Lines, rays, or segments that intersect at right angles are perpendicular.
Definition: Complementary angles
Complementary angles are two angles whose sum is 90 degrees (i.e. right angle). Each of the two angles is called the complement of the other.
Definition: Supplementary angles
Supplementary angles are two angles whose sum is 90 degrees (i.e. straight angle). Each of the two angles is called the supplement of the other.
Memorize theorems, definitions, and postulates.
Look for key words and symbols in the given information.
Think of all theorems, definitions, and postulates that involve those key words.
Decide which theorem, definition, or postulate allows you to draw a conclusion.
Draw a conclusion, and give a reason to justify the conclusion. Be certain that you have not used the reverse of the correct reason.
Theorem 4
If angles are supplementary to the same angle, then they are congruent.
Theorem 5
If angles are supplementary to congruent angles then they are congruent.
Theorem 6
If angles are complementary to the same angle, then they are congruent.
Theorem 7
If angles are complementary to congruent angles then they are congruent.
Theorem 8: Addition Property of Segments
If a segment is added to two congruent segments, the sums are congruent (Addition Property).
Theorem 9: Addition Property of Angles
If an angle is added to two congruent angles, the sums are congruent (Addition Property).
Theorem 10: Addition Property of Segments
If congruent segments are added to congruent segments, the sums are congruent (Addition Property).
Theorem 11: Addition Property of Angles
If congruent angles are added to congruent angles, the sums are congruent (Addition Property).
Theorem 12: Subtraction Property
If a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent (Subtraction Property).
Theorem 13: Subtraction Property
If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent (Subtraction Property).
An addition property is used when the segments or angles in the conclusion are greater than those in the give information.
A subtraction property is used when the segments or angles in the conclusion are smaller than those in the given information.
Theorem 14: Multiplication Property
If segments (or angles) are congruent, their like multiples are congruent. (Multiplication Property).
Theorem 15: Division Property
If segments (or angles) are congruent, their like divisions are congruent. (Division Property).
Look for a double use of the word midpoint or trisect or bisects in the given information.
The Multiplication Property is used when the segment or angles in the conclusion are greater than those in the given information.
The Division Property is used when the segments or angles in the conclusion are smaller than those in the given information.
Theorem 16: Transitive Property
If angles (or segments) are congruent to the same angle (or segment), they are congruent to each other. (Transitive Property).
Theorem 17: Transitive Property
If angles (or segments) are congruent to congruent angles (or segment), they are congruent to each other. (Transitive Property).
Definition: Substitution Property
You can solve for one variable and then substitute the value found for that variable and the variable can represent an angle or segment.
Definition: Vertical angles
Two angles are vertical angles if the rays forming the sides of one and the rays forming the sides of the other are opposite rays.
Theorem 18
Vertical angles are congruent.
Definition: Vertical angles
Two angles are vertical angles if the rays forming the sides of one and the rays forming the sides of the other are opposite rays.
Theorem 18
Vertical angles are congruent.
CPCTC - Corresponding Parts of Congruent Triangles are Congruent.
Theorem 19
All radii of a circle are congruent.
Definition: Median
A median of a triangle is a line segment drawn from any vertex of the triangle to the midpoint of the opposite side. (A median divides into two congruent segments, or bisects the side to which it is drawn).
Definition: Altitude
An altitude of a triangle is a line segment drawn from any vertex of the triangle to the opposite side, extended if necessary, and perpendicular to that side. (An altitude of a triangle forms a right angle with one of the sides.)
Auxiliary Lines
Many proofs involve lines, rays, or segments that do not appear in the original figure. These additions to diagrams are called auxiliary lines.
Postulate:
Two points determine a line (or ray segment).
Overlapping Triangles
Overlapping triangles share a side which can be helpful in proving that they are congruent so that CPCTC can be applied to prove other sides or angles are congruent.
Definition: scalene triangle
A scalene triangle is a triangle in which no two sides are congruent.
Definition: isosceles triangle
An isosceles triangle is a triangle in which at least two sides are congruent.
Definition: equilateral triangle
An equilateral triangle is a triangle in which all sides are congruent.
Definition: equilangular triangle
An equiangular triangle is a triangle in which all angles are congruent.
