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This video describes the steps to construct an angle congruent to a given angle. To follow along, you can access the worksheet here: http://goo.gl/RMnhV. Music by http://freemusicarchive.org/music/BOPD/ under http://creativecommons.org/licenses/by/3.0/

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Angles Formed by Parallel Lines Notes (BLANK)

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Angles Formed by Parallel Lines Notes with Practice

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Special Angle Pairs Notes (BLANK)

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Special Angle Pairs Review

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Constructing a Square inside a Circle

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How to Construct a Square Inscribed in a Circle

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Constructing an Equilateral Triangle inside a Circle

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Constructing an Equilateral Triangle Inscribed in a Circle

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YouTube Video (10 min.)

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Open Reference Videos

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Constructions: The drawing of various shapes using only a pair of compasses and straightedge or ruler. No measurement of lengths or angles is allowed.

The word construction in geometry has a very specific meaning: the drawing of geometric items such as lines and circles using only compasses and straightedge or ruler. Very importantly, you are not allowed to measure angles with a protractor, or measure lengths with a ruler.

Compasses

Compasses are a drawing instrument used for drawing circles and arcs. It has two legs, one with a point and the other with a pencil or lead. You can adjust the distance between the point and the pencil and that setting will remain until you change it.

This kind of compass has nothing to do with the kind used find the north direction when you are lost. A compass used to find the north direction is usually referred to in the singular - a compass. The kind we are talking about here is usually referred to in the plural - compasses. This plural reference is similar to the way we talk about scissors - with an 's' on the end.

Straightedge

A straightedge is simply a guide for the pencil when drawing straight lines. In most cases you will use a ruler for this, since it is the most likely to be available, but you must not use the markings on the ruler during constructions. If possible, turn the ruler over so you cannot see them.

Why we learn about constructions

The Greeks formulated much of what we think of as geometry over 2000 years ago. In particular, the mathematician Euclid documented it in his book titled "Elements", which is still regarded as an authoritative geometry reference. In that work, he uses these construction techniques extensively, and so they have become a part of the geometry field of study. They also provide insight into geometric concepts and give us tools to draw things when direct measurement is not appropriate.

Why did Euclid do it this way?

Why didn't Euclid just measure things with a ruler and calculate lengths? For example, one of the basic constructions is bisecting a line (dividing it into two equal parts). Why not just measure it with a ruler and divide by two?

One theory is the the Greeks could not easily do arithmetic. They had only whole numbers, no zero, and no negative numbers. This meant they could not for example divide 5 by 2 and get 2.5, because 2.5 is not a whole number - the only kind they had. Also, their numbers did not use a positional system like ours, with units, tens , hundreds etc, but more like the Roman numerals. In short, it was quite difficult to do useful arithmetic.

So, faced with the problem of finding the midpoint of a line, it was very difficult to do the obvious - measure it and divide by two. This led to the constructions using compass and straightedge or ruler. It is also why the straightedge has no markings. It is definitely not a graduated ruler, but simply a pencil guide for making straight lines. Euclid and the Greeks solved problems graphically, by drawing shapes instead of using arithmetic.

Other constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions

(C) 2011 Copyright Math Open Reference.
All rights reserved

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Constructions Introduction. Drawing shapes with compasses and straightedge - Math Open Reference

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A brief introduction to constructions - creating various geometric objects with only compasses and straightedge or ruler. History and origins

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Math is Fun

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Geometric Constructions ... Animated!

"Construction" in Geometry means to draw shapes, angles or lines accurately.

These constructions use only compass, straightedge (i.e. ruler) and a pencil.

This is the "pure" form of geometric construction: no numbers involved!

Basics

Learn these two first, they are used a lot:

Points and Lines

Angles

And it is useful to know how to do 30°, 45° and 60° angles. We can use the angle bisector method (above) to create other angles such as 15°, etc.

Triangle Basics

Triangle Centers

Circle Basics

Circles and Tangents

Polygons

And for the "Master Class":

(Note: You can also see how to Use the Protractor, Use the Drafting Triangle and Ruler, and How to construct a Triangle With 3 Known Sides, but they are not "pure" geometric constructions.)

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Geometric Constructions ... Animated! "Construction" in Geometry means to draw shapes, angles or lines accurately. These constructions use only compass, straightedge (i.e. ruler) and a pencil. This is the "pure" form of geometric construction: no numbers involved! Learn these two first, they are used a lot: And it is useful to know how to do 30°, 45° and 60° angles.

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4 minutes

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4 minutes

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2 minutes

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Notes on how to bisect a segment and an angle

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Notes on how to bisect a segment and an angle

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Construction to Bisect Segment and Angle - MathBitsNotebook (Geo - CCSS Math)

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MathBitsNotebook Geometry CCSS Lessons and Practice is a free site for students (and teachers) studying high school level geometry under the Common Core State Standards.

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to the tune of "Twist and Shout"

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Midpoint & Distance Formula Song "Math Song Guy" "The Beatles - Twist and Shout"

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To the tune of "Three Blind Mice"

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Math Open Reference Video: Steps to Draw a line Perpendicular to a given line through a given point not on the line

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Math Open Reference Video: Steps to Draw a line Perpendicular to a given line through a given point not on the line

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How to construct (draw) a perpendicular from a line through a point - Math Open Reference

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This page shows how to construct a perpendicular to a line through an external point, using only a compass and straightedge or ruler. It works by creating a line segment on the given line, then bisecting it. A Euclidean construction.

