GEOMETRY LESSONS
Whether needing help with geometry homework or reviewing for tests, Mr. X can help you, your child or your students understand Geometry. Quite simply, our math lessons help students get the point. We also have a library of sample geometry problems with examples of solved problems for each geometry lesson. Check out our free samples below, as well as the geometry curriculum.Geometry Curriculum
Angles
Title DescriptionEuclid's Four Postulates  The concepts within are what need to be known and understood. It is not important to learn these in order or by their "titles." The embedded principles are important. 

Acute and Obtuse Angles  An acute angle measures less than 90° while an obtuse angle measures between 90° and 180°.  
Classification of Angles  The magnitude of an angle is the size of that angle. A straight angle measures 180°. 

Angles Right On  Paper folding can easily make a right angle. Learn the notion of perpendicularity, that is, perpendicular rays form a 90° angle. 

A Lesson in Reading Angles  A lesson in the nature of angles, with reading integervalue angle measurements with a protractor.  
Angles on a Clock Face  More angles on a clock face help us see the values of angles as we move around the circle.  
Complements and Supplements  Two complementary angles sum to 90°. Two supplementary angles sum to 180°. 

Circles, Angles, Degrees  An introduction to angles includes a look at hands on a clock face. Angles need to be understood early in Geometry. 

Angles and Protactors  We look at equal fifths of a rotation, which are 72°. We also look at an angle greater than 180°. 

Angular Truths  A summary of angular truths includes: all straight angles are equal, all right angles are equal, vertical angles are equal, the supplements of equal angles are themselves equal, and so forth.  
Angles, Vertical  Vertical angles are congruent, that is, of the same measure. Sometimes vertical angles are termed "opposite" angles. 
Area and Perimeter
Title DescriptionArea and Perimeter Formulas  Some area formula calculations are included in this video.  
Equilateral Triangles College Exam Problem 1  Consider two equilateral triangles where one has a perimeter three times the other.  
Area and Perimeter of Triangles Practice Problems 1  A worksheet with right triangles asks us to calculate both area and perimeter.  
Equilateral Triangles College Exam Problem 2  Consider two equilateral triangles where one has an area nine times the other.  
Area and Perimeter of Squares  We show how to calculate area and perimeter of squares in this worksheet.  
Area and Perimeter of Rectangles  Almost as easy as squares, we show how to calculate area and perimeter of squares in this worksheet.  
Area and Perimeter of Parallelograms  We show how to calculate area and perimeter of parallelograms in this worksheet.  
Area and Perimeter of Trapezoids  We show how to calculate area and perimeter of trapezoids in this worksheet. 
Circles
Title DescriptionCircles  A circle is a collection of coplanar points equidistant from a center point. Every radius, for a given circle, is fixed. 

Radii, Diameter, and Circumference  We need to understand radii, diameters, and circumference. Concomitantly, we should value pi, appx. 3.14159, or expressed exactly as π. 

Euclid's Four Postulates  The concepts within are what need to be known and understood. It is not important to learn these in order or by their "titles." The embedded principles are important. 
Constructions
Title DescriptionStraightedgeandCompass: A Parallel Line (angle copy)  Another way to draw a parallel line to a given line (other than to construct a rhombus) is to copy an angle, which is shown in this lesson.  
StraightedgeandCompass: A Parallel Line  Construct a line parallel to a given line through a point not on that line, using only a straightedge and a compass.  
StraightedgeandCompass: Divide Segment into Equal Parts  Using a straightedge and compass we can divide a line segment into equal parts.  
English and Metric Drawers  Drawing line segments to given lengths. This is a very basic skill everyone needs to have. 

Constructing the Perpendicular Bisectors of a Triangle Lesson 2  This is a CompassandStraightedge Construction for the Perpendicular Bisectors of a Triangle. Those lines meet at a common point, and that point of intersection is called the Point of Concurrence. 

