Theorems, Conjectures, & Postulates

Triangle Theorems

Triangle Sum Theorem- The sum of the measures of the angles in every triangle is 180˚

 

Exterior Angle Theorem- The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

 

Corollary to the Triangle Sum Theorem- In a right triangle, the acute angles are complementary.

 

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

 

Converse of Base Angles Theorem – If two angles of a triangle are congruent, the sides opposite them are congruent.

 

Corollary to the Base Angle Theorem – If a triangle is equilateral, then it’s equiangular.

 

Corollary to the Converse of Base Angles Theorem- If a triangle is equiangular, then it is equilateral.

 

Isosceles Triangle Theorem - if a triangle is isosceles then the base angles are congruent.

 

Triangle Inequality Theorem – The sum of the lengths of any two sides of a triangle is longer than the length of the 3^rd side

 

Parallel/Proportionality Conjecture - If a line parallel to the side of a triangle passes through the other two sides, then it divides the other two sides proportionally. If a line cuts 2 sides of a triangle proportionally, then it’s parallel to the third side.

 

Extended Parallel/Proportionality Conjecture- If two or more lines pass through two sides of a triangle parallel to the third side, then they divide the two sides proportionally.

 

Different Types of Triangles

Scalene Triangle- No sides are congruent

Isosceles Triangle- At least 2 sides are congruent

Acute Triangle- All 3 angles are congruent

Right Triangle- 1 right angle

Obtuse Triangle- 1 obtuse angle

Equiangular Triangle- All 3 angles are congruent

 

 Triangle Congruence Postulates and Theorems

SSS (Side Side Side) – All three sides are congruent

SAS (Side Angle Side) – Two sides and the included angle are congruent

ASA (Angle Side Angle) – Two angles and the include side are congruent

AAS (Angle Angle Side) – Two angles and not included side are congruent

*** AAA and ASS doesn’t work

Corresponding Parts of Congruent Triangles are Congruent (CPCTC) - Two triangles are congruent if only if the corresponding parts are congruent

 

Angles

Corresponding Angles Postulate – If two parallel lines are cut by a transversal, then the corresponding angles are congruent

 

Converse of the Corresponding Angles Postulate – If two lines are cut my a transversal and the corresponding angles are congruent, then the lines are parallel.

 

Alternate Interior Angles Postulate – If two parallel lines are cut by a transversal, then the alternate interior angles are congruent

 

Converse of the Alt. Interior Angles Postulate: If two lines are cut by a transversal and the alt. interior angles are congruent the lines are parallel.

 

Alternate Exterior Angles Postulate – If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent

 

Converse of the Alt. Exterior Angles Postulate – If two lines are cut by a transversal and the alt. exterior angles are congruent then the lines are parallel.

 

Angle Bisector Theorem – If a point is on the bisector of an angle, then it is equidistant from the sides of the angle

 

Parallel Postulate – If there’s a line and a point not on the line, the there’s exactly 1 line through the point parallel to the give line.

 

Perpendicular Postulate – Given a segment or line and a point NOT on the line, there is EXACTLY one line through the point perpendicular to that point.

 

Perpendicular Bisector Theorem – If a point is on the perpendicular bisector of a segment, then it is equidistant (equally distant) from the endpoint

 

Converse of the Perpendicular Bisector Theorem – If a point in equidistant from the endpoints of a segment, then it is on the bisector of the segment

 

Shortest Distance Theorem – The shortest distance from a point to a line is measured along the perpendicular from the point to the line

 

Base Angles Theorem - If a triangle is isosceles then the base angles are congruent.

 

Converse of Base Angles Theorem – If the base angles are congruent, then the triangle is isosceles

 

Side Angle Inequality – In a triangle if one side is longer than another size, then the angle opposite the larger side is larger than the angle opposite the shorter side.

 

Vertex Angle Bisector Theorem – In an isosceles triangle, the bisector of the vertex angle is also the median and the altitude.