Definition: acute triangle
An acute triangle is a triangle in which all angles are acute.
Definition: right triangle
A right triangle is a triangle in which one of the angles is a right angle. (The side opposite the right angle is called the hypotenuse. The sides that form the right angle are called legs).
Definition: obtuse triangle
An obtuse triangle is a triangle in which one of the angles is an obtuse angle.
Theorem 20: Side-Angle Congruence
If two sides of a triangle are congruent, the angles opposite the sides are congruent.
Theorem 21: Angle-Side Congruence
If two angles of a triangle are congruent, the sides opposite the angles are congruent
Theorem 20
If two sides of a triangle are congruent, the angles opposite the sides are congruent.
Theorem 21
If two angles of a triangle are congruent, the sides opposite the angles are congruent
If at least two sides of a triangle are congruent, the triangle is isosceles.
If at least two angles of a triangle are congruent, the triangle is isosceles.
Inverse of Theorem 20
If two sides of a triangle are not congruent, then the angles apposite them are not congruent, and the larger angle is opposite the longer side.
Inverse of Theorem 21
If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.
HL Postulate
If there exists a correspondence between the vertices of two right triangles such that the hypotenuse and the leg of one triangle are congruent to the corresponding parts of the other triangle, the two right triangles are congruent. (HL).
Detour Proofs
To solve some problems, it is necessary to prove more than one pair of triangles congruent. We call the proofs we use in such cases detour proof
Determine which triangles you must prove to be congruent to reach the required conclusion.
Attempt to prove that these triangles are congruent. If you cannot do so for lack of enough given information, take a detour.
Identify the parts that you must prove to be congruent to establish the congruence of the triangles.
Find a pair of triangles that
You can readily prove to be congruent
Contains a pair of parts needed for the main roof
Prove that the triangles found in step 4 are congruent.
Use CTCTC and complete the proof panned in step 1.
Theorem 22: The Midpoint Formula
If A= (x1,y1) and B=(x2,y2), then the midpoint M=(xm,ym), of segment AB can be found by using the midpoint formula:
M = (xm,ym) = ((x1 + x2)/2, (y1 + y2)/2)
Theorem 23: Right-Angle Theorem
If two angles are both supplementary and congruent, then they are right angles
Definition: distance
The distance between two objects is the length of the shortest path between two points.
Definition: equidistant
If two points are the same distance from a third point then the third point is equidistant from the first two points.
Postulate:
A line segment is the shortest path between two points.
Definition: perpendicular bisector
The perpendicular bisector of a segment is the line that bisects and is perpendicular to the segment.
Theorem 24:
If two points are each equidistant from the end-points of a segment, then the two points determine the perpendicular bisector of that segment.
Theorem 25:
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment.
Definition: plane
A plane is a surface such that if any two points on the surface are connected by a line, all points of the line are also on the surface.
Planes have two dimensions...i.e. length and depth...but no thickness and both dimensions are infinite
Definition: coplanar
If points, lines segments and so forth, lie in the same plane, the are coplanar.
Definition: noncoplanar
Points, lines, segments that do not lie in the same plane are called noncoplanar.
Definition: noncoplanar
Points, lines, segments that do not lie in the same plane are called noncoplanar.
Definition: transversals
A transversal a line that intersects two coplanar lines in two distinct points.
Definition: interior
The region of the plane between two lines on the plane is called the interior.
Definition: exterior
The region of the plane outside two lines on a plane is called the exterior.
Definition: alternate interior angles
Alternate interior angles are a pair of angles formed by two lines and a transversal. The angles must lie in the interior, on alternate sides of the transversal, and have different vertices.
Definition: alternate exterior angles
Alternate exterior angles are a pair of angles formed by two lines and a transversal. The angles must lie in the exterior, on alternate sides of the transversal, and have different vertices.
Definition: corresponding angles
Corresponding angles are a pair of angles formed by two lines and a transversal. One angle must lie on the interior, the other on the exterior, and both on the same side of the transversal but have different vertices.
Definition: parallel lines
Parallel lines are two coplanar lines that do not intersect.