Constructing the Perpendicular Bisectors of a Triangle Lesson 1  We draw Perpendicular Bisectors of a Triangle, which are concurrent at a point that is the center of the Circumscribed Circle of the Triangle.  
StraightedgeandCompass: A 30°° Angle  With just a straightedge and compass (no protractor) construct a 30degree angle.  
StraightedgeandCompass: Bisect an Angle  With a straightedge and a compass we can divide an angle into two equal halves, that is, bisect the angle.  
StraightedgeandCompass: A 90° Angle  We can construct a 90° angle with just a straightedge and a compass.  
StraightedgeandCompass: A 45° Angle  We can construct a 45° angle with just a straightedge and a compass.  
StraightedgeandCompass: An Isosceles Triangle  We can construct a 90° angle with just a straightedge and a compass. Using only a straightedge and a compass we can construct an isosceles triangle.  
StraightedgeandCompass: An Equilateral Triangle  An easy construction is to use a straightedge and compass to draw an equilateral triangle.  
StraightedgeandCompass: A Right Triangle  Given the length of one leg and the hypotenuse, we may construct a right triangle accordingly using a straightedge and compass.  
Lesson on the Centroid and Medians of Triangles  A brief lesson on Medians in Triangles. Their point of concurrence is the Centroid of the triangle. 

Triangle Medians  Construct the medians of a triangle using just a straightedge and a compass.  
Constructing the Altitudes of a Triangle  This is a CompassandStraightedge Construction for the Altitudes of a Triangle. Those line segments meet at a common point, and that concurrent point of intersection is called the Orthocenter. 

Triangle Angle Bisectors and Inscribed Circles  A brief lesson on Angle Bisectors in Triangles. Their point of concurrence is the center of the Inscribed Circle. 

Constructing Triangle Angle Bisectors  This is a CompassandStraightedge Construction for the Angle Bisectors of a Triangle. Those rays meet at a common point, and that concurrent point of intersection is called the Incenter. 

StraightedgeandCompass: Center of a Circle  Given a circle, find the center of that circle with the basic tools of a straightedge and a compass.  
StraightedgeandCompass: Incircle  With a straightedge and compass we can construct the incircle of a triangle.  
StraightedgeandCompass: A Circle from Three Points  With only a straightedge and a compass, given three points, we can construct a circle that passes through those three points.  
StraightedgeandCompass: A Tangent to a Circle  With a straightedge and a compass we can construct a tangent to a circle; that ideal tangent intersects the circle at a single point. 
Coordinate Geometry
Title DescriptionPoints, Line Segments & Circles  We differentiate points, line segments, and circles, from their graphical representations. It's an important distinction. A point has no size; it is merely an idea. 

Lines and Line Segments  How many points can two straight lines have in common? The answer depends on whether the lines are coplanar, skew, or the same line.  
Rays  Rays are directed lines, in a way. Literally, in one direction does a ray extend. 

Directive Distance  A man walks 4.5 miles due north, and then 3.625 miles due south. How far is he from the starting point? 

A Fast Lesson in Slope  A visual demonstration and lesson on slope. We use Cartesian coordinates for this demonstration. 

Lines have No Width  Lines continue infinitely far in two directions, in one dimension. A line has no width, a ray has no width, a line segment has no width. 

Euclid's Four Postulates  The concepts within are what need to be known and understood. It is not important to learn these in order or by their "titles." The embedded principles are important. 

The Straight Line  The straight line and the plane are fundamental concepts in Geometry.  
The Distance Formula and the Pythagorean Theorem  We find the distance between two points in rectangular (or Cartesian) coordinates. We also look at the familiar Pythagorean Theorem for the relationship between sides of a right triangle. 

Distance Formula as Pythagorean Relation  A basic lesson in the Distance Formula and its close cousin, the Pythagorean Relation.  
Transformations  No words, just soothing music for a soothing, nonmathematical treatment of nonmathematical ideas on a page.  
Practice with Cartesian Coordinates  Practice (x, y) coordinate location until it is secondnature to you. 
General Topics
Title DescriptionA Basic Preface to Geometry  From Betz, Webb, and Smith, a few good words about the nature of Geometry. "Our educational troubles cannot be cured by a complete break with the past, nor by ignoring the legitimate demands of our times." 