 

Polygons

Polygon Angle Sums Theorem: 180n-360

 

Exterior Angle Sum Theorem: For any polygon, the sum of the measures of a set of exterior angles is 360.

 

Equiangular Polygon Conjecture: You can find the measure of each interior angle of an equiangular n-gon by using the formula (180n-360)/n.

 

Kite Angles Conjecture: The non vertex angles of a kite are congruent.

 

Kite Diagonals Conjecture: The diagonals of a kite are congruent.

 

Kite Diagonal Bisector Conjecture: The diagonal connecting the vertex angles of a kite is the perpendicular of the other diagonal.

 

Kite Angle Bisector Conjecture: The vertex angles of a kite are bisected by a diagonal.

 

Trapezoid Consecutive Angles Conjecture: The consecutive angles between the bases of a trapezoid are supplementary.

 

Isosceles Trapezoid Conjecture: The base angles of an isosceles trapezoid are congruent.

 

Isosceles Trapezoid Diagonals Conjecture: The diagonals of an isosceles trapezoid are congruent.

 

The Three Midsegment Conjectures: The three midsegments of a triangle divide it into 4 congruent triangles.

 

Triangle Midsegment Conjecture: A midsegment of a triangle is parallel to the third side and twice the length of the first side.

 

Trapezoid Midsegment Conjecture: The midsegment of a trapezoid is parallel to the bases and is equal in length to the average of the lengths of the bases.

 

Parallelogram Opposite Sides Theorem: The opposite sides of a parallelogram are congruent.

 

Parallelogram Opposite Angles Theorem: The opposite angles of a parallelogram are congruent.

 

Parallelogram Diagonals Theorem: The diagonals of a parallelogram bisect each other.

 

Parallelogram Consecutive Angles Theorem: The consecutive angles of a parallelogram are 180.

Double Edged Straightedge Conjecture: If two parallel lines are intersected by a second pair of parallel lines that are the same distance apart as the first pair, then the parallelogram formed is a rhombus.

 

Rhombus Diagonals Theorem: The diagonals of a rhombus are perpendicular and they bisect each other.

 

Rhombus Angles Theorem: The diagonals of a rhombus bisect the angles of the rhombus.

 

Rectangle Diagonals Theorem: The diagonals of a rectangle are congruent and bisect each other.

 

Square Diagonals Theorem: The diagonals of a square are congruent, perpendicular, and bisect each other.

 

 

Transformation

Line of Reflection Conjecture – line of reflection is perpendicular bisector of all parallel segments connecting in the original image with its corresponding point in the reflection.

 

Coordinate Transformation Conjecture -

(x,y) à (-x,y) reflection, line of reflection à y axis  

(x,y) à (x, -y) reflection – x axis

(x,y) à (-x,-y) rotation – (0,0) 180˚

(x,y) à (y,x) reflection (y=x)

 

Minimal Path Conjecture – If points A and B are on ne side of line l, then the minimal path from point A to line l to point B is found by reflecting point B over the line and drawing AB and where Point C is where AB intersects line l.

 

Reflections Across Parallel Lines Conjecture – A composition of two reflections across two parallel lines is equivalent to a single translation. In addition, the distance from any point to its second image under the two reflections is 2x the distance between the parallel lines.

 

Reflections across Intersection Lines Conjecture – A composition of two reflections across a pair of intersecting lines is equivalent to a single rotation. The angle of rotation is twice the acute angle between the parts of intersection reflection lines.

 

Composition of 2 transition – when you transform one figure using one rule then transform the new figure using a 2^nd rule

 

Conjectures:

Tangent Conjecture: a tangent to a circle is perpendicular to the radius at the point of tangent.

Tangent Segments Conjecture: Tangents segments to a circle from one point outside the circle are congruent.

Perpendicular to a Chord Conjecture: The perpendicular from the center of a circle to a chord is the bisector of the chord.

