GeoGebra-5.04/C2/Congruency-of-Triangles/English

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Visual Cue Narration
Slide Number 1

Title slide

Welcome to the spoken tutorial on Congruency of Triangles in GeoGebra.
Slide Number 2

Learning Objectives

In this tutorial we will learn to,
  • Construct congruent triangles and
  • Prove their congruency.
Slide Number 3

System Requirement

Here I am using,
  • Ubuntu Linux OS version 14.04
  • GeoGebra version 5.0.438.0-d.
Slide Number 4

Pre-requisites

www.spoken-tutorial.org

To follow this tutorial, learner should be familiar with Geogebra interface.

For the prerequisite GeoGebra tutorials, please visit our website.

www.spoken-tutorial.org

Slide Number 5

Congruency of Triangles

∆ABC ≅ ∆PQR.

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.

Slide Number 6

SSS rule of congruency

Two triangles are congruent if, three sides of one triangle are equal to the three corresponding sides of another triangle.

We will begin with the Side Side Side rule of congruency.

This is the definition of Side Side Side rule of congruency.

Point to the interface. I have already opened the GeoGebra interface on my machine.
Right-click on Graphics view >> from the sub-menu uncheck Axes check-box. For this tutorial, I will disable the axes.
Go to Options menu >> select Fontsize >> from the sub-menu select 18pt radio button. I will increase the font size to 18pt for clarity.
Click on Polygon tool >> click on Graphics view. Now let us draw a triangle ABC.

Click on the Polygon tool and a draw a triangle ABC, as explained earlier.

Cursor on triangle ABC. We will construct another triangle exactly same as triangle ABC.
Drag triangle ABC to left corner.

Click on Move tool >> drag 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 >> click on Graphics view. Click on the Circle with Center and Radius tool, then click on the Graphics view.
Point to the text box.

Type a in Radius text box >> click OK at the bottom.

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.

Point to the circle. A circle with centre D and radius a is drawn.
Click on Point tool >> click on circle. Using the Point tool, mark a point E on the circumference of circle d.
Click on Segment tool >> click points D and E.

Point to the Algebra view.

Using the Segment tool join points D and E.

Note that, in the Algebra view, segment DE is same as segment BC.

Click on Circle with Center and Radius tool >> Click on E. Select the Circle with Center and Radius tool and click on point E.
In the Radius text box type b >> click OK button at the bottom. In the Radius text box, type b and click on the OK button at the bottom.
Point to the circle. A circle with centre E and radius b is drawn.
Click on D.

In the Radius text box type c >> click OK button at the bottom

Click again on point D.

In the Radius text box, type c and click on the OK button at the bottom.

Point to the circle. A circle with centre D and radius c is drawn.
Point to the circles.

Point to circles g and e.

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 d and eas F. Click on the Intersect tool.

Click on the intersection point of circles g and e as F.

Click on intersection point of circles d and e as G. Then click on the intersection point of circles d and e as G.
Click on Segment tool >> join D, F >> join F, E. Using the Segment tool, join the points D, F and F, E.
Point to the circles g and e. Here we are using the intersection point of circles g and e to get the required triangle.
Point to the circles d and e. If we use the intersection point of circles d and e, we will not get the required triangle.
Slide Number 7

Assignment

Join the points D, G and G, E.

Compare the segment lengths in the Algebra view.

Point to the circles and triangle. Now we will hide the circles to see the triangle DEF.
Right-click on circle d.

Point to the sub-menu.

In the sub-menu >> click on Show Object check-box.

Right-click on circle d.

A sub-menu opens.

In the sub-menu, click on Show Object check-box.

Right-click on circle.

Point to the sub-menu.

In the sub-menu >> click on Show Object check-box.

Similarly I will hide circles e and g.
Cursor on Graphics view. Now we will compare the sides of the triangles ABC and DEF.
Point to Algebra view.

Under Segment right-click on a.

In the Algebra view, under Segment right-click on a.
From the sub-menu opens >> select Object Properties. From the sub-menu that opens, select Object Properties.
Point to Preferences window. The Preferences window opens.
Point to a.


