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3.1 Congruent Triangles

Learn how to prove triangles congruent using postulates and theorems like SSS, SAS, ASA, and AAS. Explore the reflexive, symmetric, and transitive properties of congruent triangles. Warning: AAA or SSA do not prove congruence.

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3.1 Congruent Triangles

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  1. 3.1 Congruent Triangles • Two triangles are congruent if one coincides with (fits perfectly over) the other. • 2 triangles are congruent if the six parts of the first triangle are congruent to the six corresponding parts of the second triangle. The symbol (↔) represents “corresponding”. • All pairs of corresponding 2. All pairs of corresponding angles are congruent: sides are congruent. M  R N  S Q  T AB  DE BC  EF AC  DF ▬ ▬ ▬ ▬ ▬ ▬ Section 3.1 Nack

  2. Constructing a Triangle whose sides are equal to given line segments Given three line segments: A A B • Draw a line segment larger than AB and then swipe an arc the length of AB. • Set your compass to the length of AC and then draw an arc above AB with your compass centered on A • Set your compass to the length of BC and draw an arc above AB with your compass set on B. See Example 2 p. 129 )B )C )C ▬ ▬ ▬ ▬ ▬ ▬ Section 3.1 Nack

  3. Relationships of Congruent Triangles and Methods for Proving Triangles CongruentPostulates and Theorems • The property of congruency of triangles is reflexive, symmetic and transitive. What does this mean? • There are four ways to prove two triangles congruent: • SSS (side, side, side): Postulate 12: If the three sides of one triangle are congruent to the three sides of a second triangle, then the triangles are congruent. Ex. 3 • SAS (side, angle, side): Postulate 13: If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent. Ex. 4 ASA (angle, side, angle): Postulate 14: 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. Ex. 5 • AAS (angle, angle, side): Theorem 3.1.1: If two angles and a nonincluded side of one triangle are congruent to two angles and a non-included side of the second triangle, then the triangles are congruent. Ex. 6 • Warning! AAA or SSA do not prove two triangles are congruent Section 3.1 Nack

  4. Proving Congruent Triangles ▬ ▬ ▬ ▬ Given: VP  TU; TP  PU Prove: ΔTVP  ΔPVU PROOF Statements Reasons • VP  TU |1. Given • 1 2 |2. If two lines are , they | must form  ’s 3. TP  PU |3. Given • VP  VP |4. Identity or Reflexive* 5. ΔTVP  ΔPVU |5. SAS *Reflexive Property of Congruence (Identity): A line segment (or an angle) that is congruent to itself. ▬ ▬ ▬ ▬ ▬ ▬ 1 2 Section 3.1 Nack

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