Definition: slope
The slope m of a nonvertical line, segment, or ray contains (x1,y1) and (x2,y2) is defined as:
m = y2 - y1 / x2 - x1 or y1 - y2 / x1 - x2 or ∆y/∆x or rise / run
Theorem 26: parallel => equal slope
If two nonvertical lines are parallel, then their slopes are equal
Theorem 27: equal slope => parallel
If the slopes of two nonvertical lines are equal, then the lines are parallel.
Theorem 28: perpendicular => opposite reciprocal
If two lines are perpendicular and neither is vertical, each line’s slope is the opposite reciprocal of the other’s.
Theorem 29: opposite reciprocal => perpendicular
If a line’s slope is the opposite reciprocal of another line’s slope, the two lines are perpendicular
List the possibilities for the conclusions
Assume that the negation of the desired conclusion is correct
Write a chain of reasons until you reach an impossibility.
This will be a contradiction of either
(a) given information
(b) a theorem, definition, or other known fact
Stat the remaining possibility as the desired conclusion.
Theorem 30: Exterior Angle Inequality Theorem
The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.
Theorem 31: Parallel by Alternate Interior Angles Congruent
If two lines are cut by a transversal such that two alternate interior angles are congruent, the lines are parallel
Theorem 32: Parallel by Alternate Exterior Angles Congruent
If two lines are cut by a transversal such that two alternate exterior angles are congruent, the lines are parallel
Theorem 33: Parallel by Corresponding Angles Congruent
If two lines are cut by a transversal such that two corresponding angles are congruent, the lines are parallel
Theorem 34: Parallel by Interior Angles on the Same Side Supplementary
If two lines are cut by a transversal such that two interior angles on the same side are supplementary, the lines are parallel.
Theorem 35: Parallel by Exterior Angles on the Same Side Supplementary
If two lines are cut by a transversal such that two exterior angles on the same side are supplementary, the lines are parallel.
Theorem 36: Parallel by Coplanar and Perpendicular to same line
If two coplanar lines are perpendicular to a third line, they are parallel.
Theorem 30: Exterior Angle Inequality Theorem
The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.
Theorem 31: Parallel by Alternate Interior Angles Congruent
If two lines are cut by a transversal such that two alternate interior angles are congruent, the lines are parallel
Theorem 32: Parallel by Alternate Exterior Angles Congruent
If two lines are cut by a transversal such that two alternate exterior angles are congruent, the lines are parallel
Theorem 33: Parallel by Corresponding Angles Congruent
If two lines are cut by a transversal such that two corresponding angles are congruent, the lines are parallel
Theorem 34: Parallel by Interior Angles on the Same Side Supplementary
If two lines are cut by a transversal such that two interior angles on the same side are supplementary, the lines are parallel.
Theorem 35: Parallel by Exterior Angles on the Same Side Supplementary
If two lines are cut by a transversal such that two exterior angles on the same side are supplementary, the lines are parallel.
Theorem 36: Parallel by Coplanar and Perpendicular to same line
If two coplanar lines are perpendicular to a third line, they are parallel.
The Parallel Postulate
Through a point not on a line there is exactly one parallel to the given line
Theorem 37: If Parallel => Alternate Interior Angles congruent
If two parallel lines are cut by a transversal, each pair of alternate interior angles are congruent.
Theorem 38: If Parallel => any pair of angles congruent/supplementary
If two parallel lines are cut by a transversal, then any pair of the angles formed are either congruent or supplementary.
Theorem 39: If Parallel => Alternate Exterior Angles Congruent
If two parallel lines are cut by a transversal, each pair of alternate exterior angles are congruent.
Theorem 40: If Parallel => Corresponding Angles Congruent
If two parallel lines are cut by a transversal, each pair of corresponding angles are congruent.
Theorem 41: If Parallel => Interior Angles on Same Side Supplementary
If two parallel lines are cut by a transversal, each pair of interior angles on the same side of the transversal are supplementary.
Theorem 42: If Parallel => Exterior Angles on Same Side Supplementary
If two parallel lines are cut by a transversal, each pair of exterior angles on the same side of the transversal are supplementary.
Theorem 43: If Perpendicular to 1 of 2 Parallel Lines => perpendicular to other
In a plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to the other.
Theorem 44: Transitive Property of Parallel Lines
If two lines are parallel to a third line, they are parallel to each other (Transitive Property of Parallel Lines)
Definition: convex
A convex polygon is a polygon in which each interior angle has a measure less than 180.
