Euclidean Geometry
Collection of Euclidean Geometry worksheets.
Table of Contents
- Proofs
- The (mathematical) logic behind the scenes - Lesson+Exploration+Practice
- Counterexamples - Lesson+Exploration
- Napoleon's Theorem - Defining Hypotheses and Thesis - Lesson+Exploration
- Triangles, Squares and Unexpected Areas
- Segments, lines and angles
- Relative Positions of Two Lines in the Plane - Lesson+Exploration+Practice
- Angles Between Two Lines and a Transversal - Lesson+Exploration+Practice
- Triangles
- Special Points of Triangles - Lesson+Exploration+Practice
- Construction of a triangle, given a side and two angles - Lesson
- Pythagorean Theorem - a "proof without words"
- Golden Triangle and Spiral - Lesson
- Dissections - Haberdasher's puzzle
- Polygons
- Convex, Concave and Self-Intersecting Polygons - Lesson+Exploration+Practice
- Exploring rotational symmetry of regular polygons - Lesson+Exploration+Practice
- Rotational symmetry and lines of symmetry of a regular polygon - Lesson+Exploration+Practice
- Areas
- Triangle Area = (b ∙ h)/2 Why? Visual proof (1)
- Triangle Area = (b ∙ h)/2 Why? Visual proof (2)
- Pythagoras, semicircles and lunes - Lesson+Exploration
- Misc.
- Kaleidoscope
- Ruotiamo...
The (mathematical) logic behind the scenes - Lesson+Exploration+Practice
Material implication: ⇒
Let's consider the statement [math]p:x=3[/math], and the statement [math]q:x+1=4[/math].[br]We say that [math]p[/math] [i]implies [/i][math]q[/math], and we write [math]p\Longrightarrow q[/math] to mean that if [math]p[/math] is true, then also [math]q[/math] is true. [br](If [math]p[/math]is true, [math]q[/math] is [i]necessarily [/i]true). [br][br]The symbol [math]\Longrightarrow[/math] [i]connects [/i]a premise [math]\left(p\right)[/math] and a conclusion [math]\left(q\right)[/math] and is very used in proofs, because it's a symbolic way to show deductive reasoning.[br][br]The statement "[math]p[/math] implies [math]q[/math]" is also written "if [math]p[/math] then [math]q[/math]" or sometimes "[math]q[/math] if [math]p[/math]".[br][br]Does this sound complicated? No... let's see a few examples of implications.[br][list][*][i]If [/i]you score 68% or more in this problem, [i]then [/i]you will pass the exam.[/*][*]Your head will hurt [i]if[/i] you bang it against a wall.[/*][/list]
Exploring implications in geometry
Given a quadrilateral [i]Q[/i], use the applet below to find out the reciprocal implications between the following statements:[br][i]a[/i]: [i]Q[/i] has an obtuse angle.[br][i]b[/i]: [i]Q[/i] has three acute angles.[br][i]c[/i]: [i]Q[/i] has no right angles. [br][br](drag the orange points to explore different quadrilaterals)
Which are the implications between statements [i]a[/i], [i]b[/i] and [i]c[/i]?
Implication is confused by fake guys
Consider this example: [math]10=11\Longrightarrow10\cdot0=11\cdot0[/math][br]We started with a false premise and implied a true conclusion.[br][br]Now consider this: [math]10=11\Longrightarrow10-1=11-1[/math][br]We started - again - with a false premise, and implied a wrong conclusion.[br][br]Implication doesn't like false premises. If we start with a false premise, the conclusion obtained by implication can be anything.
Showing why things go wrong
In the example above, we had the following three statements about a quadrilateral [i]Q[/i]:[br][i]a[/i]: [i]Q[/i] has an obtuse angle.[br][i]b[/i]: [i]Q[/i] has three acute angles.[br][i]c[/i]: [i]Q[/i] has no right angles. [br][br]We can say that:[br][list][*][i]a[/i] doesn't imply [i]b [/i]because a rhombus (that is not a square) has an obtuse angle, but not 3 acute ones.[/*][*][i]a[/i] doesn't imply [i]c[/i] because a right trapezoid (that is not a rectangle) contains an obtuse angle, and two right angles.[/*][*][i]c[/i] doesn't imply [i]b[/i] because a rhombus (that is not a square) has no right angles, but doesn't have three acute angles.[/*][/list][br]We explained that implication doesn't hold using a "tool" that in mathematics is named [i]counterexample[/i].[br]Click [url=https://www.geogebra.org/m/aexepwzu]here [/url]to discover more about counterexamples.
