# Constructions, Definitions, Postulates, and Theorems

## Constructions

• Acute Angle
• Acute Triangle
• Altitude
• Angle Bisector
• Centroid
• Circle through Three Points
• Circumcenter
• Circumscribed Circle
• Congruent Angles
• Congruent Line Segments
• Congruent Triangles
• Equiangular Triangle
• Equilateral Triangle
• Incenter
• Inscribed Circle
• Isosceles Triangle
• Median
• Obtuse Angle
• Obtuse Triangle
• Orthocenter
• Parallel Line
• Perpendicular Bisector
• Perpendicular Line
• Right Angle
• Right Triangle
• Scalene Triangle
• Segment
• Segment Bisector
• Tangents to Circles

## Definitions

(Note: The following are the most commonly used definitions in proofs. For a complete list of definitions for the course, refer to the glossary.)

• Acute Angle: An angle that measures less than 90°.
• Altitude: A perpendicular segment from a vertex of a triangle to its opposite side.
• Angle: A figure consisting of two non-collinear rays or segments with a common endpoint.
• Bisector: A line or segment that divides another figure into two congruent parts.
• Circle: A set of all points in a plane that are a given distance from a point identified as the center.
• Complementary Angles: Two angles that add up to 90°.
• Congruent Angles: Two angles with exactly the same degree measure.
• Congruent Segments: Two segments with exactly the same length.
• Congruent Triangles: Two triangles are congruent if and only if their corresponding parts are congruent: Corresponding Parts of Congruent Triangles are Congruent (CPCTC).
• Line: A one-dimensional figure that consists of an infinite number of points extending in opposite directions.
• Line Segment: A part of a line that has two endpoints.
• Median: A segment from a vertex of a triangle to the midpoint of the opposite side.
• Midpoint: A point that divides a segment into two equal parts. The ordered pair of this point can be found by the midpoint formula: .
• Obtuse Angle: An angle that measures more than 90°, but less than 180°.
• Parallel Lines: Two lines that lie within the same plane and never intersect. Parallel lines have the same slope.
• Perpendicular Lines: Two lines that intersect at 90° angles. Perpendicular lines have slopes that are the negative reciprocals of one another.
• Perpendicular Bisector: A line that divides a segment into two halves and intersects the segment at a 90° angle.
• Plane: An infinite flat surface that has length and width but no depth.
• Point: A location that has no dimension.
• Ray: Part of a line that has one endpoint and continues in one direction infinitely.
• Right Angle: An angle that measures exactly 90°.
• Similar Triangles: Two triangles are similar if corresponding angles are congruent and  corresponding sides are proportional.
• Supplementary Angles: Two angles that add up to 180°.
• Straight Angle: An angle that measures exactly 180°.
• Vertex: The common endpoint of two segments or rays that form a “corner” of an angle.

## Theorems and Postulates

For real numbers a, b, and c, if a = b, then a + c = b + c.

The sum of any real number and zero is that same real number. In other words, for any real number a, a + 0 = a.

Alternate Exterior Angles Theorem
If a transversal intersects two parallel lines, then alternate exterior angles are congruent.

Alternate Interior Angles Theorem
If a transversal intersects two parallel lines, then alternate interior angles are congruent.

The measure of an angle created by two adjacent angles may be found by adding the measures of the two adjacent angles.

Angle-Angle (AA) Similarity Postulate
If two corresponding angles of two or more triangles are congruent, the triangles are similar.

Angle-Angle-Side (AAS) Postulate
If two angles and a non-included side are congruent to the corresponding two angles and side of a second triangle, the two triangles are congruent.

Angle-Side-Angle (ASA) Postulate
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.

The measure of an arc created by two adjacent arcs may be found by adding the measures of the two adjacent arcs.

Area of a Square
The area of a square is a measurement representing the space within the interior of a square. It is found by the formula A = s2 where A is the area and s is the length of a side.

Area of a Triangle
The area of a triangle is a measurement representing the space within the interior of a triangle. It is found by the formula A = bh where A is the area, b is the length of the base, and h is the length of the height.

For real numbers a, b, and c, a + (b + c) = (a + b) + c.

Associative Property of Multiplication
For real numbers a, b, and c, a • (b • c) = (a • b) • c.

Bisecting Diagonal Theorem
The diagonal of a kite connecting the vertex angles bisects the diagonal connecting the nonvertex angles.

