Y. Davis Geometry Notes Chapter 4

Classifications of Triangles Angles Sides Acute Scalene Right Isosceles Obtuse Equilateral Equiangular

Theorem 4.1 Triangle Angle-Sum Theorem The sum of the measures of the interior angles of a triangle is 180 degrees.

Auxiliary Line an extra line or segment drawn in a figure to help analyze geometric relationships.

Exterior Angles Are formed by one side of a triangle and the extension of an adjacent side. Each exterior angle has 2 remote interior angles (non-adjacent interior angles).

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.

Flow Proof Has statements written in boxes (each reason is written below the boxes.) The boxes are connected by arrows to show the logical progression of the argument.

Corollary A theorem that follows easily from a previously proven theorem.

Corollaries Triangle Angle-Sum Theorem Corollary 4.1—The acute angles of a right triangle are complementary. Corollary 4.2—There can be at most one right or obtuse angle in a triangle.

Congruent Figures Two geometric figures that have the same shape and size.

Congruent Polygons All corresponding parts (sides and angles) are congruent.

C.P.C.T.C Corresponding Parts of Congruent Triangles are Congruent. (Definition of congruent Triangles) (Can only be used once two triangles have been proven congruent)

Theorem 4.3 Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the 3rd set of angles of the triangles are congruent.

Theorem 4.4 Properties of Triangle Congruence Reflexive— Symmetric— Transitive—

Postulate 4.1 Side-Side-Side (SSS) Congruence If three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent.

Included Angle An angle formed by two adjacent sides of a polygon.

Postulate 4.2 Side-Angle-Side (SAS) Congruence If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Included Side The side located between two consecutive angles.

Postulate 4.3 Angle-Side-Angle (ASA) 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 two triangles are congruent.

Postulate 4.4 Angle-Angle-Side (AAS) Congruence If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the two triangles are congruent.

Parts of an Isosceles Triangle Base—the non-congruent side Legs—the congruent sides Vertex Angle—The included angle of the legs. Base Angles—The angles that include the base.

Theorem 4.10 Base Angles Theorem If two sides of a triangle are congruent, then the angels opposite them are congruent.

Theorem 4.11 Converse of Base Angles Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Corollaries of Equilateral Triangles Corollary 4.3—A triangle is equilateral if and only if it is equiangular. Corollary 4.4—Each angle of an equilateral triangle measure 60 degrees.

Transformation An operation that maps a preimage (original figure), onto an image (new figure).

Isometry A rigid transformation (congruent transformation)

Reflection A flip over a line (line of reflection) Each point of a preimage and image are equidistant from the line of reflection.

Translation A slide All points of the preimage are moved the same distance and same direction.

Rotation A turn about a fixed point (center of rotation) Each point of the image and preimage are equidistant from the center of rotation.

Coordinate Proof Uses figures in a coordinate plane and algebra (Midpoint, Distance, and Slope Formulas) as well as postulates, definitions and theorems to prove geometric concepts.