Relative Positions of Two Lines in the Plane - Lesson+Exploration+Practice
In the app below, select the relative position that you want to explore, and change the displayed straight lines by dragging the points that define them.
Answer the following questions
There is only one line passing through three distinct points in a plane.
Given two points in a plane, there are infinitely many distinct lines passing through them.
Any two lines intersect at a point.
Special Points of Triangles - Lesson+Exploration+Practice
Explore...
Drag the vertices of the triangle and explore how the special points position changes, depending on the characteristics of the triangle (acute, obtuse, right, isosceles,...)
Build...
Use the GeoGebra tools to build the main special points of the triangle, then verify your construction by selecting the related check box.[br]
Check and measure...
Check the [i]Euler line[/i] box to explore the alignment property of orthocenter, barycenter and circumcenter.[br]Use the measuring tools of GeoGebra to verify the property [math]length\left(OG\right)=2\cdot length\left(GK\right)[/math]
Answer...
The circumcenter can be a point outside the triangle?[br]And a point on the perimeter?
The orthocenter can be a point outside the triangle?[br]And a point on the perimeter?
Convex, Concave and Self-Intersecting Polygons - Lesson+Exploration+Practice
Simple and self-intersecting polygons
In the app below you can see a [i]polygon: [/i]polygon is a word derived from Greek, and means "many angles".[br][br]A polygon can be [i]simple[/i], when its boundary does not cross itself, otherwise it's [i]self-intersecting[/i].[br][br]Drag the vertices of the polygon (green points) and create [i]simple[/i] and [i]self-intersecting[/i] figures. The intersection points will be shown in orange.[br]
Convex and Concave Polygons
When polygons are [i]simple[/i], we can characterize them further, as [i]convex [/i]or [i]concave[/i].[br][br]Explore the applet below, create a few different [i]convex [/i]and [i]concave[/i] polygons, observe the measure of the angles and the position of the diagonals with respect to the portion of the plane enclosed within its sides.[br][br]Formulate your own conjecture about how to differentiate a [i]convex [/i]and a [i]concave [/i]polygon, depending on these properties of the figure.
Convex polygons and interior angles
What can you say about the measures of the interior angles of a [i]convex [/i]polygon?[br]Do they all have a common characteristic?
Concave polygons and interior angles
What can you say about the measures of the interior angles of a [i]concave[/i] polygon?[br]Do some of them have a common characteristic?
Convex polygons, concave polygons and their diagonals
Describe the [i]difference [/i]between the [i]position[/i] of the [i]diagonals[/i] of a [i]convex [/i]and of a [i]concave [/i]polygon, with respect to the portion of the plane enclosed within its sides.
Triangle Area = (b ∙ h)/2 Why? Visual proof (1)
Let's define first the objects we will be working with.[br]We say that two figures [math]F_1[/math] and [math]F_2[/math] are [i][color=#1e84cc]equidecomposable [/color][/i]if they can be dissected into a finite number of parts, respectively congruent.[br][br]This means that if we have two figures, and we are able to cut each of them such that each cut out piece is the same for both figures, then the two of them are said to be "[i]equidecomposable[/i]".[br]The figures that you can build using all the pieces of a [url=https://www.geogebra.org/m/kyzukdsr#material/hue9x3yf]Tangram game[/url] are an example of [i]equidecomposable [/i]figures.[br][br]In the app below you will discover why, given any triangle, you can calculate its area by multiplying its base by half of its height.[br][br]Use the [color=#ff7700]slider [/color]to dissect (cut) the triangle and obtain an equivalent parallelogram.[br][br]Move the vertices [i][color=#6aa84f]A[/color][/i], [i][color=#6aa84f]B[/color][/i] and [i][color=#6aa84f]C[/color][/i] of the triangle to explore different configurations.
Kaleidoscope
Drag the red dots and the central red small circle to obtain a nice kaleidoscope effect.