Chord Distance to Center Conjecture: Two congruent chords in a circle are equidistant from the center of the circle.

Perpendicular Bisector of a Chord Conjecture: The perpendicular bisector of a chord interest at the center.

Tangent Chord Conjecture: The measure of an angle formed by tangent and chord is half the measure of the intercepted arc.

Intersecting Chord Conjecture: The measure of the angle formed by two intersecting chord is half the measure of the sum of the intercepted arcs.

Chord Central Angles Conjecture: If two chords in a circle are congruent, then the determine two central angles that are congruent.

Chords Arcs Conjecture: if two chords in a circle are congruent, then their intercepted arcs are congruent.

Inscribed Angle Conjecture: The measure of an angle inscribed in a circle is half the measure of the intercepted arcs.

 Inscribed Angles Intercepting Arcs Conjecture: Inscribed angles that intercept the same arc are congruent.

Angles Inscribed in a Semicircle Conjecture: Angles inscribed in a semicircle is 90 degrees.

Cyclic Quadrilateral Conjecture: The opposite angles of a cyclic quadrilateral are supplementary.

Parallel Lines Intercepted Arcs Conjecture: Parallel lines intercept congruent arcs on  a circle.

Tangent Secant Conjecture: The measure of the angle formed by an intersecting tangent and secant to a circle is half the different between the intercepted arcs.

Intersecting Secants Conjecture: The measure of an angle formed by two secants that intersect outside a circle is half the measure of the difference of the intercepted arcs.

Intersecting Tangents Conjecture: The measure of the angle formed by two intersecting tangents to a circle is supplementary to the center angle.

Dilation Similarity Conjecture: if one polygon is a dilated image of another polygon then the polygons are similar 

AA Similarity Conjecture: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

SSS: If the three sides of one triangle are proportional to the three sides of another triangle, then the two triangles are similar. 

SAS: If two dies of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar

Parallel/Proportionality Conjecture: If a line parallel to one side of a triangle passes through the other two sides, then it divides the other two sides proportionally. If a line cuts 2 sides of a triangle proportionally, then it's parallel to the third side 

Extended parallel/proportionality Conjecture: If two or more lines pass through two sides of a triangle parallel to the third side, then they divide the two sides proportionally 

Parallel/Proportionality Conjecture- If a line parallel to the side of a triangle passes through the other two sides, and then it divides the other two sides proportionally. If a line cuts 2 sides of a triangle proportionally, then it’s parallel to the third side.

Extended Parallel/Proportionality Conjecture- If two or more lines pass through two sides of a triangle parallel to the third side, then they divide the two sides proportionally.

 

Sine= opposite/hypotenuse; cosecant= hypotenuse/opposite; sinA=cosB

Cosine= adjacent/hypotenuse; secant= hypotenuse/adjacent; secA=cscB

Tangent= opposite/adjacent; cotangent= adjacent/opposite; tanA=cotB

 

the sine of theta: opposite leg/hypotenuse
the cosine of theta: adjacent leg/hypotenuse
the tangent of theta: opposite/adjacent
cofunctions:
- secent (sec) -> 1/cos theta (hypot/adj)
- cosecant (csc) -> 1/sin theta (hypot/oppo)
- cotangent (cot) -> 1/tan theta (adj/oppo)
Area of triangle: base (height) (1/2)
Area of Parallelogram: Base (height)
Area of Trapezoid: (h/2) (base 1 + base 2)
Distance formula:
Area of any polygon: [(base)(height)]/2 (# of triangles)
Area of a sector = x/360 (pi (r)squared
Law of cosine

Law of Sines: For a triangle with angles A, B, and C and sides of lengths a, b, and c (a opposite A, b opposite B, and c opposite C), (sinA/a) = (sinB/b) = (sinC/c).

