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					Section 4.4   Use the SAS Congruence Postulate


Posutlate 20 Side-Angle- Side (SAS) Congruence
IF two sides and included angle of one triangle
are congruent to two sides and the included
angle of a second triangle, then the two triangles
are congruent.
If:
Side RS  UV
Angle R  U
Side RT  UW
RST  UVW
EXAMPLE 1          Use the SAS Congruence Postulate

Write a proof.

GIVEN        BC     DA, BC AD

 PROVE            ABC         CDA

         STATEMENTS                      REASONS

         S    1.    BC        DA         1. Given
              2.    BC        AD         2. Given
         A 3.           BCA        DAC   3. Alternate Interior
                                            Angles Theorem
         S 4.       AC        CA         4. Reflexive Property of
                                            Congruence
EXAMPLE 1         Use the SAS Congruence Postulate


                 STATEMENTS         REASONS
            5.      ABC       CDA    5. SAS Congruence
                                        Postulate
EXAMPLE 1   Use the SAS Congruence Postulate

   Given: MP  NP, OP bisects MPN
   Prove: MOP  NOP
EXAMPLE 2      Use SAS and properties of shapes

 In the diagram, QS and RP pass through
 the center M of the circle. What can you
 conclude about      MRS and     MPQ?

 SOLUTION

Because they are vertical angles, PMQ        RMS. All
points on a circle are the same distance from the center,
so MP, MQ, MR, and MS are all equal.

 ANSWER

   MRS and   MPQ are congruent by the SAS
Congruence Postulate.
GUIDED PRACTICE           for Examples 1 and 2

In the diagram, ABCD is a square with four
congruent sides and four right angles. R,
S, T, and U are the midpoints of the sides
of ABCD. Also, RT SU and SU VU .

 1.   Prove that         SVR      UVR
        STATEMENTS                  REASONS

        1.   SV     VU              1. Given
        2.    SVR         RVU       2. Definition of   line
        3.   RV     VR              3. Reflexive Property of
                                       Congruence
        4.        SVR      UVR      4. SAS Congruence
                                       Postulate
GUIDED PRACTICE         for Examples 1 and 2

 2.    Prove that      BSR      DUT

      STATEMENTS               REASONS

      1.   BS     DU            1. Given
      2.    RBS        TDU      2. Definition of   line
      3. RS       UT            3. Given
      4.    BSR        DUT      4. SAS Congruence
                                   Postulate
EXAMPLE 2   Use SAS and properties of shapes

   In the diagram R is the center of the circle.
   If angle SRT is congruent to angle URT,
   what can you conclude about triangle SRT
   and Triangle URT?
Section 4.4   Right Angles


                  Right Triangles:
      In a right triangles the sides adjacent
       to the right angle are called the legs.

        The side opposite the right angle is
         called the hypotenuse of the right
                       angle
Section 4.4   Right Angles

Theorem 4.5 Hypotenuse-Leg(HL) Congruence
If the hypotenuse and a leg of a right triangle
are congruent to the hypotenuse and a leg of a
second right triangle, then the two triangles are
congruent.




                             ABC  DEF
EXAMPLE 3        Use the Hypotenuse-Leg Congruence Theorem

Write a proof.

GIVEN    WY      XZ, WZ    ZY, XY   ZY

 PROVE      WYZ           XZY

 SOLUTION

Redraw the triangles so they are
side by side with corresponding
parts in the same position. Mark
the given information in the
diagram.
EXAMPLE 3      Use the Hypotenuse-Leg Congruence Theorem

   STATEMENTS                   REASONS

     1.   WY      XZ           1. Given
     2. WZ ZY, XY       ZY     2. Given
     3.     Z and Y are        3. Definition of lines
            right angles
     4.     WYZ and XZY are 4. Definition of a right
            right triangles.   triangle
   L 5. ZY        YZ           5. Reflexive Property of
                                  Congruence
     6.     WYZ        XZY     6. HL Congruence
                                  Theorem
EXAMPLE 3   Use the Hypotenuse-Leg Congruence Theorem


      Given YW  XZ , XY  ZY
      Prove: XYW  ZYW
EXAMPLE 4      Choose a postulate or theorem

 Sign Making

You are making a canvas sign to hang on the triangular
wall over the door to the barn shown in the picture. You
think you can use two identical triangular sheets of
canvas. You know that RP QS and PQ PS . What
postulate or theorem can you use to conclude that
    PQR        PSR?
EXAMPLE 4     Choose a postulate or theorem

 SOLUTION

You are given that PQ PS . By the Reflexive Property, RP
RP . By the definition of perpendicular lines, both
  RPQ and RPS are right angles, so they are congruent.
So, two sides and their included angle are congruent.

 ANSWER

 You can use the SAS Congruence Postulate to conclude
 that   PQR       PSR.
EXAMPLE 4    Choose a postulate or theorem

   If you know that AB  BC and
   ABD  CBD, what postulate or theorem can you use
   conclude that ABD and CBD are congruent?
GUIDED PRACTICE        for Examples 3 and 4

Use the diagram at the right.

 3.   Redraw     ACB and    DBC side by
      side with corresponding parts in the
      same position.
GUIDED PRACTICE        for Examples 3 and 4

Use the diagram at the right.

 4.    Use the information in the diagram to
       prove that    ACB       DBC

      STATEMENTS                 REASONS

       1.   AC   DB             1. Given
       2. AB BC, CD     BC      2. Given
       3.    C    B             3. Definition of   lines
       4.   ACB and DBC are 4. Definition of a right
            right triangles.   triangle
GUIDED PRACTICE      for Examples 3 and 4

   STATEMENTS                 REASONS

  L 5. BC       CB             5. Reflexive Property of
                                  Congruence
     6.   ACB        DBC       6. HL Congruence
                                  Theorem
Summary          Summary


   Summary: Summarize the major points
  How can you use two sides and an angle to
        prove triangles congruent?
                    Day 1
                  p.243-246
          1, 2 ,4-even, 9-18, 25-27
                    Day 2
                  p.243-246
          19-24, 31-39, 42-48 even

				
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