Geometry Midterm vocab, formulas, theorems, postulates.

Simplifying radicals

Rewrite radicals by factoring inside the radical, pulling out perfect squares, and simplifying fully using order of operations.

Adding/Subtracting multipliers in front of same radicals

Multiplication/Division inside and outside of simplified radicals

Plane

A flat surface that extends infinitely in all directions

Coplanar vs Colinear

Points or lines that lie in the same plane vs points on the same line

Line

A straight path that extends infinitely in two directions

Line segment

A part of a line with two endpoints

Ray

A part of a line with one endpoint that extends infinitely in one direction

Opposite rays

Two rays that share an endpoint and form a straight line

Intersection

The point or set of points where figures meet

Congruent

Figures that have the same shape and size

Midpoint

The point that divides a segment into two equal parts

Perpendicular

Lines that intersect at a 90° angle

Parallel

Lines that never intersect and are always the same distance apart

Midpoint formula

Distance formula

Angle

A figure formed by two rays with a common endpoint

Right angle

An angle that measures exactly 90°

Obtuse angle

An angle greater than 90° but less than 180°

Acute angle

An angle less than 90°

Alternate interior angles

Angles on opposite sides of a transversal and inside parallel lines

Alternate exterior angles

Angles on opposite sides of a transversal and outside parallel lines

Same-side interior angles

Interior angles on the same side of a transversal that sum to 180°

Triangle

A polygon with three sides

Right triangle

A triangle with one 90° angle

Isosceles triangle

A triangle with at least two congruent sides

Equilateral triangle

A triangle with all sides congruent

Obtuse triangle

A triangle with one obtuse angle

Acute triangle

A triangle with all acute angles

Scalene triangle

A triangle with no congruent sides

Complementary angles

Two angles whose measures add to 90°

Supplementary angles

Two angles whose measures add to 180°

Bisect

To divide into two equal parts

Bisects

Divides a segment or angle into two congruent parts

Perpendicular bisector

A line that cuts a segment in half at 90°

Angle bisector

A ray that divides an angle into two congruent angles

Construction

A geometric drawing made using only a compass and straightedge

Line of reflection

The line over which a figure is flipped

Reflection

A transformation that flips a figure over a line

Translation

A transformation that slides a figure without turning or flipping

Rotation

A transformation that turns a figure around a fixed point

Dilation

A transformation that enlarges or reduces a figure proportionally

Directed line segment

A segment with a starting point and direction

Sequence of transformations

Two or more transformations applied in order

Symmetry

When a figure can map onto itself

Line of symmetry

A line that divides a figure into mirror images

Rotational symmetry

A figure that matches itself after rotation. Found by 360°÷n

SSS congruence

Triangles are congruent if all three sides are congruent

SAS congruence

Triangles are congruent if two sides and the included angle are congruent

AAS congruence

Triangles are congruent if two angles and a non-included side are congruent

HL congruence

Right triangles are congruent if the hypotenuse and one leg are congruent

Rigid motions

Transformations that preserve distance and angle measures

Rigid transformations

Reflections, rotations, and translations

Parallelogram proof

Show both pairs of opposite sides are parallel or congruent

Diagonals bisect each other, not perpendicular.

Rectangle proof

Show a parallelogram has four right angles

Diagonals bisect each other, though not perpendicular. All diagonal parts congruent.

Rhombus proof

Show a parallelogram has four congruent sides

Rhombus diagonals are perpendicular and bisect each other.

Square proof

Show a rectangle with all sides congruent

Diagonals all congruent to each other, bisect each other, and perpendicular.

Perimeter

The distance around a figure

Area

The amount of space inside a figure

Finding dilated perimeter and area

Multiply perimeter by scale factor. Multiply area by scale factor²

Proportions

Two ratios that are equal

Geometric mean

used for right triangles and similar figures

Scale factor

The ratio used to enlarge or reduce a figure

Triangle similarity

Triangles with equal angles and proportional sides

AA similarity

Triangles are similar if two angles are congruent

SSS similarity

Triangles are similar if corresponding sides are proportional

SAS similarity

Triangles are similar if two sides are proportional and included angle congruent

SAA similarity

Triangles are similar if two angles and a side are proportional

HL similarity

Right triangles are similar if hypotenuse and one leg are proportional

Pythagorean triples

Sets of whole numbers that satisfy 𝑎²+𝑏²=𝑐²

(3,4,5)

A common Pythagorean triple 3, 4, 5

(5,12,13)

A common Pythagorean triple 5, 12, 13

(8,15,17)

A common Pythagorean triple 8, 15, 17

(7,24,25)

A common Pythagorean triple 7, 24, 25

Commutative property of addition

a+b=b+a

Commutative property of multiplication

𝑎𝑏=𝑏𝑎

Commutative property of subtraction

Order changes the result (not commutative)

Commutative property of division

Order changes the result (not commutative)

Associative property of addition

(a+b)+c=a+(b+c)

Associative property of multiplication

(ab)c=a(bc)

Distributive property

a(b+c)=ab+ac

Reflexive property of equality

A quantity equals itself

Symmetric property of equality

If a=b, then b=a

Transitive property of equality

If a=b, and b=c, then a=c

Addition property of equality

Adding the same value preserves equality

Subtraction property of equality

Subtracting the same value preserves equality

Multiplication property of equality

Multiplying by the same value preserves equality

Division property of equality

Dividing by the same nonzero value preserves equality

Reflexive property of congruence

A figure is congruent to itself

Symmetric property of congruence

If A ≅ B, then B ≅ A

Transitive property of congruence

If A ≅ B and B ≅ C, then A ≅ C

Segment addition postulate

The sum of parts equals the whole segment

Angle addition postulate

The sum of adjacent angles equals the whole angle

Definition of congruence

Figures with equal measures

Definition of complementary angles

Two angles that sum to 90°

Definition of supplementary angles

Two angles that sum to 180°

Triangle angle sum theorem

The angles of a triangle sum to 180°

Definition of linear pair

Two adjacent supplementary angles forming a straight line

Definition of vertical angles

Opposite angles formed by intersecting lines

Linear pair postulate

Linear pairs are supplementary

Exterior angle theorem

An exterior angle equals the sum of the two remote interior angles

Corresponding angles theorem

Corresponding angles are congruent if lines are parallel