Press Ctrl key >> click on b, c, f, h and i.

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. In Show Label drop-down, choose Name & Value option.
Click on x button to close. Close the Preferences window.
Point to the sides. Notice that AB = DF, BC = DE and AC = EF.
Click on Move tool >> drag points A, B or C.

Observe the drag.

Using the Move tool, let us move the points A, B or C.


Note that all the lengths change accordingly, as we drag.

Point to the triangles ABC and DEF. This proves that, triangles ABC and DEF are congruent.
Slide Number 8

ASA Rule of Congruency

Two triangles are congruent if,

If two angles and an included side of a triangle are equal to two corresponding angles and an included side of another triangle.

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.

Click on File >> Select New Window Let us open a new GeoGebra window.


Click on File and select New Window.

Click on Polygon tool >> click on Graphics view. I will draw a triangle using the Polygon tool.
Cursor on triangle ABC.

Click on the Angle tool and click on C B A and A C B.

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.

Point to the Algebra view. The values of the angles alpha and beta are displayed in the Algebra view.
Drag triangle ABC to a corner. Using the Move tool, I will drag triangle ABC to the left side.

This will create some space to construct the congruent triangle.

Click on Segment with Given Length tool >> click in Graphics view. Click on Segment with Given Length tool and click in the Graphics view.
Point to the text box. Segment with Given Length text box opens.
Type Length as a >> click OK button at the bottom. Type Length as a and click on the OK button at the bottom.
Point to segment DE Segment DE is drawn.
Point to the lengths in Algebra view. Notice that the length of segment DE is the same as segment BC.
Point to alpha and beta. Now we will construct angles which are same as alpha and beta for the congruent triangle.
Click on Angle with Given Size tool >> click on point E >> click on D. Click on the Angle with Given Size tool, click on point E and then on point D.
Point to the text box. Angle with Given Size text box opens.
Delete 45 degrees in the text box.

Select alpha from the symbols table.

Click on OK button at the bottom.

In the text box delete 45 degrees.

Select alpha from the symbols table.

Click on the OK button at the bottom.

Point to the values in Graphics view and Algebra View. Notice that angle gamma equal to alpha is constructed at D.
Click on point D >> point E Next click on point D and then on point E.
Point to the text box.

Delete 45 degrees.

In the Angle with Given Size text box delete 45 degrees.
Select beta from symbols table. Select beta from the symbols table.
Choose clockwise radio button >> click OK button. This time choose clockwise radio button and click on OK button.
Point to the values in Graphics view and Algebra View. Notice that angle delta equal to beta is constructed at E.
Point to the points E' and D' Observe that, points E' and D' are drawn when angles gamma and delta are constructed.
Click on Line tool >> join points D, E' and E, D'. Using the Line tool, we will join the points D, E'(E prime) and E, D'(D prime).
Click on Move tool to deactivate the tool.

Point to Graphics view.

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.

Point to the intersection point.


Click on Intersect tool >> click on point of intersection point.

The lines g and h intersect at a point.

Using the Intersect tool, mark the point of intersection as F.

Point to the lines h and j. We will hide the lines g and h, as we need only the intersection point of the lines.
Right-click on line g.

From the sub-menu, click on Show Object check-box.

Right-click on line g and click on Show Object check-box.
Right-click on line h.

From the sub-menu click on Show Object check-box.

Similarly hide the line h.
Click on Segment tool >> join points D, F >> join points F, E.

Point to the triangles ABC and DEF.

Now using the Segment tool join D, F and F, E.

The formed triangle DEF is congruent to triangle ABC.

Point to the values of angles and lengths in Algebra view. In the Algebra view, compare the values of lengths and angles of the triangles.
Point to the values. The values indicate that the angles and side are congruent.

This proves the Angle Side Angle rule of congruency.

Press Ctrl+A keys to select all the objects.


Press Delete key on the Keyboard.

Now let us delete all the objects.

Press Ctrl+A keys to select all the objects.

Then press Delete key on the keyboard.

