4.5 Proving ?s are ? : ASA and AAS

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4.5 Proving ?s are ? ASA and AAS
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Objectives

• Use the ASA Postulate to prove triangles
  congruent
• Use the AAS Theorem to prove triangles congruent

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Postulate 4.3 (ASA)Angle-Side-Angle 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 (AAS) Angle-Angle-Side 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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Proof of the Angle-Angle-Side (AAS) Congruence
Theorem

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

D
A
B
F
C
Paragraph Proof 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.
E
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Example 1

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

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Example 1

• 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.
  Thus, you can use the AAS Congruence Theorem to
  prove that ?EFG ? ?JHG.

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Example 2

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

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Example 2

• 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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Example 3

• Given ADEC, 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
• Given
• Given
• If lines, then alt. int. ?s are ?
• Vertical Angles Theorem
• ASA Congruence Postulate

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Assignment

• Geometry Pg. 210 9 14, 25 28
• Pre-AP Geometry Pg. 211 10 18 evens, 25 -
  28