 

Formulas:

Surface Area of Cone: ∏r (r+l)

Volume of a Pyramid/Cone: 1/3 (area of base) height

SAS Triangle Area Formula: ½ cb (sinA)

Volume of Sphere: 4/3 ∏r³

Surface Area of Sphere: 4∏r²

Arc Length: measure/360 (circumference)

Volume of any prism: Base of area (height)

 

45-45-90:

 

 

DEFINITIONS

Point - has a location but no width, height, length, or depth. It is represented with a dot and capital letter.

 

Lines – has a length, no width, no height. It’s straight, infinitely long and has two directions. It is represented as a segment with 2 arrows.

 

Plane – infinite length or width and has no height. It is represent as a 4 sided figure and named with cursive capital letters.

 

Line segments have length.

 

Midpoints – to find the midpoint of a line segment with endpoints (x1, y1) and (x2,y2).

 

Angle: two rays that project or start at the same point

 

Diagonal: a line segment that connects two vertices that are not consecutive.

 

Equilateral polygon – a figure that has all congruent sides and no closing and is not concave.

 

Convex: one diagonal that is outside

 

Equiangular polygon – a figure with all congruent vertices

 

Regular Polygon – a figure with all congruent sides and vertices.

 

Chord – any segment that touches both ends of a circle

 

Diameter – a chord that goes through the center point

 

Tangent – any lives that touches the circle on the outside

 

Arc – a piece of the circle ( half a circle – semicircle )

 

Minor Arc – less that 1/2 the circle named with only 2 endpoints.

 

Major Arc – more than ½ a circle named with the endpoints and one middle letter

 

Inductive Reasoning – the process of observing date, recognizing patterns and making generalizations about those patterns

 

Conjecture – a generalization based on inductive reasoning

 

The converse of the conditional statement reverses the hypothesis

 

The inverse of the conditional statement negates both the hypothesis and conclusion

 

The contrapositive of the statement both reverses and negates they hypothesis and conclusion

 

Bioconditional statement – when a conditional statement and its converse is true, it is a biconditional statement. A biconditional statement contains the phrase “if and only if”.

 

Law of Detachment –the hypothesis of a true conditional statement then the conclusion is true.

 

Law of Syllogism – If A, then B. If b, then C. If A, then C.

 

Circumcenter – the concurrent point of the perpendicular bisectors of any triangle ; equidistance to the vertices

 

Intcenter – point of concurrence of angle bisectors of a triangle.; equidistant to the sides of a triangle

 

Orthocenter – the point of concurrence of the altitude of a triangle.

 

Centroid – the point of concurrence of the medians of the triangle aka center of gravity

 

Properties of Equalities

Reflexive Property  (AB = AB)

Symmetric Property (A=B, B=A)

Transitive Property  (A=B and B=C, then A=C)

 

Triangle Congruence – Two triangles are congruent if only if the corresponding parts are congruent

 

Linear Pair of Angles- 2 or more angles that make 1 or more straight line(s)

 

Median Triangle – Connects a vertex wit the midpoint of the opposite side

Mid Segment – connects the midpoints of two sides of a triangle

 

Translation – whole figure is moved along parallel paths and same distance/each point is equidistance/ has a magnitude (translation vector)

 

Rotations – all points in original figure rotate, then the same number of degrees about a fixed center point/ defined by center point , # of degree, if if the rotation is clockwise or counter clockwise

 

Reflections – mirror image/line where the mirror is placed is the called the ‘line of reflection

 

Rhombus – a parallelogram with all congruent sides

 

Parallelogram – a quadrilateral that has 2 pair of parallel sides

 

Quadrilateral – a 4 sided polygon

 

Rectangle – a parallelogram with 90˚ angles

 

Square – a rhombus with 90˚ angles

 

Trapezoid – a quadrilateral that has one pair of parallel lines

 

Kite – A quadrilateral with 2 pair of consecutive congruent sides

 

Circle – all points that are a given radius from the center point