A Basic Preliminary Lesson  Geometry originated in Egypt. Geometry means "earth measurement." 

A Geometric Supplement: Algebra  How many degrees in an angle that is 12° less than it supplement? How many degrees in an angle that is 18° greater than its complement?  
A Lesson with The Protractor  Everyone needs to be able to measure accurately with a protractor.  
Acute and Obtuse Angles  An acute angle measures less than 90° while an obtuse angle measures between 90° and 180°.  
Angles Right On  Paper folding can easily make a right angle. Learn the notion of perpendicularity, that is, perpendicular rays form a 90° angle. 

Angles, Vertical  Vertical angles are congruent, that is, of the same measure. Sometimes vertical angles are termed "opposite" angles. 

Angular Truths  A summary of angular truths includes: all straight angles are equal, all right angles are equal, vertical angles are equal, the supplements of equal angles are themselves equal, and so forth.  
Basic Lesson  Angles  An angle is formed by two distinct rays that share a common endpoint. You should learn the three ways to identify an angle. 

Basic Lesson  Classification of Angles  The magnitude of an angle is the size of that angle. A straight angle measures 180°. 

Basic Lesson  Directed Distance  A man walks 4.5 miles due north, and then 3.625 miles due south. How far is he from the starting point? 

Basic Lesson  English and Metric Drawers  Drawing line segments to given lengths. This is a very basic skill everyone needs to have. 

Basic Lesson  Lines and Line Segments  How many points can two straight lines have in common? The answer depends on whether the lines are coplanar, skew, or the same line.  
Basic Lesson  Opposite Directions  Get comfortable moving in opposite directions before tackling negative values.  
Basic Lesson  Positive and Negative Angles  In looking ahead toward higher math, we observe that the direction of rotation matters for a dynamic movement of a ray. We distinguish static from dynamic situations. 

Basic Lesson  Rays  Rays are directed lines, in a way. Literally, in one direction does a ray extend. 

Basic Lesson  Solid Values  We know what "space" is, but enclosed spaces are the items of interest in Geometry. We are not concerned with the matter or material of a space, but rather its size and shape. 

Basic Lesson  The Straight Line  The straight line and the plane are fundamental concepts in Geometry.  
Classification of Triangles  Let us master the classification of triangles.  
Complements and Supplements  Two complementary angles sum to 90°. Two supplementary angles sum to 180°. 

Degrees, Minutes, Seconds  We measure angles in degrees, to be sure. Finer measurements include minutes, which are 1/60 of a degree. Even finer measurements include seconds, which are 1/60 of a minute. 

Geom. 1011: Tree of Life, Categories  Just as we compartmentalize the animal kingdom intro group with like (or similar) properties, we end up doing the same thing with polygons in geometry.  
Geom. 1012: Properties of Polygons  We label and categorize polygons according to their properties. Shared properties bear a common name, a shared label. It is good to focus on the properties of parallograms. 

Geom. 1013: Properties of Squares and Rectangles  We want to assign properties to the polygons we study in Geometry.  
Geom. 1014: Properties of Squares and Rectangles  Diagonals are congruent in both rectangles and squares. The diagonals also bisect one another in rectangles (and squares). 

Geom. 1015: Properties of Parallelograms  Congruence, parallel, and bisection is discussed with parallelograms. Diagonals bisect one another. 

Geom. 1016: Properties of Parallelograms  When two parallel lines are cut by a transversal, alternate interior angles are congruent.  
Geom. 1017: Properties of Parallelograms  The classic Mr. X "BS" presentation. Seriously. When two parallel lines are cut by a transversal, alternate interior angles are congruent, and alternate exterior angles are congruent. 

Geom. 1018: Point, Line Segment, Circle  We differentiate points, line segments, and circles, from their graphical representations. It's an important distinction. A point has no size; it is merely an idea. 

