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# Chapter Four

### Congruent Triangles

TermDefinition
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.
Created by: amgeometry