1. M  R N  S Q  T AB  DE BC  EF AC  DF
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 ▬ ▬ ▬ ▬ ▬ ▬
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2. Constructing a Triangle whose sides are equal to given line segments
Given three line segments: 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 ▬ ▬ ▬ ▬ ▬ ▬
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3. There are four ways to prove two triangles congruent:
Relationships of Congruent Triangles and Methods for Proving Triangles Congruent Postulates 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
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4. Proving Congruent Triangles
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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.
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