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Math Open Reference Video: Steps to Draw a Parallel Line through a point not on the given line

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Math Open Reference Video: Steps to Draw a Parallel Line through a point not on the given line

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How to construct a parallel through a point with compass and straightedge or ruler - Math Open Reference

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This page shows how to construct a line parallel to a given line that passes through a given point with compass and straightedge or ruler. It is called the 'angle copy method' because it works by using the fact that a transverse line drawn across two parallel lines creates pairs of equal corresponding angles.

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Math Open Reference Video: Steps to Bisect an Angle

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Math Open Reference Video: Steps to Bisect an Angle

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How to bisect an angle with compass and straightedge or ruler - Math Open Reference

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How to bisect an angle with compass and straightedge or ruler. To bisect an angle means that we divide the angle into two equal (congruent) parts without actually measuring the angle. This Euclidean construction works by creating two congruent triangles. See the proof below for more on this.

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Math Open Reference Video: Steps to Copy an Angle

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Math Open Reference Video: Steps to Copy an Angle

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How to copy an angle with compass and straightedge or ruler - Math Open Reference

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Given an angle formed by two lines with a common vertex, this page shows how to construct another angle from it that has the same angle measure using a compass and straightedge or ruler. It works by creating two congruent triangles. A proof is shown below. A Euclidean construction

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Math Open Reference Video: Steps to Bisect a Segment

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Math Open Reference Video: Steps to Bisect a Segment

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How to bisect a segment with compass and straightedge or ruler - Math Open Reference

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This construction shows how to draw the perpendicular bisector of a given line segment with compass and straightedge or ruler. This both bisects the segment (divides it into two equal parts), and is perpendicular to it. Finds the midpoint of a line segmrnt. The proof shown below shows that it works by creating 4 congruent triangles.

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Math Open Reference Video: Steps to Copy a Segment

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Math Open Reference Video: Steps to Copy a Segment

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Copying a line segment with compass and straightedge or ruler - Math Open Reference

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How to copy a line segment with compass and straightedge or ruler. Given a line segment, this shows how to make another segemnt of the same length. A Euclidean construction.

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Constructing Parallel Lines through a Given Point Video

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Constructing Parallel Lines through a Given Point Video

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how to construct a parallel line - Bing video

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Steps to Constructing Parallel Lines A Line Given to a Parallel Line Drawing How to Construct a Line Segment Construct Parallel Lines Worksheet Construct a Line Parallel to Given Line How to Make a Parallel Line How to Draw Parallel Lines with Compass Making Parallel Lines with a Compass How to Make a Parallel Ruler

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How to Make Angle Bisectors - Bing video

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Constructing an Angle Bisector Angle Bisector Kahn Angle Bisector of a Triangle Perpendicular Bisector of a Triangle Copy an Angle Creating Angle Bisector How to Find the Length of an Angle Bisector Segment Bisector Construct Angle Bisector with a Slide Compass Different Types of Angles Angle Bisectors and In-Center Angle Bisector Theorem

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How to Make Angle Bisectors - Bing video

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Constructing an Angle Bisector Angle Bisector Kahn Angle Bisector of a Triangle Perpendicular Bisector of a Triangle Copy an Angle Creating Angle Bisector How to Find the Length of an Angle Bisector Segment Bisector Construct Angle Bisector with a Slide Compass Different Types of Angles Angle Bisectors and In-Center Angle Bisector Theorem

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Using a compass and straightedge to bisect an angle - Bing video

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Constructing an angle bisector using a compass and straightedge | Geometry | Khan Academy

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Writing the Equation of a line PERPENDICULAR to another line through a given point

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Writing the Equation of a line PERPENDICULAR to another line through a given point

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Equation of a Line Perpendicular to a Given Line and Passing Through a Given Point 127-4.26

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Basics of Graphing Points - 10 min Video

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Algebra Basics: Graphing On The Coordinate Plane - Math Antics

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Slope-intercept Form of a Line Video

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Slope from an Equation (Simplifying Math)

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Introduction Video of Slope - 7 min

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7:30 min

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4 min Video

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Using the Distance Formula & Midpoint Formula

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Segment Addition Postulate Including Midpoint Examples (Video)

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Segment Addition Postulate Including Midpoint Examples (Video)

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Proof Involving Segment Addition Postulate (Video)

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Proof Involving Segment Addition Postulate (Video)

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Beginning Geometry Proof Using Segment Addition Postulate

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Segment Addition and Midpoint Example Video

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Segment Addition and Midpoint Example Video

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Segment Addition Postulate Example - 8 min

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Segment Addition Postulate Example - 8 min

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Segment Addition Postulate Teaching Video 10 min

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Segment Addition Postulate Teaching Video 10 min

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Distance and Midpoint Formulas with Practice

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Distance and Midpoint Formulas with Practice

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The steps are listed at the top of the page.

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Distance and Midpoint Formulas with Practice

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Segment Addition Postulate with Midpoint Practice

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Segment Addition Postulate Example Video 3:40

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Segment Addition Postulate Example Video 3:40

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Given information about the lengths of line segments, Sal forms and solves an equation in order to find the lengths of the segments.

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Notes Part 1 (point, line, plane, ray, segment, angle)

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Notes Part 1 (point, line, plane, ray, segment, angle)

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Notes Part 2 (parallel, perpendicular, collinear, coplanar)

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Notes Part 2 (parallel, perpendicular, collinear, coplanar)