Geom. 1019: Cake, Circles, Fractions  Latch onto the relationship between reciprocals early in your study of Geometry. Reciprocals make study a piece of cake, even for Picasso. 

Geom. 1020: Circles, Angles, Degrees  An introduction to angles includes a look at hands on a clock face. Angles need to be understood early in Geometry. 

Geom. 1021: Angles on a Clock Face  More angles on a clock face help us see the values of angles as we move around the circle.  
Geom. 1022: Planes  Plain planes. Technically, no one has ever seen a mathematical plane. 

Geom. 1023: Parallel  Parallel lines are not only straight, but in the same plane (coplanar).  
Geom. 1024: Parallel Planes  Get the idea of parallel planes, mathematically, before embarking on a course in Geometry. Bookshelves and cookies help visualize the idea. 

Geom. 1025: Parallel Postulate  A basic and simple explanation of the powerful Parallel Postulate from Euclid.  
Geom. 1026: Intersecting Lines in One Plane  As we define lines in Plane Geometry, intersecting lines must lie in a single plane.  
Geom. 1027: Lines Have No Width  Lines continue infinitely far in two directions, in one dimension. A line has no width, a ray has no width, a line segment has no width. 

Geom. 1028: Finite vs. Infinite  Appreciate infinity and the infinitesimal as you delve in Geometry. "Uncountable" and "infinite" will be different ideas for us in Geometry. 

Geom. 1029: Rays, Congruence, and Vertical Angles  Rays are infinite. Angles are formed by two rays. Congruence means "of the same measure." Vertical Angles are congruent. 

Geom. 1030: Sum of Angles 180º in Triangle  A simple proof: the sum of the angles in a triangle is 180º.  
Geom. 1031: Circles  A circle is a collection of coplanar points equidistant from a center point. Every radius, for a given circle, is fixed. 

Geom. 1032: Radii, Diameter, and Circumference  We need to understand radii, diameters, and circumference. Concomitantly, we should value pi, appx. 3.14159, or expressed exactly as π. 

Geom. 1033: Euclid's Four Postulates  The concepts within are what need to be known and understood. It is not important to learn these in order or by their "titles." The embedded principles are important. 

Geom. 1034: Cartesian Coordinates  I am thinking; ergo, I exist. Practice (x, y) coordinate location until it is secondnature to you. 

StraightedgeandCompass: A 30° Angle  With just a straightedge and compass (no protractor) construct a 30degree angle.  
StraightedgeandCompass: A 45° Angle  We can construct a 45° angle with just a straightedge and a compass.  
StraightedgeandCompass: A 90° Angle  It is a straightforward task to construct a 90° angle with just a straightedge and a compass.  
StraightedgeandCompass: A Circle from Three Points  With only a straightedge and a compass, given three points, we can construct a circle that passes through those three points.  
StraightedgeandCompass: A Parallel Line  Construct a line parallel to a given line through a point not on that line, using only a straightedge and a compass.  
StraightedgeandCompass: A Parallel Line (angle copy)  Another way to draw a parallel line to a given line (other than to construct a rhombus) is to copy an angle, which is shown in this lesson.  
StraightedgeandCompass: A Perpendicular Line  We may construct a perpendicular line through a point not on a given line.  
StraightedgeandCompass: A Right Triangle  Given the length of one leg and the hypotenuse, we may construct a right triangle accordingly using a straightedge and compass.  
StraightedgeandCompass: A Tangent to a Circle  With a straightedge and a compass we can construct a tangent to a circle; that ideal tangent intersects the circle at a single point.  
StraightedgeandCompass: An Equilateral Triangle  An easy construction is to use a straightedge and compass to draw an equilateral triangle.  
StraightedgeandCompass: An Isosceles Triangle  Using only a straightedge and a compass we can construct an isosceles triangle.  
StraightedgeandCompass: Bisect an Angle  With a straightedge and a compass we can divide an angle into two equal halves, that is, bisect the angle.  
StraightedgeandCompass: Center of a Circle  Given a circle, find the center of that circle with the basic tools of a straightedge and a compass.  
StraightedgeandCompass: Divide Segment into Equal Parts  Using a straightedge and compass we can divide a line segment into equal parts.  
StraightedgeandCompass: Foci of an Ellipse  Given the graph of an ellipse we find the foci with straightedge and compass.  
StraightedgeandCompass: Incircle  With a straightedge and compass we can construct the incircle of a triangle.  
StraightedgeandCompass: Triangle Medians  Construct the medians of a triangle using just a straightedge and a compass.  
Surface Area of a Right Circular Cone  This lesson comes from a Problem Set. We want to understand the formula for the lateral surface area of a cone. 
Parallel and Perpendicular Lines
Title DescriptionIdentifying Parallel, Perpendicular, & Intersecting Lines Problem Set 1  A worksheet for beginning geometry where we identify a basic relationship between lines that are either parallel (distinct lines in the same plane that never meet), perpendicular (lines that intersect at 90 degrees, or a right angle) o...  
A Fast Lesson in Slope  A visual demonstration and lesson on slope. We use Cartesian coordinates for this demonstration. 

