Geometry Definitions, Postulates, and Theorems

Flashcards for Geometry Review

Complementary Angles

Two angles whose measures have a sum of 90 degrees

Supplementary Angles

Two angles whose measures have a sum of 180 degrees

Theorem

A statement that can be proven

Vertical Angles

Two angles formed by intersecting lines and facing in the opposite direction

Transversal

A line that intersects two lines in the same plane at different points

Corresponding angles

Pairs of angles formed by two lines and a transversal that make an F pattern

Same-side interior angles

Pairs of angles formed by two lines and a transversal that make a C pattern

Alternate interior angles

Pairs of angles formed by two lines and a transversal that make a Z pattern

Congruent triangles

Triangles in which corresponding parts (sides and angles) are equal in measure

Similar triangles

Triangles in which corresponding angles are equal in measure and corresponding sides are in proportion (ratios equal)

Angle bisector

A ray that begins at the vertex of an angle and divides the angle into two angles of equal measure

Segment bisector

A ray, line, or segment that divides a segment into two parts of equal measure

Legs of an isosceles triangle

The sides of equal measure in an isosceles triangle

Base of an isosceles triangle

The third side of an isosceles triangle

Equiangular

Having angles that are all equal in measure

Perpendicular bisector

A line that bisects a segment and is perpendicular to it

Altitude

A segment from a vertex of a triangle perpendicular to the line containing the opposite side

Angle Addition Postulate

For any angle, the measure of the whole is equal to the sum of the measures of its non-overlapping parts

Linear Pair Theorem

If two angles form a linear pair, then they are supplementary

Congruent supplements theorem

If two angles are supplements of the same angle, then they are congruent

Congruent complements theorem

If two angles are complements of the same angle, then they are congruent

Right Angle Congruence Theorem

All right angles are congruent

Vertical Angles Theorem

Vertical angles are equal in measure

Theorem

If two congruent angles are supplementary, then each is a right angle

Angle Bisector Theorem

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

Converse of the Angle Bisector Theorem

If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle

Segment Addition postulate

For any segment, the measure of the whole is equal to the sum of the measures of its non-overlapping parts

Postulate

Through any two points there is exactly one line

Postulate

If two lines intersect, then they intersect at exactly one point

Common Segments Theorem

Given collinear points A,B,C and D arranged as shown, if AB = CD, then AC = BD

Corresponding Angles Postulate

If two parallel lines are intersected by a transversal, then the corresponding angles are equal in measure

Converse of Corresponding Angles Postulate

If two lines are intersected by a transversal and corresponding angles are equal in measure, then the lines are parallel

Geometric mean

The value of x in proportion a/x=x/b where a, b, and x are positive numbers (x is the geometric mean between a and b)

Sine, sin

For an acute angle of a right triangle, the ratio of the side opposite the angle to the measure of the hypotenuse. (opp/hyp)

Cosine, cos

For an acute angle of a right triangle the ratio of the side adjacent to the angle to the measure of the hypotenuse. (adj/hyp)

Tangent, tan

For an acute angle of a right triangle, the ratio of the side opposite to the angle to the measure of the side adjacent (opp/adj)

Addition Prop. Of equality

If the same number is added to equal numbers, then the sums are equal

Subtraction Prop. Of equality

If the same number is subtracted from equal numbers, then the differences are equal

Multiplication Prop. Of equality

If equal numbers are multiplied by the same number, then the products are equal

Division Prop. Of equality

If equal numbers are divided by the same number, then the quotients are equal

Reflexive Prop. Of equality

A number is equal to itself

Symmetric Property of Equality

If a = b then b = a

Substitution Prop. Of equality

If values are equal, then one value may be substituted for the other

Transitive Property of Equality

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

Distributive Property

a(b+c) = ab+ac

Reflexive Property of Congruence

If AB, then BA

Symmetric Property of Congruence

If AB, then BA

Transitive Property of Congruence

If A = B and B = C then A = C

Proportional Perimeters and Areas Theorem

If the similarity ratio of two similar figures is a/b, then the ratio of their perimeter is a/b and the ratio of their areas is (a/b)^2 or a^2/b^2.

