Definitions for Proofs in Math Geometry 24'-25'

Congruent Triangles

2 Figures (Triangles) are congruent, if 6 parts of 1 triangle are congruent to 6 parts of another triangle.

Collinear

Points in or on a straight line.

Betweenness

A point is between 2 points if and only if the 3 points are collinear.

Bisects

A point (or segment, ray, or line) that divides a segment into 2 congruent segments.

Midpoint (and Formula)

The bisection point is called the midpoint of the segment.

Formula: Mx = x₁+x₂/2 ; y₁+y₂/2

Trisection

Trisection points divide segments into 3 congruent segments.

Angle Bisector

The bisector of an angle is the ray that divides the angle into two congruent angles.

Angle Trisection

2 lines/rays divide an angle into 3 congruent angles.

Perpendicularity

If 2 lines/segments are perpendicular, then they form right angles.

Converse of Perpendicularity

If a right angle has been formed, then it has been formed by perpendicular lines/segments.

Complementary Angles

2 angles whose sum is 90 degrees.

Supplementary Angles

2 angles whose sum is 180 degrees.

Adjacent Angles

Angles next to each other that share a common ray.

Linear Pair

2 angles that form a straight line.

Vertical Angles

The angles that are opposite each other.

Scalene Triangle

A triangle with all sides of different lengths.

Isosceles Triangle

A triangle with at two sides of equal length.

Equilateral Triangle

A triangle with all three sides of equal length.

Equiangular Triangle

A triangle in which all three angles are equal.

Are equiangular and equilateral triangles reversible?

YES

Acute Triangle

A triangle where all three angles are less than 90 degrees (all 3 angles are acute in a triangle.)

Right Triangle

A triangle in which one angle measures exactly 90 degrees (Triangle with only 1 right angle.)

Obtuse Triangle

A triangle in which one angle measures more than 90 degrees (Triangle with one obtuse angle).

Perpendicular Bisector

An object (seg, line, ray) that perpendicularly bisects another segment.

Parallelogram

A parallelogram is a quadrilateral in which

2 pairs of opposite sides are parallel.

Rhombus

A rhombus is a parallelogram in which at

least 2 consecutive sides are congruent.

Square

A square is a parallelogram that is both a

rectangle and a rhombus.

Isosceles Trapezoid

Trapezoid where the not parallel sides are congruent

Quadrilateral

4 sided figure

Rectangle

A rectangle is a parallelogram in which at

least 1 angle is a right angle.

Kite

A kite is a quadrilateral in which 2 dis¬

joint pairs of consecutive sides are con¬

gruent.

Trapezoid

A trapezoid is a quadrilateral with exactly

1 pair of parallel sides.

Properties of Parallelogram (5)

• The opposite sides are parallel by def.

• The opposite sides are congruent

• The opposite angles are congruent

• The diagonals bisect each other

• Any pair of consecutive angles are supplementrary

Properties of Rectangle (3)

• All properties of parallelogram apply by def.

• All angles are right angles

• The diagonals are congruent

Properties of Kite (5)

• 2 disjoint pairs of consecutive sides are congruent by def.

• The diagnols are perpendicular

• 1 diagonal is the perpendicular bisector of the other

• 1 of the diagonals bisects a pair of opposite sides

• 1 pair of opposite angles are congruent

Properties of Rhombus (6)

• All properties of a parallelogram apply by def.

• All the properties of a kite apply (half properties become full)

• All sides are congruent

• The diagonals bisect the angles

• The diagonals are perpendicular bisectors of each other

• The diagonals divide the rhombus into 4 congruent right angles

Properties of Squares (3)

• All properties of a rectangle apply by def.

• All properties of a rhombus apply by def.

• The diagonals form 4 isosceles right triangles

Properties of Isosceles Trapezoids (6)

• The legs are congruent by def.

• The bases are parallel by def.

• The lower base angles are congruent

• The upper base angles are congruent

• The diagonals are congruent

• Any lower base angle is supplementary to any upper base angle

A regular polygon is both _____ and _____.

Equilateral and equiangular

Ratio

The quotient of 2 numbers

Proportion

An equation stating that two or more ratios are equal

Axium…

Postulate

Similar Polygons are polygons in which:

1) The ratios of the measure of corresponding sides are =

2) Corresponding angles are congruent

To enlarge =

To make smaller =

Dilation

Reduction

Dialation

A transformation in which all points of a figure are enlarged or

reduced by a scale factor K, about a center of rotation C such that:

CP′

CP = |K|

If _____________ then the dilation

is called an ___________________________

• If _____________ then the dilation

is called ______________________________

|K| > 1

Enlargement

|K| < 1

Reduction

Arc

Arc is made up of 2 pts. on a circle and all the pts. of the circle needed to connect them on a single path.

Measure

m AB (Arc symbol over AB)

Length

AB (Arc symbol over AB)

Sector of a Circle

Portion bounded by 2 radii and an arc of a circle

Chord

Line segment joining 2 pts. on a circle

Inscribed Angle

An angle formed by 2 chords

Prism Def.

A 3-D solid w/ 2 congruent bases and connecting sides

A rectangular prism has ____ congruent diagnols

4

Two circles are congruent if…

their radii are congruent

Two coplanar circles with the same center are called

Concentric