## Ch. 4 Definitions, Postulates, and Theorems

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trismalhotra  on November 26, 2011

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# Ch. 4 Definitions, Postulates, and Theorems

 Acute Triangea triange that has three acute angles
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#### Definitions

Acute Triange a triange that has three acute angles
Equiangular Triangle A triangle with 3 congruent angles
Obtuse Triangle a triangle with one obtuse angle
Right Triangle a triangle with 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
Auxiliary Line An extra line or line segment drawn in a figure to help in a proof.
Exterior Angle an angle formed by one side of a triangle and the extension of another side
Remote Interior Angles the two nonadjacent interior angles corresponding to each exterior ange of a triangle
Flow Proof a type of proof that uses arrows to show flow of a logical argument statements are connected by arrows to show how each statement comes before it. and each reason is written below the statement it justifies
Corollary a theorem with a proof that follows as a direct result of another theorem
Congruent Having the same size and shape
Congruent Polygons two polygons whose corresponding sides and angles are congruent
Corresponding Parts The parts of congruent figures that match.
Definition of Congruent Polygons two triangles are congruent, if and only if their corresponding parts are equal.
Included Angle the angle formed by two adjacent sides of a polygon
Included Side a side included between 2 angles
Legs of an Isosceles Triangle The congruent sides of the isosceles triangle
Vertex Angle the angle between the two sides of equal length
Base Angles the angles whose vertices are the endpoints of the base of the triangle
Transformation an operation that maps an original geometric figure onto a new figure
Preimage the original figure
Image the new figure
Congruence Transformation transformation changes the position of a figure without changing its size or shape
Isometry same as a congruence transformation
Reflection A transformation that "flips" a figure over a mirror or reflection line.
Translation A transformation that "slides" each point of a figure the same distance in the same direction.
Rotation A transformation that turns a figure about a fixed point at a given angle and a given direction.
Coordinate Proof proof that uses figures in the coordinate plane & algebra to prove geometric concepts.
Placing Triangles on a Coordinate Plane Step 1: Use origin as vertex
Step 2: Place at least one side on an axis
Step 3: Keep triangle in the first quadrant
Step 4: Use simple coordinates
SSS Congruence when three sides of one triangle are congruent to three sides of another triangle
SAS Congruence 2 sides, 1 angle
ASA Congruence two triangles are congruent if two angles and the included side of one triangles are congruent, respectively, to two angles and the included side of the other
Triangle Angle-Sum the rule that all angles of a triangle must add up to 180˚
Exterior Angle Theorem measure of the exterior angle is equal to the sum of the remote interior angles
Triangle Angle-Sum Corollary #1 the acute angles of a right angle are complementary
Triangle Angle-Sum Corollary #2 there can be at most one right or obtuse angle in a triangle
Third Angle Theorem If 2 angles of 1 triangle are congruent to 2 angles of a second triangle then the third angles of the triangles are congruent
Properties of Triangle Congruence Reflexive, Symmetric, and Transitive Properties
AAS Congruence If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent.
Leg-Leg Congruence If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent
Hypotenuse-Angle Congruence If the hypotenuse and 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.
Leg-Angle Congruence 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.
Hypotenuse-Leg Congruence if the hypotenuse and a leg of a right triangle are congruent to the hypotenuse of another right triangle then the right triangle then the triangle are congruent
Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent
Converse of Isosceles Triangle Theorem if two angles of a triangle are congruent, then the sides opposite those angles are congruent
Equilateral Triangle Corollary #1` a triangle is equilateral iff it is equiangular
Equiangular Triangle Corollary #2 each acute angle of an equiangular triangle is 60 degrees

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