Welcome to the spoken tutorial on Congruency of Triangles in GeoGebra.
In this tutorial we will learn to, Construct congruent triangles and Prove their congruency.
Here I am using, Ubuntu Linux OS version 14.04 GeoGebra version 5.0.438.0-d.
To follow this tutorial, learner should be familiar with Geogebra interface.
For the prerequisite GeoGebra tutorials, please visit our website.
First I will explain about congruency of triangles.
Two triangles are congruent if, they are of the same size and shape.
All the corresponding sides and interior angles are congruent.
We will begin with the Side Side Side rule of congruency.
This is the definition of Side Side Side rule of congruency.
I have already opened the GeoGebra interface on my machine.
For this tutorial, I will disable the axes.
I will increase the font size to 18pt for clarity.
Now let us draw a triangle ABC.
Click on the Polygon tool and a draw a triangle ABC, as explained earlier.
We will construct another triangle exactly same as triangle ABC.
Using the Move tool, I will drag triangle ABC to the left side.
This will create some space, for the new construction.
Click on the Circle with Center and Radius tool, then click on the Graphics view.
A Circle with Center and Radius text box opens.
In the Radius text box, type a and click on the OK button at the bottom.
A circle with centre D and radius a is drawn.
Using the Point tool, mark a point E on the circumference of circle d.
Using the Segment tool join points D and E.
Note that, in the Algebra view, segment DE is same as segment BC.
Select the Circle with Center and Radius tool and click on point E.
In the Radius text box, type b and click on the OK button at the bottom.
A circle with centre E and radius b is drawn.
Click again on point D. In the Radius text box, type c and click on the OK button at the bottom.
A circle with centre D and radius c is drawn.
Now we have three circles in the Graphics view.
We will mark the intersection points of the circles g and e and circles d and e.
Click on the Intersect tool.
Click on the intersection point of circles g and e as F.
Then click on the intersection point of circles d and e as G.
Using the Segment tool, join the points D, F and F, E.
Here we are using the intersection point of circles g and e to get the required triangle.
If we use the intersection point of circles d and e, we will not get the required triangle.
Join the points D, G and G, E.
Compare the segment lengths in the Algebra view.
Now we will hide the circles to see the triangle DEF.
Right-click on circle d. A sub-menu opens.
In the sub-menu, click on Show Object check-box.
Similarly I will hide the circles e and g.
Now we will compare the sides of the triangles ABC and DEF.
In the Algebra view, under Segment right-click on a.
From the sub-menu that opens, select Object Properties.
The Preferences window opens.
Notice that a is already selected.
While holding the Ctrl key, click on b, c, f, h and i to select them.
In Show Label drop-down, choose Name & Value option.
Close the Preferences window.
Notice that AB = DF, BC = DE and AC = EF.
Using the Move tool, let us move the points A, B or C.
Note that all the lengths change accordingly, as we drag.
This proves that, triangles ABC and DEF are congruent.
Now we will learn to construct and prove Angle Side Angle rule of congruency.
This is the definition of Angle Side Angle rule of congruency.
Let us open a new GeoGebra window.
Click on File and select New Window.
I will draw a triangle using the Polygon tool.
Next we will measure two angles of the triangle.
Click on the Angle tool and click on the points C B A and A C B.
The values of the angles alpha and beta are displayed in the Algebra view.
Using the Move tool, I will drag the triangle ABC to the left side.
This will create some space to construct the congruent triangle.
Click on Segment with Given Length tool and click in the Graphics view.
Segment with Given Length text box opens.
Type Length as a and click on the OK button at the bottom.
Segment DE is drawn.
Note that the length of segment DE is the same as segment BC.
Now we will construct angles which are same as alpha and beta for the congruent triangle.
Click on the Angle with Given Size tool, click on point E and then on point D.
Angle with Given Size text box opens.
In the text box delete 45 degrees.
Select alpha from the symbols table. Click on the OK button at the bottom.
Notice that angle gamma equal to alpha is constructed at D.
Next click on point D and then on point E.
In the Angle with Given Size text box delete 45 degrees.
Select beta from the symbols table.
This time choose clockwise radio button and click on OK button.
Notice that angle delta equal to beta is constructed at E.
Observe that, points E' and D' are drawn when angles gamma and delta are constructed.
Using the Line tool, we will join the points D, E prime and E, D prime.
After using a particular tool, click on the Move tool to deactivate it.
This will prevent the drawing of unnecessary points in the Graphics view.
The lines g and h intersect at a point.
Using the Intersect tool, mark the point of intersection as F.
We will hide the lines g and h, as we need only the intersection point of the lines.
Right-click on line g and click on Show Object check-box.
Similarly hide the line h.
Using the Segment tool join D, F and F, E.
The formed triangle DEF is congruent to triangle ABC.
In the Algebra view, compare the values of lengths and angles of the triangles.
The values indicate that the angles and side are congruent.
This proves the Angle Side Angle rule of congruency.
Now let us delete all the objects. Press Ctrl+A keys to select all the objects.
Then press Delete key on the keyboard.
Now we learn to construct and prove Side Angle Side rule of congruency.
Here is the definition of Side Angle Side rule of congruency.
Using the Polygon tool, draw a triangle ABC.
Let us measure the angle A C B. Click on the Angle tool and click on the points A C B.
Let us draw the base of the congruent triangle.
Click on Segment with Given Length tool and click in the Graphics view.
In the Segment with Given Length text box, type length as a. Then click on the OK button.
Segment DE is drawn.
Let us copy angle alpha(ACB) at point E.
Click on the Angle with Given Size tool.
Click on point D and then on point E.
Angle with Given Size text box opens.
In the Angle text box, delete 45 degrees and select alpha from the symbols table.
Choose clockwise radio button and click on the OK button.
Angle beta which is same as angle alpha is constructed at point E.
Using the Line tool, let us join points E, D' prime.
Now we need to construct two segments with lengths same as b and c.
Click on the Segment with Given Length tool, and then click on point D.
Segment with Given Length text box opens.
In the Length text box type c and click on the OK button.
Segment DF with length same as AB is drawn in the horizontal direction.
Now click on the Circle with Centre through Point tool.
Click on point D and then click on point F.
A circle with centre at D and passing through F, is drawn.
Observe that circle d intersects line g at two points.
Click on the Intersect tool and click on the points of intersection.
Now we will hide circle d, line g, points D prime and F and segment h, to complete our drawing.
To hide, click on the blue dots corresponding to the objects in the Algebra view.
Using the Segment tool , click on points D G, G, E and D, H to join them.
Here we see the two triangles DGE and DHE.
Notice from the Algebra view that triangle DGE is matching triangle ABC.
Now we will compare the lengths of the sides.
Click on the Distance or Length tool. And then click on the segments AB, BC, AC, DG, DE and GE.
Observe that AB = DG, BC=DE, AC=GE.
This indicates that all sides are congruent And angle alpha is equal to angle beta.
The triangles ABC and DGE are congruent using SAS rule of congruency.
Let us summarise what we have learnt.
In this tutorial we have learnt to, Construct congruent triangles and prove their congruency.
As an assignment, Construct two triangles and prove, 1. Angle Angle Side rule of congruency 2. Hypotenuse Leg rule of congruency
Your assignments should look as follows.
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This is Madhuri Ganapathi from, IIT Bombay signing off. Thank you for watching.