Bisecting Vertex Angles Theorem
The diagonal of a kite connecting the vertex angles is an angle bisector of these vertex angles.

Central Angle Theorem
The measure of a central angle is equal to the measure of the arc it intercepts.

For real numbers a and b, a + b = b + a.

Commutative Property of Multiplication
For real numbers a and b, a • b = b • a.

Concurrency of Altitudes Theorem
The lines containing the altitudes of a triangle are concurrent.

Concurrency of Angle Bisectors Theorem
The angle bisectors of a triangle are concurrent at a point equidistant from the sides of the triangle.

Concurrency of Medians Theorem
The medians of a triangle are concurrent at a point that is two-thirds the distance from each vertex to the midpoint of the opposite side.

Concurrency of Perpendicular Bisectors Theorem
The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

Congruent Arcs and Chords Theorem
Two minor arcs within the same circle or between congruent circles are congruent if and only if their corresponding chords are congruent.

Congruent Arcs Theorem
Two arcs are congruent if the central angles that intercept them are also congruent.

Congruent Inscribed Angles Theorem
Two or more distinct inscribed angles that intercept the same arc, or congruent arcs, are congruent.

Coplanar Points Postulate
Through any three non-collinear points, there is exactly one plane.

Corresponding Angles Postulate
If a transversal intersects two parallel lines, then corresponding angles are congruent.

Cross Product Property
For real numbers a, b, c, and d,  is equivalent to a • d = b • c or ad = bc.

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

Distance between Two Points Postulate
The distance between two points can be found by taking the absolute value of the difference between the coordinates of the two points.

Distributive Property
For real numbers a, b, and c, a(b + c) = ab + ac.

Division Property of Equality
For real numbers, a, b, and c, if a = b and c ≠ 0, then .

Exterior Angle to a Circle Theorem
If two secants, two tangents, or a secant and a tangent intersect outside a circle, the measure of the created angle between them is one-half the absolute value of the difference of the measures of their intercepted arcs.

Exterior Angle Sum Theorem
The sum of the exterior angles of any polygon total 360°.

• Corollary to the Exterior Angle Sum Theorem
The angle measure of any single exterior angle to a regular polygon may be found by dividing 360° by the number of sides.

The Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.

• The Converse of the Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer third side.

Hypotenuse-Leg (HL) Theorem
If two right triangles have congruent hypotenuses and corresponding, congruent legs, the two right triangles are congruent.

Inscribed Angle to a Semicircle Theorem
An inscribed angle that intercepts a semicircle is a right angle.

Inscribed Angle Theorem
The measure of an inscribed angle is equal to half the measure of its intercepted arc.

The opposite angles of an inscribed quadrilateral to a circle are supplementary.

Interior Angle Sum Theorem
If n represents the number of sides in any polygon, the expression to find the sum of the interior angles in a polygon is (n – 2) • 180°.

• Corollary to the Interior Angle Sum Theorem
A single interior angle measure of a regular polygon may be found by dividing by the number of sides. The expression for finding a single interior angle measure in a regular polygon is .

Intersecting Lines Postulate
If two lines intersect, then they intersect in exactly one point.

Intersecting Planes Postulate
If two distinct planes intersect, then they intersect in exactly one line.

Isosceles Trapezoid Theorem
If the legs of a trapezoid are congruent, then the base angles are congruent.

Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

• The Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent making the triangle an isosceles triangle.

Midsegment of a Triangle Theorem
A segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.

Midsegment of a Trapezoid Theorem
The length of the midsegment of a trapezoid is half of the total of the lengths of the bases.

Multiplication Property of Equality
For real numbers a, b, and c, if a = b, then ac = bc.

Nonvertex Angles Theorem
The nonvertex angles of a kite are congruent.

Opposite Angle Theorem
If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.

Opposite Side Theorem
If two sides of a triangle are not congruent, then the larger angle lies opposite the larger side.

Parallel Postulate
Given a line and a point not on that line, there exists only one line through the given point parallel to the given line.

Perpendicular Diagonals Theorem
The diagonals of a kite are perpendicular.

Perpendicular Diameters and Chords Theorem
If a diameter is perpendicular to a chord, then the diameter bisects the chord and the minor arc between the endpoints of the chord.