Area Addition Postulate

The area of a region is equal to the sum of the areas of its nonoverlapping parts

Theorem

If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency

Theorem

If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle

Theorem

If two segments are tangent to a circle from the same external point then the segments are congruent

Arc Addition Postulate

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs

Theorem

In a circle or congruent circles: congruent central angles have congruent chords, congruent chords have congruent arcs and congruent arcs have congruent central angles

Theorem

In a circle, if a radius (or diameter) is perpendicular to a chord, then it bisects the chord and its arc

Theorem

In a circle, the perpendicular bisector of a chord is a radius (or diameter)

Inscribed Angle Theorem

The measure of an inscribed angle is half the measure of its intercepted arc

Corollary

If inscribed angles of a circle intercept the same are or are subtended by the same chord or arc, then the angles are congruent

Theorem

If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc

Theorem

If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of the intercepted arcs

Theorem

If a tangent and a secant, two tangents or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measure of its intercepted arc

Chord-Chord Product Theorem

If two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal

Secant-Secant Product Theorem

If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment

Secant-Tangent Product Theorem

If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared

Equation of a Circle

The equal of a circle with center (h, k) and radius r is (x-h)²+(y-k)² = r²

Theorem

An inscribed angle subtends a semicircle IFF the angle is a right angle

Theorem

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary

Postulate

Through a point not on a given line, there is one and only one line parallel to the given line

Alternate Interior Angles Theorem

If two parallel lines are intersected by a transversal, then alternate interior angles are equal in measure

Alternate Exterior Angles Theorem

If two parallel lines are intersected by a transversal, then alternate exterior angles are equal in measure

Same-side Interior Angles Theorem

If two parallel lines are intersected by a transversal, then same-side interior angles are supplementary

Converse of Alternate Interior Angles Theorem

If two lines are intersected by a transversal and alternate interior angles are equal in measure, then the lines are parallel

Converse of Alternate Exterior Angles Theorem

If two lines are intersected by a transversal and alternate exterior angles are equal in measure, then the lines are parallel

Converse of Same-side Interior Angles Theorem

If two lines are intersected by a transversal and same-side interior angles are supplementary, then the lines are parallel

Theorem

If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular

Theorem

If two lines are perpendicular to the same transversal, then they are parallel

Perpendicular Transversal Theorem

If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one

Perpendicular Bisector Theorem

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

Converse of the Perpendicular Bisector Theorem

If a point is the same distance from both the endpoints of a segment, then it lies on the perpendicular bisector of the segment

Parallel Lines Theorem

In a coordinate plane, two nonvertical lines are parallel IFF they have the same slope

Perpendicular Lines Theorem

In a coordinate plane, two nonvertical lines are perpendicular IFF the product of their slopes is -1

Two-Transversals Proportionality Corollary

If three or more parallel lines intersect two transversals, then they divide the transversals proportionally

Theorem

If two angles of one triangle are equal in measure to two angles of another triangle, then the two triangles are similar

Theorem

If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar

Theorem

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

Theorem

If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent

Theorem

If two sides and the included angle of one triangle are equal in measure to the corresponding sides and angle of another triangle, then the triangles are congruent

Theorem

If three sides of one triangle are equal in measure to the corresponding sides of another triangle, then the triangles are congruent

Theorem

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent

Theorem

The sum of the measure of the angles of a triangle is 180°

Theorem

The acute angles of a right triangle are complementary

Theorem

An exterior angle of a triangle is equal in measure to the sum of the measures of its two remote interior angles

Triangle Proportionality Theorem

If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally

Converse of Triangle Proportionality

If a line divides two sides of a triangle proportionally, then it is parallel to the third side

Triangle Angle Bisector Theorem

An angle bisector of a triangle divides the opposite sides into two segments whose lengths are proportional to the lengths of the other two sides.

Theorem

If two angles and a non-included side of one triangle are equal in measure to the corresponding angles and side of another triangle, then the triangles are congruent

Theorem

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent

Theorem

If two sides of a triangle are equal in measure, then the angles opposite those sides are equal in measure

Theorem

If two angles of a triangle are equal in measure, then the sides opposite those angles are equal in measure