Lesson 4 3 and 4 4 proving triangles are congruent
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Lesson 4.3 and 4.4 Proving Triangles are Congruent. p. 212. Learning Target. I can list the conditions (SAS, SSS) to prove triangles are congruent. I can identify and use reflexive, symmetric and transitive property in my proof. How To Find if Triangles are Congruent.

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Lesson 4.3 and 4.4 Proving Triangles are Congruent

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Lesson 4 3 and 4 4 proving triangles are congruent

Lesson 4.3 and 4.4 Proving Triangles are Congruent

p. 212


Learning target

Learning Target

I can list the conditions (SAS, SSS) to prove triangles are congruent.

I can identify and use reflexive, symmetric and transitive property in my proof.


How to find if triangles are congruent

How To Find if Triangles are Congruent

  • Two triangles are congruent if they have:

    • exactly the same three sides and

    • exactly the same three angles.

  • But we don't have to know all three sides and all three angles ...usually three out of the sixis enough.

  • There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.


  • 1 sss side side side

    1. SSS   (side, side, side)

    If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

    SSS stands for "side, side, side“

    and means that we have two triangles

    with all three sides equal.

    For example:

    is congruent to:


    2 sas side angle side

    2. SAS (side, angle, side)

    If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.

    SAS stands for "side, angle, side"

    and means that we have two triangles

    where we know two sides and the included angle are equal.

    For example:

    is congruent to:


    3 asa angle side angle

    3. ASA   (angle, side, angle)

    If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

    ASA stands for "angle, side, angle“

    and means that we have two triangles

    where we know two angles and the

    included side are equal.

    For example:

    is congruent to:


    4 aas angle angle side

    4. AAS   (angle, angle, side)

    If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

    AAS stands for "angle, angle, side“

    and means that we have two triangles

    where we know two angles and the

    non-included side are equal.

    For example:

    is congruent to:


    5 hl hypotenuse leg

    5. HL   (hypotenuse, leg)

    HL applies only to right angled-triangles!

    HL stands for "Hypotenuse, Leg" (the longest side of the triangle is called the "hypotenuse", the other two sides are called "legs")

    and


    5 hl hypotenuse leg1

    5. HL   (hypotenuse, leg)

    If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

    • It means we have two right-angled triangles with

      • the same length of hypotenuseand

      • thesame length for one of the other two legs.

    • It doesn't matter which leg since the triangles could be rotated.

    • For example:

      is congruent to 


    Caution don t use aaa

    Caution ! Don't Use "AAA" !

    Without knowing at least one side, we can't be sure if two triangles are congruent..

    AAA means we are given all three

    angles of a triangle, but no sides.

    This is not enough information to decide if two triangles are congruent!

    Because the triangles can have the same angles but be different sizes:

    For example:

    is congruent to 


    Goal 2

    Goal 2

    Proving Triangles are Congruent

    E

    B

    D

    A

    F

    C

    DEF

    DEF

    ABC

    DEF

    ABC

    ABC

    DEF

    JKL

    ABC

    JKL

    If  , then  .

    L

    If  and  , then .

    J

    K

    You have learned to prove that two triangles are congruent by the definition of congruence – that is, by showing that all pairs of corresponding angles and corresponding sides are congruent.

    THEOREM

    Theorem 4.4Properties of Congruent Triangles

    Reflexive Property of Congruent Triangles

    Every triangle is congruent to itself.

    Symmetric Property of Congruent Triangles

    Transitive Property of Congruent Triangles


    Lesson 4 3 and 4 4 proving triangles are congruent

    Using the SAS Congruence Postulate

    Prove that

    AEBDEC.

    1

    2

    1

    2

    Statements

    Reasons

    AE  DE, BE  CEGiven

    1  2Vertical Angles Theorem

    3

    AEBDEC SAS Congruence Postulate


    Lesson 4 3 and 4 4 proving triangles are congruent

    Proving Triangles Congruent

    ARCHITECTURE You are designing the window shown in the drawing. You

    want to make DRAcongruent to DRG. You design the window so that

    DRAG and RARG.

    D

    A

    G

    R

    GIVEN

    DRAG

    RARG

    DRADRG

    PROVE

    MODELING A REAL-LIFE SITUATION

    Can you conclude that DRADRG?

    SOLUTION


    Lesson 4 3 and 4 4 proving triangles are congruent

    Proving Triangles Congruent

    GIVEN

    RARG

    DRADRG

    PROVE

    1

    2

    6

    3

    4

    5

    Statements

    Reasons

    Given

    DRAG

    If 2 lines are, then they form

    4 right angles.

    DRA and DRG

    are right angles.

    Right Angle Congruence Theorem

    DRADRG

    DRAG

    Given

    RARG

    DRDR

    Reflexive Property of Congruence

    SAS Congruence Postulate

    DRADRG

    D

    A

    R

    G


    Given sp qr qp pr prove spq spr

    Given: SP  QR; QP  PRProve  SPQ  SPR

    S

    StatementsReasons

    1. Given

    1. SP  QR; QP  PR

    2. QPS and RPS are right ’s.

    2. Def. of 

    3. QPS  PRS

    3. Rt.   Thm.

    4. SP  SP

    4. Reflexive POC

    5.  SPQ  SPR

    5. SAS  Post.

    Q

    R

    P


    Pair share

    Pair-share

    Work on classwork on “Congruence Triangle”

    Sage and Scribe on #21 to #24


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