Definition: diagonal
A diagonal of a polygon is any segment that connects two non-consecutive (non-adjacent) vertices of the polygon.
Definition: quadrilaterals
A quadrilateral is a four-sided polygon.
Definition: parallelogram
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
Definition: rectangle
A rectangle is a parallelogram in which at least one angle is a right angle.
Definition: rhombus
A rectangle is a parallelogram in which at least at least two consecutive sides are congruent.
Definition: kite
A kite is a quadrilateral in which two disjoint pairs of consecutive sides are congruent.
Definition: square
A square is a parallelogram that is both a rectangle and a rhombus
Opposite sides are parallel (by definition)
Opposite sides are congruent
Opposite angles are congruent
Diagonals bisect each other
Any pair of consecutive angles are supplementary
All the properties of a parallelogram apply (by definition).
All angles are right angles
Diagonals are congruent
Two disjoint pairs of consecutive sides are congruent (by definition).
Diagonals are perpendicular
One diagonal is the perpendicular bisector of the other
One of the diagonals bisects a pair of opposite angles
One pair of opposite angles are congruent
Properties 3-5 are half properties
All properties of a parallelogram apply (by definition)
All properties of a kite apply (half properties become full properties)
All sides are congruent
Diagonals bisect the angles
Diagonals are perpendicular bisectors of each other
Diagonals divide the rhombus into 4 congruent triangles
All properties of a rectangle apply (by definition)
All properties of a rhombus apply (by definition)
Diagonals form 4 isosceles right triangles
The legs are congruent (by definition)
The bases are parallel (by definition)
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 angles
Both pairs of opposite sides are parallel
Both pairs of opposite sides are congruent
One pair of opposite sides are both parallel and congruent
Diagonals bisect each other
Both pairs of opposite angles are congruent
First prove Parallelogram, then...
Contains at least one right angle
Diagonals are congruent
Or, all 4 angles are right angles
Two disjoint pairs of consecutive sides are congruent
One of the diagonals is perpendicular bisector of the other diagonal
First, show Parallelogram, then…
Contains a pair of consecutive sides that are congruent
Either diagonal bisects two angles of the parallelogram
Or...prove diagonals are perpendicular bisectors of each other
Prove, both a rectangle and a rhombus
NonParallel sides are congruent
Lower or upper base angles are congruent
Diagonals are congruent
Definition: foot
The point of intersection of a line and a plane is called the foot of a line.
Postulate
Three noncollinear points determine a plane
Theorem 45
A line and a point not on a line determine a plane.
Theorem 46
Two intersecting lines determine a plane.
Theorem 47
Two parallel lines determine a plane.
Postulate
If a line intersects a plane not containing it, then the intersection is exactly one point.
Postulate
If two planes intersect, their intersection is exactly one line.
A line is perpendicular to a plane if it is perpendicular to every one of the lines in the plane that pass through its foot.
Theorem 48
if a line is perpendicular to two distinct lines that lie in a plane and that pass through its foot, then it is perpendicular to the plane.
Definition: Parallel Line & Plane
A line and a plane are parallel if they do not intersect.
Definition: Parallel Planes
Two planes are parallel if they do not intersect.
Definition: Skew Lines
Two lines are skew if they are not coplanar.
Theorem 49
If a plane intersects two parallel planes the lines of the intersection are parallel.
Properties Relating Parallel Lines and Planes
If two planes are perpendicular to the same line, they are parallel to each other
If a line is perpendicular to one of two parallel planes, it is perpendicular to the other plane as well.
If two planes are parallel to the same plane, they are parallel to each other.
if two lines are perpendicular to the same plane, they are parallel to each other.
If a plane is perpendicular to one of two parallel lines, it is perpendicular to the other line as well.
The sum of the measure of the three angles of a triangle is 180.
Theorem 51
The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.
Theorem 52: Midline Theorem
A segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is one-half the length of the third side.
If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent.
Theorem 54: AAS
If there exists a correspondence between the vertice of two triangles such that two angles and a non-included side of one are congruent to the corresponding parts of the other, then the triangles are congruent.
The sum, Si, of the measure of the three angles of the angles of a polygon with n sides is given by the formula Si = (n - 2)180.
