( SAS rule )

    1.2 CONGRUENCE AND TRANSFORMATION

    1.3 SIMILAR FIGURES

  • a)The corresponding angles are equal and

    b)The corresponding sides are in the same ratio .

    Two similar figures have the same shape but not necessarily the same size.

    TIP FOR STUDENTS:

    When two figures are congruent, they are also similar . However the converse is not true.

    SIMILAR TRIANGLES:

    1. Two triangles are similar if

    a) Their corresponding angles are equal and

    b) Their corresponding sides are in the same ratio.

    2. If ABC is similar to PQR, then

    A = P

    B = Q and

    =

    =

    C = R

  • P

    A

    B C Q R

    TEST FOR SIMILARITY BETWEEN TWO TRIANGLES:

    One of the following conditions is sufficient for two triangles to be similar.

    1. Two triangles are similar if two of their corresponding angles are equal.

    A P

    B C

    Q R

    If A = P and

    B = Q then

    ABC