Slide Number 9

SAS rule of Congruency

Two triangles are congruent if,

two sides and an included angle of a triangle are equal to corresponding two sides and an included angle of another triangle.

Now we learn to construct and prove Side Angle Side rule of congruency.


Here is the definition of Side Angle Side rule of congruency.

Click on Polygon tool >> draw a triangle ABC. Using the Polygon tool, draw a triangle ABC.
Point to ACB.

Click on Angle tool >>click on the points A C B'.

Let us measure the angle A C B.

Click on the Angle tool and click on the points A C B.

Point segment BC. Let us draw the base of the congruent triangle.
Click on Segment with Given Length tool >> click on Graphics view. Click on Segment with Given Length tool and click in the Graphics view.
Point to the text box.

Type length as a >> click OK button.

Point to segment DE.

In the Segment with Given Length text box, type length as a.

Then click on the OK button.

Segment DE is drawn.

Point to angle alpha. Let us copy angle alpha(ACB) at point E.
Click on Angle with Given Size tool.

Click on point D >> on point E.

Click on the Angle with Given Size tool.

Click on point D then on point E.

Point to the text box. Angle Given Size text box opens.
Delete 45 degrees in the Angle text box >> select alpha from symbols table. In the Angle text box, delete 45 degrees and select alpha from the symbols table.
Choose clockwise radio button >> click OK. Choose clockwise radio button and click on the OK button.
Point to angle beta. Angle beta which is same as angle alpha is constructed at point E.
Click on Line tool>>click E and D' Using the Line tool, let us join points E, D'.
Point to the segments b and c. Now we need to construct two segments with lengths same as b and c.
click on Segment with Given Length tool, click on D. Click on the Segment with Given Length tool, and then click on point D.
Point to the text box. Segment with Given Length text box opens.
Type c as length >> click OK In the Length text box type c and click on the OK button.
Point to segment DF. Segment DF with length same as AB is drawn in the horizontal direction.
Click on Circle with Centre through Point tool >> click on D >> on F. Now click on the Circle with Centre through Point tool.

Click on point D and then click on point F.

Point to the circle. A circle with centre at D and passing through F, is drawn.
Cursor on the intersection points. Observe that circle d intersects line g at two points.
Click on Intersect tool >> click on points of intersection. Click on the Intersect tool and click on the points of intersection.
Point to all the objects. Now we will hide circle d, line g, points D' and F and segment h, to complete our drawing.
Click the blue dots in Algebra view. To hide, click on the blue dots corresponding to the objects in Algebra view.
Click on Segment tool >> click on points E, G

D, G and D, H.

Using the Segment tool , click on points D G, G, E and D, H to join them.
Point to the triangles. Here we see the two triangles DGE and DHE.
Point to the triangles. Notice from the Algebra view that triangle DGE is matching triangle ABC.
Point to the lengths. Now we will compare the lengths of the sides of these triangles.
Click on Distance or Length tool, click on the segments,


AB, BC, AC, DG, DE and GE.

Click on the Distance or Length tool.


And then click on the segments AB, BC, AC, DG, DE and GE.

Point to AB = DG,

BC=DE, AC=GE.

Point to angles alpha and beta.

Observe that AB = DG,

BC=DE, AC=GE.

This indicates that all sides are congruent

And angle alpha is equal to angle beta.

Point to the triangles ABC and DGE. The triangles ABC and DGE are congruent using SAS rule of congruency.
Let us summarise what we have learnt.
Slide Number 10

Summary

In this tutorial we have learnt to,
  • Construct congruent triangles and
  • prove their congruency.
Slide Number 9


Assignment

As an assignment,

Construct two triangles and prove,

1. Angle Angle Side rule of congruency

2. Hypotenuse Leg rule of congruency

Show the assignments. Your assignments should look as follows.
Slide Number 11

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Slide Number 12

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Slide Number 13

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Slide Number 14

Acknowledgement

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This is Madhuri Ganapathi from, IIT Bombay signing off.

Thankyou for watching.

Contributors and Content Editors

Madhurig, Nancyvarkey