Definition: Interior angles
Sometimes the angles of a polygon are referred to as the interior angles.
Theorem 56
If one exterior angle is taken at each vertex, the sum Se of the measures of the exterior angles of a polygon is given by the formula Se = 360.
Theorem 57
The number d of diagonals that can be drawn in a polygon of n sides is given by the formula
d = n(n-3)/2
Definition: Regular Polygons
A regular polygon is a polygon that is both equilateral and equiangular.
Theorem 58
The measure E of each exterior angle of an equiangular polygon of n sides is given by the formula
E = 360/n
Definition: Ratio
A ratio is a quotient of two numbers. Ratios can be represented in the following formats:
5/3
5:3
5 to 3
5÷3
Definition: Proportion
A proportion is an equation stating that two or more ratios are equal.
Definition: Extremes
The first and fourth terms of a proportion are called the extremes.
Definition: Means
The second and third terms are called the means.
Theorem 59: Means-Extremes Products
In a proportion, the product of the means is equal to the product of the extremes.
Theorem 60: Means-Extremes Ratio
If the product of a pair of nonzero numbers is equal to the product of another pair of nonzero numbers, then either pair of numbers may be made the extremes, and the other pair the means, of a proportion.
Definition: Geometric Mean (Mean Proportional)
If the means in a proportion are equal, either mean is called a geometric mean, or mean proportional, between the extremes.
Definition: Arithmetic Mean
The average of two numbers is another type of mean between two numbers, called the arithmetic mean.
Dilation → enlargement
Reduction → opposite of dilation
Definition: Similar polygons
Similar polygons are polygons in which:
the ratios of the measures of the corresponding sides are equal.
Corresponding angles are congruent
Theorem 61: Perimeter Ratio of Similar Polygons
The ratio of the perimeters of two similar polygons equals the ratio of any pair of corresponding sides.
Postulate: AAA
If there exists a correspondence between the vertices of two triangles such that the three angles of one triangle are congruent to the corresponding angles of the other triangle, then the triangles are similar. (AAA)
Theorem 62: AA
If there exists a correspondence between the vertices of two triangles such that the two angles of one triangle are congruent to the corresponding angles of the other triangle, then the triangles are similar. (AA)
Theorem 63: SSS~
If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of corresponding sides are equal, then the triangles are similar. (SSS~)
Theorem 64: SAS~
If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of the two corresponding sides and equal and the included angles are congruent, then the triangles are similar. (SAS~)
Theorem 65: Side-Splitter Theorem
If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally. (Side-Splitter Theorem)
Theorem 66: Side-Splitter Theorem
If three or more parallel lines are intersected by two traversals, the parallel lines divide the transversals proportionally.
Theorem 67: Angle Bisector Theorem
If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides. (Angle Bisector Theorem).
Theorem 68: Altitude-On-Hypotenuse Theorem
If an altitude is drawn to the hypotenuse of a right triangle then
The two triangles formed are similar to the given right triangle and to each other
The altitude to the hypotenuse is the mean proportional between the segments of the hypotenuse
x/h = h/y or h2 = xy
Either leg of the given right triangle is the mean proportional between the hypotenuse of the give right triangle and the segment of the hypotenuse adjacent to that leg (i.e. the projection of that leg on the hypotenuse)
x/a = a/c or a2 = yc
x/b = b/c or b2 = xc
Theorem 69: Pythagorean Theorem
The square of the measure of the hypotenuse of a right triangle is equal to the sum of the squares of the measures of the legs.
Theorem 70: Converse of Pythagorean Theorem
If the squares of the measure of one side of a triangle equals the sum of the squares of the measures of the other two sides, then the angle opposite the longest side is a right angle.
Theorem 71: Distance Formula
If P = (x1, y1) and Q = (x2, y2) are any two points, then the distance between them can be found with the formula:
PQ = √((x2 - x1) + (y2 - y1)) OR √((∆x)2 + (∆y)2)
Definition: Pythagorean triple
Any three whole numbers that satisfy the equation a2 +b2 = c2 form a Pythagorean triple.
Principle of the Reduced Triangle
Reduce the difficulty of the problem by multiplying or dividing the three lengths by the same number to obtain a similar but simpler, triangle in the same family
Solve for the missing side of this easier triangle.