Intersecting Lines in One Plane  As we define lines in Plane Geometry, intersecting lines must lie in a single plane.  
Parallel  Parallel lines are not only straight, but in the same plane (coplanar). 
Pythagorean Theorem
Title DescriptionThe Classic Ladder Before and After Moving the Base  Alternate Approach  In this solution to the Ladder Before and After problem, we use a Pythagorean relation instead of a trig function to derive the answer.  
Pythagorean Relations with Irrational Lengths  We solve right triangles with various lengths of the sides. The Pythagorean relation holds, where the sum of the squares of the legs (the perpendicular sides) is equal to the square of the hypotenuse (the square of the longest side). 

Pythagorean Triples  Basic right triangles are solved with integer values for the lengths of the sides. We call these Pythagorean Triples. No irrational numbers are required. 

Distance Formula as Pythagorean Relation  A basic lesson in the Distance Formula and its close cousin, the Pythagorean Relation.  
The Distance Formula and the Pythagorean Theorem  We find the distance between two points in rectangular (or Cartesian) coordinates. We also look at the familiar Pythagorean Theorem for the relationship between sides of a right triangle. 

The Pythagorean Theorem  The Pythagorean Theorem is a pretty important thing. At some point, you should learn it. 
Quadrilaterals & Polygons
Title DescriptionTree of Life  Categories  Just as we compartmentalize the animal kingdom intro group with like (or similar) properties, we end up doing the same thing with polygons in geometry.  
Properties of Squares and Rectangles II  Diagonals are congruent in both rectangles and squares. The diagonals also bisect one another in rectangles (and squares). 

Properties of Squares and Rectangles  We concentrate on more specialized parallelograms on this lesson: squares and rectangles.  
Properties of Polygons  We label and categorize polygons according to their properties. Shared properties bear a common name, a shared label. It is good to focus on the properties of parallograms. 

Properties of Parallelograms  We return to parallelograms on this lesson. Congruence, parallel, and bisection is discussed with parallelograms. Diagonals bisect one another. 

Properties of Parallelograms III  The classic Mr. X "BS" presentation. Seriously. When two parallel lines are cut by a transversal, alternate interior angles are congruent, and alternate exterior angles are congruent. 

Properties of Parallelograms II  When two parallel lines are cut by a transversal, alternate interior angles are congruent.  
Area and Perimeter of Trapezoids  We show how to calculate area and perimeter of trapezoids in this worksheet.  
Area and Perimeter of Squares  We show how to calculate area and perimeter of squares in this worksheet.  
Area and Perimeter of Rectangles  Almost as easy as squares, we show how to calculate area and perimeter of rectangles in this worksheet.  
Area and Perimeter of Parallelograms  We show how to calculate area and perimeter of parallelograms in this worksheet. 
Transformations
Title DescriptionTransformations  No words, just soothing music for a soothing, nonmathematical treatment of nonmathematical ideas on a page. 
Triangles
Title DescriptionTriangle Facts Lesson  An overview of the Triangles Handout from MathAids.com.  
Classification of Triangles  Let us master the classification of triangles.  
Area and Perimeter of Triangles Practice Problems 5  A worksheet with isosceles triangles for which we are asked to calculate both area and perimeter.  
Area and Perimeter of Triangles Practice Problems 4  Two flavors of triangles comprise this worksheet: right triangles and equilateral triangles. We are to calculate both area and perimeter. 

