## 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 Sample Lesson 1
Geometry Sample Lesson 2
Geometry Sample Lesson 3

# Geometry Curriculum

## Angles

Title Description
 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. 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 integer-value 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 Description
 Area 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 Description
 Circles A circle is a collection of co-planar 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 Description
 Straightedge-and-Compass: 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. Straightedge-and-Compass: 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. Straightedge-and-Compass: 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 Compass-and-Straightedge 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. Straightedge-and-Compass: A 30°° Angle With just a straightedge and compass (no protractor) construct a 30-degree angle. Straightedge-and-Compass: Bisect an Angle With a straightedge and a compass we can divide an angle into two equal halves, that is, bisect the angle. Straightedge-and-Compass: A 90° Angle We can construct a 90° angle with just a straightedge and a compass. Straightedge-and-Compass: A 45° Angle We can construct a 45° angle with just a straightedge and a compass. Straightedge-and-Compass: 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. Straightedge-and-Compass: An Equilateral Triangle An easy construction is to use a straightedge and compass to draw an equilateral triangle. Straightedge-and-Compass: 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 Compass-and-Straightedge 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 Compass-and-Straightedge 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. Straightedge-and-Compass: Center of a Circle Given a circle, find the center of that circle with the basic tools of a straightedge and a compass. Straightedge-and-Compass: Incircle With a straightedge and compass we can construct the incircle of a triangle. Straightedge-and-Compass: 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. Straightedge-and-Compass: 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 Description
 Points, 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, non-mathematical treatment of non-mathematical ideas on a page. Practice with Cartesian Coordinates Practice (x, y) coordinate location until it is second-nature to you.

## General Topics

Title Description
 A 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 "B-S" 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 second-nature to you. Straightedge-and-Compass: A 30° Angle With just a straightedge and compass (no protractor) construct a 30-degree angle. Straightedge-and-Compass: A 45° Angle We can construct a 45° angle with just a straightedge and a compass. Straightedge-and-Compass: A 90° Angle It is a straightforward task to construct a 90° angle with just a straightedge and a compass. Straightedge-and-Compass: 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. Straightedge-and-Compass: 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. Straightedge-and-Compass: 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. Straightedge-and-Compass: A Perpendicular Line We may construct a perpendicular line through a point not on a given line. Straightedge-and-Compass: A Right Triangle Given the length of one leg and the hypotenuse, we may construct a right triangle accordingly using a straightedge and compass. Straightedge-and-Compass: 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. Straightedge-and-Compass: An Equilateral Triangle An easy construction is to use a straightedge and compass to draw an equilateral triangle. Straightedge-and-Compass: An Isosceles Triangle Using only a straightedge and a compass we can construct an isosceles triangle. Straightedge-and-Compass: Bisect an Angle With a straightedge and a compass we can divide an angle into two equal halves, that is, bisect the angle. Straightedge-and-Compass: Center of a Circle Given a circle, find the center of that circle with the basic tools of a straightedge and a compass. Straightedge-and-Compass: Divide Segment into Equal Parts Using a straightedge and compass we can divide a line segment into equal parts. Straightedge-and-Compass: Foci of an Ellipse Given the graph of an ellipse we find the foci with straightedge and compass. Straightedge-and-Compass: Incircle With a straightedge and compass we can construct the incircle of a triangle. Straightedge-and-Compass: 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 Description
 Identifying 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 Description
 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. 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.

Title Description
 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. 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 "B-S" 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 Description
 Transformations No words, just soothing music for a soothing, non-mathematical treatment of non-mathematical ideas on a page.

## Triangles

Title Description
 Triangle Facts Lesson An overview of the Triangles Handout from Math-Aids.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 Math-Aids.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_Triangles-Angle-Bisectors-Lesson_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 Description