Convert back to the original problem
Theorem 72: 30-60-90 Triangle Theorem
In a triangle whose angles have the measures 30, 60, 90, the lengths of the sides opposite these angles can be represented by x, x√3, 2x respectively.
Theorem 73: 45-45-90 Triangle Theorem
In a triangle whose angles have the measures 45, 45, 90, the lengths of the sides opposite these angles can be represented by x, x, x√2 respectively
Six Common Families of Right Triangles
30-60-90 ⇔ (x, x√3, 2x)
45-45-90 ⇔ (x, x, x√2)
(3, 4, 5)
(5, 12, 13)
(7, 24, 25)
(8, 15, 17)
Definition: Faces, Edges, Diagonals of Rectangular Solid
Given the rectangular solid, ABCDEFGH, the solid has 6 rectangular faces, where AB is one of the 12 edges, and HB is one of the 4 diagonals.
Definition: Base, Vertex, Altitude and Slant Height of Rectangular Square Pyramid
Given the rectangular square pyramid, JKMOP, the square is called its base, the top point is called its vertex, the altitude a line from the vertex to the base that is perpendicular to the base, and the slant height is a line drawn from the vertex to the side of the base that is perpendicular to that side of the base.
Definition: Cube
A cube is a rectangular solid in which all edges are congruent.
Definition: 3 Trigonometry Ratios
Sine: sin ⦣A = opposite leg / hypotenuse
Cosine: cos ⦣A = adjacent leg / hypotenuse
Tangent: tan ⦣A = opposite leg / adjacent leg
Definition: Angle of Elevation
If an observer at a point P looks upward toward an object at A, the angle the line of sight, PA makes with the horizontal PH is called the angle of elevation.
Definition: Angle of Depression
If an observer at a point P looks upward toward an object at B, the angle the line of sight, PB makes with the horizontal, PH, is called the angle of depression.
Definition: Circle
A cube is the set of all points in a plane that are a given distance from the given point in the plane. The given point is the center of the circle, and the given distance is the radius. A segment that joins the center to a point on the circle is also called a radius.
Definition: Concentric
Two or more coplanar circles with the same center are called concentric circles.
Definition: Congruent Circles
Two circles are congruent if the have congruent radii.
Definition: Interior
A point is inside (in the interior) if its distance from the center is less than the radius.
Definition: Exterior
A point is outside (in the exterior) if its distance from the center is greater than the radius.
Definition: On the Circle
A point is on the circle if its distance from the center is equal to the radius.
Definition: Chord
A chord of a circle is a segment joining any two points on the circle.
Definition: Diameter
A diameter of a circle is is a chord that passes through the center of the circle.
Formula: Area of a Circle
A area of a circle can be found with the formula:
A = 𝜋r2
Formula: Circumference of a Circle
A circumference(perimeter) of a circle can be found with the formula:
C = 𝜋d
Definition: Chord Distance
The distance from the center of a circle to a chord is the measure of the the perpendicular segment from the center to the chord.
Theorem 74: If r ┴ ⇒ bisects chord
If a radius is perpendicular to a chord, then it bisects the chord.
Theorem 75: If r bisects chord ⇒ ┴
If a radius of a circle bisects a chord that is not a diameter, then it is perpendicular to that chord.
Theorem 76: ┴ bisector passes through a circle’s center
The perpendicular bisector of a chord passes through the center of the circle.
Theorem 77: If chord distances ≃ ⇒ chords ≃
If two chord of a circle are equidistant from the center, then they are congruent.
Theorem 78: If chord distances ≃ ⇒ chords ≃
If two chord of a circle are equidistant from the center, then they are congruent.
Definition: Arc
An arc consists of two points on a circle and all points on the circle needed to connect the points by a single path.
Definition: Arc Center
An center of an arc is the center of the circle of which the arc is part.
Definition: Central Angle
A central angle is an angle whose vertex is at the center of the circle.
Definition: Minor Arc
A minor arc is an arc whose points are on or between the sides of a central angle.
Definition: Major Arc
A major arc is an arc whose points are outside the sides of a central angle.
Definition: Semicircle
A semicircle is an arc whose endpoints are the endpoints of a diameter.