Area and Perimeter of Triangles Practice Problems 3  Common triangles may be termed scalene, when no two sides or angles are congruent. The worksheet from MathAids.com asks for calculations of both area and perimeter. 

Area and Perimeter of Triangles Practice Problems 1  A worksheet with right triangles asks us to calculate both area and perimeter.  
Area and Perimeter of Triangles Practice Problems 2  A worksheet with right triangles asks us to calculate both area and perimeter.  
Equilateral Triangles College Exam Problem 2  Consider two equilateral triangles where one has an area nine times the other.  
Equilateral Triangles College Exam Problem 1  Consider two equilateral triangles where one has a perimeter three times the other.  
Triangle Inequalities of Sides Problem Set  The largest angle in a triangle is opposite the longest side; the smallest angle is opposite the smallest side.  
Sum of Angles 180° in Triangle  A simple proof: the sum of the angles in a triangle is 180°.  
Geometry_TrianglesAngleBisectorsLesson_W14  A brief lesson on Angle Bisectors in Triangles. Their point of concurrence is the center of the Inscribed Circle. 

Lesson on the Centroid and Medians of Triangles  A brief lesson on Medians in Triangles. Their point of concurrence is the Centroid of the triangle. 
Trigonometry
Title DescriptionReciprocal Trigonometric Ratios  Quickly learn the pairs of reciprocal trig functions: sine and cosecant are reciprocal trig functions; cosine and secant are reciprocal trig functions; tangent and cotangent are reciprocal trig functions.  
Cofunctions  The names of the six basic trig functions make it easy to follow: sine and cosine functions are cofunctions; tangent and cotangent functions are cofunctions; secant and cosecant functions are cofunctions.  
The Basic TRIG Functions  While we have six basic trigonometric functions, in this lesson we concentrate on "the first three," namely, sine, cosine, and tangent.  
Tangent is Just Like Slope  We introduce x, y, and r around a circle centered at the origin. the radius of the circle is r, and points around the circle are the familiar (x, y). 

Inverse TRIG Functions  Inverse TRIG functions return an angle.  
The Distance Formula and the Pythagorean Theorem  We find the distance between two points in rectangular (or Cartesian) coordinates. We also look at the familiar Pythagorean Theorem for the relationship between sides of a right triangle. 

The Classic LadderAgainsttheWall  Every trig course includes at least one problem of a ladder of fixed length resting against a vertical wall.  
Pythagorean Relations with Sines and Cosines  We solve for the sine and cosine of an acute angle within a right triangle.  
Pythagorean Relations with Irrational Lengths  We solve right triangles with various lengths of the sides. The Pythagorean relation holds, where the sum of the squares of the legs (the perpendicular sides) is equal to the square of the hypotenuse (the square of the longest side). 

The Classic Ladder Before and After Moving the Base  Just as we always have a static ladder problem in every trig course, we always have a problem where the base of the ladder is moved in a second scenario. How far does the top of the ladder rise when we move the ladder's base a fixed distance toward the wall? 

The Classic Ladder Before and After Moving the Base  Alternate Approach  In this solution to the Ladder Before and After problem, we use a Pythagorean relation instead of a trig function to derive the answer.  
Sines and Cosines of Acute Angles in Right Triangles  We practice with irrational numbers as we find the sine and cosine of acute angles within right triangles.  
Area of a Triangle  There are many ways to calculate the area of a triangle. Trigonometry helps when the altitude of the triangle is unknown. We look at Heron's (Hero's) Formula, as well as a 3x3 determinant. 