4.4 Proving Triangles are Congruent: ASA and AAS

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4.4 Proving Triangles are Congruent ASA and AAS

• Geometry
• Mrs. Spitz
• Fall 2004

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Objectives

1. Prove that triangles are congruent using the ASA
   Congruence Postulate and the AAS Congruence
   Theorem
2. Use congruence postulates and theorems in
   real-life problems.

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Assignment

• 4.4 pp. 223-225 1-22 all
• Quiz after this section

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Postulate 21 Angle-Side-Angle (ASA) Congruence
Postulate

• If two angles and the included side of one
  triangle are congruent to two angles and the
  included side of a second triangle, then the
  triangles are congruent.

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Theorem 4.5 Angle-Angle-Side (AAS) Congruence
Theorem

• If two angles and a non-included side of one
  triangle are congruent to two angles and the
  corresponding non-included side of a second
  triangle, then the triangles are congruent.

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Theorem 4.5 Angle-Angle-Side (AAS) Congruence
Theorem

• Given ?A ? ?D, ?C ? ?F, BC ? EF
• Prove ?ABC ? ?DEF

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Theorem 4.5 Angle-Angle-Side (AAS) Congruence
Theorem

• You are given that two angles of ?ABC are
  congruent to two angles of ?DEF. By the Third
  Angles Theorem, the third angles are also
  congruent. That is, ?B ? ?E. Notice that BC is
  the side included between ?B and ?C, and EF is
  the side included between ?E and ?F. You can
  apply the ASA Congruence Postulate to conclude
  that ?ABC ? ?DEF.

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Ex. 1 Developing Proof

• Is it possible to prove the triangles are
  congruent? If so, state the postulate or theorem
  you would use. Explain your reasoning.

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Ex. 1 Developing Proof

• A. In addition to the angles and segments that
  are marked, ?EGF ??JGH by the Vertical Angles
  Theorem. Two pairs of corresponding angles and
  one pair of corresponding sides are congruent.
  You can use the AAS Congruence Theorem to prove
  that ?EFG ? ?JHG.

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Ex. 1 Developing Proof

• Is it possible to prove the triangles are
  congruent? If so, state the postulate or theorem
  you would use. Explain your reasoning.

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Ex. 1 Developing Proof

• B. In addition to the congruent segments that
  are marked, NP ? NP. Two pairs of corresponding
  sides are congruent. This is not enough
  information to prove the triangles are congruent.

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Ex. 1 Developing Proof

• Is it possible to prove the triangles are
  congruent? If so, state the postulate or theorem
  you would use. Explain your reasoning.
• UZ WX AND UW
• WX.

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Ex. 1 Developing Proof

• The two pairs of parallel sides can be used to
  show ?1 ? ?3 and ?2 ? ?4. Because the included
  side WZ is congruent to itself, ?WUZ ? ?ZXW by
  the ASA Congruence Postulate.

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Ex. 2 Proving Triangles are Congruent

• Given AD EC, BD ? BC
• Prove ?ABD ? ?EBC
• Plan for proof Notice that ?ABD and ?EBC are
  congruent. You are given that BD ? BC
• . Use the fact that AD EC to identify a pair of
  congruent angles.

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Proof

• Statements
• BD ? BC
• AD EC
• ?D ? ?C
• ?ABD ? ?EBC
• ?ABD ? ?EBC

• Reasons
• 1.

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Proof

• Statements
• BD ? BC
• AD EC
• ?D ? ?C
• ?ABD ? ?EBC
• ?ABD ? ?EBC

• Reasons
• 1. Given

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Proof

• Statements
• BD ? BC
• AD EC
• ?D ? ?C
• ?ABD ? ?EBC
• ?ABD ? ?EBC

• Reasons
• Given
• Given

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Proof

• Statements
• BD ? BC
• AD EC
• ?D ? ?C
• ?ABD ? ?EBC
• ?ABD ? ?EBC

• Reasons
• Given
• Given
• Alternate Interior Angles

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Proof

• Statements
• BD ? BC
• AD EC
• ?D ? ?C
• ?ABD ? ?EBC
• ?ABD ? ?EBC

• Reasons
• Given
• Given
• Alternate Interior Angles
• Vertical Angles Theorem

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Proof

• Statements
• BD ? BC
• AD EC
• ?D ? ?C
• ?ABD ? ?EBC
• ?ABD ? ?EBC

• Reasons
• Given
• Given
• Alternate Interior Angles
• Vertical Angles Theorem
• ASA Congruence Theorem

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Note

• You can often use more than one method to prove a
  statement. In Example 2, you can use the
  parallel segments to show that ?D ? ?C and ?A ?
  ?E. Then you can use the AAS Congruence Theorem
  to prove that the triangles are congruent.