Definition: Measure of an Minor Arc
A measure of a minor arc or a semicircle is the same as the measure of the central angle that intercepts the arc.
Definition: Measure of an Major Arc
A measure of a major arc or 360 minus the measure of the minor arc with the same endpoints.
Definition: Congruent Arcs
Two arcs are congruent whenever they have the same measure and are parts of the same circle or congruent circles.
Theorem 79: If central ∠ ≃ ⇒ ⌒≃
If two central angles of a circle (or of congruent circles) are congruent, then their intercepted arcs are congruent.
Theorem 80: If ⌒ ≃ ⇒ ∠ ≃
If two arcs of a circle (or of congruent circles) are congruent, then the corresponding central angles are congruent.
Theorem 81: If ∠ ≃ ⇒ chords ≃
If two central angles of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent.
Theorem 82: If chords ≃ ⇒ ∠ ≃
If two chords of a circle (or of congruent circles) are congruent, then the corresponding central angles are congruent.
Theorem 83: If ⌒ ≃ ⇒ chords ≃
If two arcs of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent.
Theorem 84: If chords ≃ ⇒ ⌒ ≃
If two chords of a circle (or of congruent circles) are congruent, then the corresponding arcs are congruent.
Definition: Secant
A secant is a line that intersects a circle at exactly two points (every secant contains a chord).
Definition: Tangent
A tangent is a line that intersects a circle at exactly one point....called the point of tangency or point of contact.
Postulate: Tangent ┴ to r
A tangent line is perpendicular to the radius drawn to the point of contact.
Postulate: If ┴ to r ⇒ Tangent
If a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle.
Definition: Tangent Segment
A tangent segment is the part of a tangent line between the point of contact and a point outside the circle.
Definition: Secant Segment
A secant segment is the part of a secant that joins a point outside the circle to the farther intersection point of the secant and the circle.
Definition: External Part
The external part of a secant segment is the part of a secant line outside point to the neaer intersection point.
Theorem 85: Two-Tangent Theorem
If two tangent segments are drawn to a circle from an external point, then those segments are congruent.
Definition: Tangent Circles
Tangent Circles are circles that intersect each other at exactly one point.
Definition: Externally Tangent
Two circles are externally tangent if each of the tangent circles lies outside the other.
Definition: Internally Tangent
Two circles are internally tangent if one of the tangent circles lies inside the other.
Definition: Common Tangent
A common tangent is a line tangle to two circles. It is a common internal tangent if it lies between the circles. It is a common external tangent if it is not between the circles (i.e. does not intersect the line of centers.
Procedure: Common-Tangent Procedure
To solve problems involving common tangents…
Draw the segment joining the centers
Draw the radii to the points of contact
Through the center of the smaller circle, draw a line parallel to the common tangent
Observe that this line will intersect the radius of the larger circle (extended if necessary) to form a rectangle and a right triangle.
Use the Pythagorean Theorem and properties of a rectangle
Definition: Inscribed Angle
An inscribed angle is an angle whose vertex is on a circle and whose sides are determined by two chords.
Definition: Tangent-Chord Angle
A tangent-chord angle is an angle whose vertex is on a circle and whose sides are determined by a tangent and a chord that intersect a the tangent’s point of contact.
Theorem 86: Inscribed or Tangent-Chord ∠ = ½ ⌒
The measure of an inscribed angle or a tangent-chord angle (vertex on a circle) is one-half the measure of its intercepted arc.
Definition: Chord-Chord Angle
A chord-chord angle is an angle formed by two chords that intersect inside a circle but not at the center.
Theorem 87: Chord-Chord Angle ∠ = ½ ∑⌒’s
The measure of a chord-chord angle is one-half the sum of the measures of the arcs intercepted by the chord-chord angle and its vertical angle (i.e. same as the average)
Definition: Secant-Secant Angle
A secant-secant angle is an angle whose vertex is outside a circle and whose sides are determined by two secants.
Definition: Secant-Tangent Angle
A secant-tangent angle is an angle whose vertex is outside a circle and whose sides are determined by a secant and a tangent.
Definition: Tangent-Tangent Angle
A tangent-tangent angle is an angle whose vertex is outside a circle and whose sides are determined by two tangents.