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# Chapter Four

### Congruent Triangles

Term | Definition |
---|---|

acute triangle | 3 acute angles in a triangle |

equiangular triangle | 3 congruent acute angles in the triangle |

obtuse triangle | 1 obtuse angle in the triangle |

right triangle | a triangle containing one right angle |

equilateral triangle | a triangle with three congruent sides |

isosceles triangle | a triangle with at least two congruent sides |

scalene triangle | a triangle with no congruent sides |

Theorem 4.1 Triangle Angle-Sum Theorem | The sum of the measure of the angles of a triangle is 180. |

auxiliary line | an extra line or segment drawn in a figure to help analyze geometric relationships. |

Theorem 4.2 Exterior Angle Theorem | The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. |

corollary | a theorem with a proof that follows as a direct result of another theorem. |

Triangle Angle-Sum Corollary 4.1 | The acute angles of a right triangle are complementary |

Triangle Angle Sum Corollary 4.2 | There can at most be one right or obtuse angle in a triangle. |

Congruent | two geometric figures with the same shape and size. |

congruent polygons | all the parts of one polygon are congruent to the corresponding parts or matching parts of the other polygon. |

corresponding parts | include corresponding angles and corresponding sides |

Theorem 4.3 Third Angles Theorem | If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent. |

Reflexive Property of Congruence | Triangle ABC is congruent to Triangle ABC |

Symmetric Property of Triangle Congruence | If triangle ABC is congruent to triangle EFG, then triangle EFG is congruent to triangle ABC. |

Transitive Property of Triangle Congruence | If triangle ABC is congruent to EFG and EFG is congruent to JKL, then ABC is congruent to JKL. |

Postulate 4.1 Side by SIde by Side Congruence | If three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent. |

Postulate 4.2 Side-Angle-Side Congruence | 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. |

Postulate 4.3 Angle-Side-Angle Congruence | If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. |

Theorem 4.10 Isosceles Triangle Theorem | If two sides of a triangle are congruent, then the angles opposite those sides are congruent. |

Theorem 4.11 Converse of Isosceles Triangle Theorem | If two angles of a triangle are congruent, then the sides opposite those angles are congruent. |

Equilateral Triangle Corollary 4.3 | A triangle is equilateral if and only if it is equiangular, |

Equilateral Triangle Corollary 4.4 | Each angle of an equilateral triangle measures 60. |

transformation | an operation that maps an original geometric figure |

preimage | the original geometric figure in a transformation |

image | new geometric figure in a transformation |

congruence/rigid transformation/isometry | the position of the image may differ from that of the preimage, but the two figures remain congruent. |

reflection/flip | transformation over a line called the line of reflection. Each point of the preimage and its image are the same distance from the line of reflection. |

translation/slide | transformation that moves all points of the original figure in the same distance in the same direction. |

rotation/turn | transformation around a fixed point called a center of rotation through a specific angle, in a specific direction. Each point of the original figure and its image are the same distance from the center. |

coordinate proofs | figures in the coordinate plan and algebra used to prove geometric concepts. |

flow proof | uses statements written in boxes and arrows to show the logical progession of an argument. |

base angles | the two angles formed by the base and the congruent sides |

vertex angle | the angle with sides that are the leg |

legs | the two congruent sides of an isoscles triangle |

Theorem 4.5: Angle-Angle-Side Congruence | If two angles and the nonincluded side of one triangle are congruent to the corresponding two angles and a side of a second triangle, then the two triangles are congruent. |

Theorem 4.6 Right Angle Congruence Leg/Leg LL | If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent. |

Theorem 4.7 Hypotenuse-Angle Congruence HA | If the hypotenuse and the acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the two triangles are congruent. |

Theorem 4.8 Leg-Angle Congruence LA | If one leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent. |

Theorem 4.9 Hypotenuse-Leg Congruence HL | If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. |

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amgeometry