Deductive Reasoning
"The process of showing that certain statements follow logically from agreed-upon assumptions and proven facts"-Discovering Geometry
Inductive Reasoning
"The process of observing data, recognizing patterns, and making generalizations about those patterns"-Discovering Geometry
Recognizing Patterns and Modeling Them with Equations
x | 1 | 2 | 3 | 4 | 5 | x |
y | 1 | -2 | -5 | -8 | -11 | -3x+4 |
Special Angle Relationships
Vertical Angles: 2 angles opposite each other are congruent when formed by intersecting lines.
Linear Pair: If two angles sharing a vertex are to be a linear pair, then their angles add up to 180.
Corresponding angles: If parallel lines are cut by a transversal then the corresponding angles are congruent
Alternate Interior Angles: If parallel lines are cut by a transversal then the alternate interior angles are congruent.
Alternate Exterior Angles: If parallel lines are cut by a transversal then the alternate exterior angles are congruent.
Points of Concurrency of Triangles
Incenter: The point of concurrency of angle bisectors of a triangle.
Circumcenter: The point of concurrency of the perpendicular bisectors of a triangle.
Orthocenter: The point of concurrency of the altitudes of a triangle.
Centroid: The point of concurrency of the medians of a triangle.
Constructions with Compass and Straight Edge
Incenter: The point of concurrency of angle bisectors of a triangle.
Circumcenter: The point of concurrency of the perpendicular bisectors of a triangle.
Orthocenter: The point of concurrency of the altitudes of a triangle.
Centroid: The point of concurrency of the medians of a triangle.
Constructions with Compass and Straight Edge
Discovering and Proving Triangle Properties
Triangle Sum Conjecture: the sum of the measures of the angle in every triangle is 180.
Isosceles Triangle: If a triangle is isosceles then its base angles are congruent.
Triangle Inequality: the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Side Angle Inequality: In a triangle the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.
Triangle Sum Conjecture: the sum of the measures of the angle in every triangle is 180.
Isosceles Triangle: If a triangle is isosceles then its base angles are congruent.
Triangle Inequality: the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Side Angle Inequality: In a triangle the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.
Exterior Angle: The measure of an exterior angle of a triangle is equal to the sum of the remote interior angle measures.
Triangle Congruence shortcuts: side-side-side, side-angle-side, angle-side-angle, side-angle-side
Discovering and Proving Polygon Properties
Polygon Sum: The sum of the interior angles of the polygon add up to 180(n-2)
Equilangular Polygon: Each interior angle of an equilangular "n"-gon =180-360/n
Exterior Angles: in any polygon, the sum of a set of exterior angles adds up to 360.
Recognizing Patterns and Modeling Them with Equations
I enjoyed learning about this topic because it was easy and we had done it last year, so I didn't have to put in a lot of effort.
"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful."
I enjoy learning about the recognizing patterns because I feel that it is elegant in that it visualizes an abstract equation.
Triangle Congruence shortcuts: side-side-side, side-angle-side, angle-side-angle, side-angle-side
Discovering and Proving Polygon Properties
Polygon Sum: The sum of the interior angles of the polygon add up to 180(n-2)
Equilangular Polygon: Each interior angle of an equilangular "n"-gon =180-360/n
Exterior Angles: in any polygon, the sum of a set of exterior angles adds up to 360.
Recognizing Patterns and Modeling Them with Equations
x | 1 | 2 | 3 | 4 | 5 | x |
y | 1 | -2 | -5 | -8 | -11 | -3x+4 |
"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful."
I enjoy learning about the recognizing patterns because I feel that it is elegant in that it visualizes an abstract equation.
