Download Proofs - Midthun Math

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)
Algebraic Proof Tools (Reasons)
Example
• Algebraic Example:
Solve: -5 = 3n + 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 180o
• 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
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–
–
–
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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