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
- Angles, Segments and Lines
- Construcing a 30° Angle + Properties of Rhombi
- Construcing a 60° Angle + Properties of 30°-60°-90° Triangles
- Constructing the Angle Bisector + Activities
- Constructing the Copy of an Angle + Angles Activities
- Constructing the Midpoint and the Perpendicular Bisector of a Segment
- Relative Positions of Two Lines in the Plane - Lesson+Exploration+Practice
- Angles Between Two Lines and a Transversal - Lesson+Exploration+Practice
- Constructing the Perpendicular Line Through a Point on the Line
- Constructing the Perpendicular Line Through a Point Not on the Line
- Triangles
- Special Points of Triangles - Lesson+Exploration+Practice
- Constructing an Isosceles Triangle with Given Side + Activities
- Constructing Equilateral Triangles + Activities
- Constructing an Inscribed Equilateral Triangle + Activities
- Constructing a triangle, given a side and two angles
- 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
- Constructing a Regular Hexagon + 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.
Construcing a 30° Angle + Properties of Rhombi
Use compass and ruler to draw on paper the construction described in the app below.
Try It Yourself...
The following app is the same as the previous one, but now includes GeoGebra tools.
Explore the entire construction in the app above, then use the GeoGebra tools to draw the segments [math]AC,CG,GD[/math] and [math]DA[/math]. [br]What geometric figure is obtained? Why?[br][br](Use the [i]Undo [/i]and [i]Redo [/i]buttons at the top right of the toolbar, or refresh the browser page to delete possible objects you have created but that are not useful or correct).
Now draw segments [math]CD[/math] and [math]AG[/math].[br]What do they represent for the polygon created previously?[br][br]
Describe the properties of the diagonals of a rhombus.
Consider the triangle [math]ACD[/math].[br]What type of triangle is it? Explain why you classified it as such.
Complete the following sentence.[br][br]Let [math]H[/math] be the point of intersection of [math]CD[/math] and [math]AG[/math].[br]Then [math]CH[/math] _________ [math]DH[/math] and the triangle [math]ACH[/math] is a right triangle with one side __________ the other, therefore the angle [math]CAH[/math] measures ___________ .
True or False?
If a statement is false, correct it to make it true, or provide a counterexample.[br][br][list=1][*]If a parallelogram has two congruent consecutive sides, therefore it is a rhombus.[/*][*]Some rhombi are rectangles.[/*][*]A quadrilateral whose diagonals are perpendicular to each other is a rhombus.[/*][*]The diagonals of a rhombus are congruent.[/*][*]A rhombus is a figure symmetrical with respect to one of its diagonals.[/*][*]A rhombus is a figure symmetrical with respect to the point of intersection of its diagonals.[/*][*]A rhombus has two pairs of congruent sides.[/*][*]The sum of all the interior angles of a rhombus is equivalent to 2 right angles.[/*][*]In every rhombus the minor diagonal is the same length as the side.[br][/*][/list]
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.