Overview

Features and Tools Examples Activities

Lines and Angles

Triangles & Polygons 

Circles

More Activities

Angle bisector

Division of segment

Parallel lines

Perpendicular bisector

Building equal angles

Angles between lines

Area of triangle

Angles of a triangle

Triangles and circles

Equilateral triangle

Isosceles trapezoid

Polygons 

Inscribed angles

Inscribed right angles

Chords

Tangent and chord

Circles and lines

Common tangents

Symmetry to y-axis

Translation

Reflection

Rotation

Proportional circles

Proportional pentagons 

The following are some examples generated from Pythagoras and its Graphic window (Arena).
When using Pythagoras software, it is possible to present dynamic examples and demonstrations, as well as to experience or explore mathematical concepts and functions.

Angle Bisector

Construction 
1. Using the Compass button draw an arc centered  at A and create intersection points F and G.
2. With F and G as centers, draw arcs with equal radii that intersect at J.
3. Draw a ray through A and J. This is the desired angle bisector.

Verification and proof
A. Verify the above by measuring the angles with the compass.
B. Prove the ray AJ is an angle bisector, using logical geometric arguments.

 

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Dividing a Segment into Equal Parts

 

 

Construction
1. Draw a ray with A as center. With the compass, mark off equal segments starting at A: AE=EF=FH=HC.
2. Draw CB. Using the parallel button draw parallels to CB through points E,F,H.
The intersection points of the parallels with AB, divide AB into four equal parts.

Verification and proof
A. Verify the above by measuring the segments.
B. Prove the segments are equal using logical geometric arguments.

 


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Parallel to a Given Line

 

 

Construction
1. With C as center draw an arc that intersects the line at points E and F.
2. Draw lines through CE and CF. With C as center draw an arc which intersects CE at I and CF at K. With I and K as centers draw equal arches that intersect at L.
3. Draw a line through C and L. This is the desired parallel.

Verification and proof
A. Verify the above by measuring the appropriate angles.
B. Prove the lines are parallel using logical geometric arguments.

 

 


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Perpendicular Bisector to a Given Segment

 

 

Construction
1. Using the compass button, draw an arc with E as center that intersects AB at points G and F.
2. With the intersection points G and F as centers, draw equal arcs that intersect at the other side of AB at point J. Connect E to J. This is the desired perpendicular.

Verification and proof
A. Verify the above by measuring the angles EDF and EDG.
B. Prove CJ is a perpendicular using logical geometric arguments.


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Constructing an Angle that is Equal to a Given One

 

 

Construction
1. With A as center draw an arc that intersects the angle sides at points E and F.
2.
Draw ray GH.
3.
With G as center draw an arc with the same radius as before that intersects GH at point J.
4.
With J as center draw an arc with radius EF intersecting the previous arc at point K.
Draw GK. Angle HGK will equal angle BAC.

Verification and proof
A. Verify the above by measuring both angles.
B. Prove the angles are equal using logical geometric arguments.


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Angles Between Lines

Construction
1. Draw line AB and then line CD parallel to AB. Draw transversal EF.
Measure and mark all the formed angles. Find all vertical angles, supplementary angles and corresponding angles.
2. Explore connections between the angles when the line EF is moved, by dragging points E or F.
3.
Explore connections between the angles when the line AB is moved, by dragging point B so that the lines are not parallel.

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