Proofs Geometry Chapter 2 Proof Step by step

Proofs Geometry - Chapter 2

Proof • Step – by – step from given to answer • Each step has a reason – Reasons are theorems, postulates or definitions • Two types: – Algebraic – Solving mathematical equations – Geometric – Proving or finding measures

Algebraic Proof Tools (Reasons)

Example • Algebraic Example: Solve: -5 = 3 n + 1 Solve for x:

Geometric Proofs Geometry 2. 6 • Typically used to establish relationships or prove congruence – Then used to solve or work with Algebraic aspects • Definitions – Statements that describe a mathematical object or relationship • Postulates – Relationships that have always proven accurate (accepted without proof) • Theorems – Statements that have been proven

Geometric Proofs - Example Geometry 2. 6

Triangles • Attributes: What do we know about triangles?

Triangles • Types - Classifications – By Angles: • Acute • Obtuse • Right – By Congruent Parts: • Scalene • Isosceles • Equilateral

Triangles – Angle Relationships Geometry 4. 3 • Triangle Sum Theorem – Sum of the angle measures of a triangle is 180 o • Example: p. 232 • Exterior Angle Theorem – Measure of the exterior angle of a triangle = sum of the opposite interior angles • Example: p. 233 • Third Angle Theorem – If 2 angles of one triangle are congruent to 2 angles of another – their third angles are congruent • Example: p. 234

Congruence - Congruent Figures Geometry 4. 4 • Same Size – Same Shape – All angles are congruent (same measure) – All lengths are congruent (same length) • Any shape, 2 or 3 dimensional • Labeling: – Congruent parts in the same order in both figures

Triangle Congruence (4. 5 -4. 7) • Several ways to prove congruence – SSS – Three sides congruent: p. 250 – SAS – Two sides & included angle: p. 251 – ASA – Two angles & included side: p. 261 – AAS – Two angles and an adjoining side: p. 262 – HL (Right Triangles only) – Hypotenuse/Leg: p. 263 – CPCTC • Corresponding Parts of Congruent Triangles are Congruent • P. 268

Parallelograms • Attributes - Quadrilateral – Both pairs of opposite sides are Parallel – Both pairs of opposite sides are congruent – One pair opposite sides are congruent AND parallel – Diagonals of quadrilateral bisect each other – Both pairs of opposite angles are congruent • Special Cases – Rectangle – Rhombus – Square

Parallelograms • Using Attributes in Proofs • To prove a parallelogram • To prove triangles