Pieces of Right Triangles Similarity Theorem
If an altitude is drawn from the right angle of a right triangle, the two smaller triangles created are similar to one another and to the larger triangle.

• First Corollary to the Pieces of Right Triangles Similarity Theorem
The length of the altitude from the right angle of a right triangle is the geometric mean between the segments of the hypotenuse created by the intersection of the altitude and the hypotenuse.
• Second Corollary to the Pieces of Right Triangles Similarity Theorem
Each leg of a right triangle is the geometric mean between the hypotenuse and the segment of the hypotenuse created by the altitude adjacent to the given leg.

Points Postulate
Through any two points there exists exactly one line.

Properties of Kites

• Two pairs of adjacent, congruent sides.
• Nonvertex angles are congruent.
• Diagonals are perpendicular.
• The diagonal connecting the vertex angles bisects the diagonal connecting the nonvertex angles.
• The diagonal connecting the vertex angles bisects those angles.
• The angles created by the diagonal connecting the nonvertex angles are congruent above and below the diagonal.

Properties of Parallelograms

• Opposite sides are congruent and parallel.
• The diagonals bisect each other.
• Opposite angles are congruent.
• Consecutive angles are supplementary.

Properties of Rectangles

• Opposite sides are congruent and parallel.
• The diagonals bisect each other.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• Each angle is a right angle.
• The diagonals are congruent.

Properties of a Rhombus

• Opposite sides are congruent and parallel.
• The diagonals bisect each other.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• All four sides are congruent.
• The diagonals are perpendicular.
• The diagonals are angle bisectors.

Properties of a Square

• Opposite sides are congruent and parallel.
• The diagonals bisect each other.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• Each angle is a right angle.
• The diagonals are congruent.
• All four sides are congruent.
• The diagonals are perpendicular.
• The diagonals are angle bisectors.

Properties of a Trapezoid

• One pair of parallel sides.
• Consecutive angles between the bases are supplementary.

Proportional Perimeter Theorem
If two triangles are similar, the perimeters of each triangle are proportional to their corresponding sides.

Pythagorean Theorem
If a right triangle has sides a and b and hypotenuse c then a2 + b2 = c2.

• Converse of the Pythagorean Theorem
In a triangle with sides a, b, and c, if a2 + b2 = c2, then the triangle is a right triangle.

Reflexive Property of Equality
For real number a, a = a.

Same-Side Interior Angles
If a transversal intersects two parallel lines, then same-side interior angles are supplementary.

Secant Interior Angle Theorem
The measure of a secant angle is equal to half the sum of the arcs it and its vertical angle intercept.

Secant-Tangent Intersection Theorem
When a secant and tangent intersect at the point of tangency, the angles created at the point of intersection are half the measurement of the arcs they intersect.

If point C is between points A and B, then AC + CB = AB.

Side-Angle-Side (SAS) Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Side-Angle-Side Similarity Postulate
If two or more triangles have corresponding, congruent angles and the sides that make up these angles are proportional, then the triangles are similar.

Side-Side-Side (SSS) Postulate
If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

Side-Side-Side Similarity Theorem
If two or more triangles have three corresponding, proportional sides, then the triangles are similar.

Square Root Property of Equality
For any real number a,  = a.

Substitution Property of Equality
For real numbers a and b, if a = b, then a can replace b in any expression and vice versa.

Subtraction Property of Equality
For real numbers a, b, and c, if a = b, then a − c = b − c.

Supplementary Angles of a Trapezoid Theorem
Consecutive angles between the two bases of a trapezoid are supplementary.

Symmetric Property of Equality
For real numbers a and b, if a = b, then b = a.

Transitive Property of Equality
For real numbers a, b, and c, if a = b and b = c, then a = c.

Triangle Altitude Similarity Theorem
If two triangles are similar, the corresponding altitudes are proportional to each set of corresponding sides.

Triangle Exterior Angle Theorem
The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the third side.

• Corollary to the Triangle Inequality Theorem
The length of the third side of a triangle is less than the total and greater than the absolute value of the difference of the other two sides.

Triangle Proportionality Theorem
If a line is parallel to one side of a triangle and also intersects the other two sides, the line divides the sides proportionally.

Triangle Sum Theorem
The sum of the measures of the angles in a triangle is 180°.

Vertical Angles Theorem
Vertical angles are congruent.