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

Madhurig at 06:15, 25 June 2020

2020-06-25T06:15:44Z

Nancyvarkey at 17:42, 16 August 2018

2018-08-16T17:42:07Z

Madhurig at 12:51, 10 August 2018

2018-08-10T12:51:40Z

Madhurig: Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- || '''Slide Number 1 ''' '''Title slide ''' || Welcome to the spoken tutorial on '''Congruency of Triangles''' in '''GeoGe..."

2018-08-10T09:49:15Z

Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- || '''Slide Number 1 ''' '''Title slide ''' || Welcome to the spoken tutorial on '''Congruency of Triangles''' in '''GeoGe..."

New page

{|border=1
||'''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 '''e'''as '''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 the 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'''<nowiki>=</nowiki>'''DE''', '''AC'''<nowiki>=</nowiki>'''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'''

'''About Spoken Tutorial project'''
|| The video at the following link summarises the Spoken Tutorial project.

Please download and watch it.

|-
|| '''Slide Number 12'''

'''Spoken Tutorial workshops'''
|| The '''Spoken Tutorial Project '''team:

conducts workshops and gives certificates For more details, please write to us.

|-
|| '''Slide Number 13'''

'''Forum for specific questions:'''

* Do you have questions in THIS '''Spoken Tutorial'''?

* Please visit this site

* Choose the minute and second where you have the question.</nowiki>

*Explain your question briefly</nowiki>

* Someone from our team will answer them.
|| Please post your timed queries in this forum.

|-
|| '''Slide Number 14'''

'''Acknowledgement'''
|| Spoken Tutorial Project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.

|-
||
|| This is Madhuri Ganapathi from, IIT Bombay signing off.

Thankyou for watching.
|-
|}

@@ Line 105: / Line 105: @@
 |-
 || Point to the circle.
-|| A circle with centre '''D'''and radius '''a''' is drawn.
+|| A circle with centre '''D''' and radius '''a''' is drawn.
 |-
@@ Line 507: / Line 507: @@
 |-
 || Click on '''Line''' tool>>click '''E''' and '''D''''
-|| Using the Line tool, let us join points '''E''', '''D''''.
+|| Using the '''Line tool,''' let us join points '''E''', '''D''''.
 |-

@@ Line 177: / Line 177: @@
 '''Assignment'''
-|| Join the points '''D''', '''G''' and '''G''', '''E'''.
+|| Pause the video and do the following assignment on your own.
 Compare the segment lengths in the '''Algebra view'''.

@@ Line 105: / Line 105: @@
 |-
 || Point to the circle.
-|| A circle with centre''' D '''and radius '''a''' is drawn.
+|| 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.'''
+|| Using the '''Point''' tool, mark a point '''E''' on the circumference of circle '''d'''.
 |-
@@ Line 115: / Line 115: @@
 Point to the '''Algebra view'''.
-|| Using the''' Segment''' tool join points '''D '''and '''E'''.
+|| Using the''' Segment''' tool join points '''D''' and '''E'''.
 Note that, in the '''Algebra view''', segment '''DE''' is same as segment '''BC'''.
@@ Line 147: / Line 147: @@
 Point to circles '''g''' and '''e.'''
-|| Now we have three circles in the '''Graphics view.'''
+|| 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'''.
@@ Line 162: / Line 162: @@
 |-
-|| Click on Segment tool>>join '''D''', '''F''' >> join '''F''', '''E'''.
+|| Click on Segment tool >> join '''D''', '''F''' >> join '''F''', '''E'''.
 || Using the '''Segment '''tool, join the points '''D''', '''F''' and '''F''', '''E'''.
@@ Line 228: / Line 228: @@
 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.
 || In '''Show Label''' drop-down, choose '''Name & Value''' option.
@@ Line 242: / Line 241: @@
 |-
 || Point to the sides.
-|| Notice that '''AB= DF''', '''BC= DE''' and '''AC= EF'''.
+|| Notice that '''AB = DF''', '''BC  = DE''' and '''AC = EF'''.
-−
 |-
-|| Click on '''Move''' tool >> drag points A,B or C.
+|| Click on '''Move''' tool >> drag points '''A''', '''B''' or '''C'''.
 Observe the drag.
@@ Line 333: / Line 331: @@
 || Delete 45 degrees in the text box.
-Select alpha from the symbols table.
+Select '''alpha''' from the symbols table.
 Click on OK button at the bottom.
@@ Line 357: / Line 355: @@
 |-
-|| Select '''beta '''from the Symbols table.
+|| Select '''beta ''' from symbols table.
-|| Select '''beta '''from the symbols table.
+|| Select '''beta ''' from the symbols table.
-−
 |-
 || Choose '''clockwise''' radio button >> click '''OK''' button.
@@ Line 369: / Line 366: @@
 |-
-|| Point to the points '''E' '''and '''D''''
+|| Point to the points '''E' ''' and '''D''''
-|| Observe that, points '''E'''' and '''D'''' are drawn when angles '''gamma''' and '''delta '''are constructed.
+|| Observe that, points '''E'''' and '''D'''' are drawn when angles '''gamma''' and '''delta''' are constructed.
 |-
@@ Line 400: / Line 397: @@
 || Right-click on line '''g'''.
-From the sub-menu click on '''Show Object''' check-box.
+From the sub-menu, click on '''Show Object''' check-box.
 || Right-click on line '''g''' and click on '''Show Object''' check-box.
 |-
@@ Line 481: / Line 478: @@
 |-
-|| Point to angle alpha.
+|| Point to angle '''alpha'''.
 || Let us copy angle '''alpha'''(ACB) at point '''E'''.
@@ Line 497: / Line 494: @@
 |-
-|| Delete 45 degrees in the Angle text box>>select '''alpha''' from symbols table.
+|| 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.
@@ Line 509: / Line 506: @@
 |-
-|| Click on '''Line''' tool>>click E and D'
+|| Click on '''Line''' tool>>click '''E''' and '''D''''
-|| Using the Line tool, let us join points E, D'.
+|| Using the Line tool, let us join points '''E''', '''D''''.
 |-
-|| Point to the segments '''b''' and '''c.'''
+|| Point to the segments '''b''' and '''c'''.
-|| Now we need to construct two segments with lengths same as '''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 '''Segment with Given Length''' tool, click on '''D'''.
 || Click on the '''Segment with Given Length''' tool, and then click on point '''D'''.
@@ Line 561: / Line 558: @@
 || Click on '''Segment''' tool >> click on points '''E''', '''G'''
-'''D,G''' and '''D,H.'''
+'''D''', '''G''' and '''D''', '''H'''.
-|| Using the '''Segment ''' tool , click on points '''D G''', '''G, E '''and '''D, H''' to join them.
+|| 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.'''
+|| 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.'''
+|| Notice from the '''Algebra''' view that triangle '''DGE '''is matching triangle '''ABC'''.
 |-
@@ Line 580: / Line 577: @@
-'''AB, BC, AC, DG, DE''' and '''GE'''.
+'''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.'''
+And then click on the segments '''AB''', '''BC''', '''AC''', '''DG''', '''DE''' and '''GE'''.
 |-
@@ Line 592: / Line 589: @@
 Point to angles '''alpha''' and''' beta.'''
-|| Observe that '''AB = DG,'''
+|| Observe that '''AB = DG''',
-'''BC=DE, AC=GE.'''
+'''BC=DE''', '''AC=GE.'''
 This indicates that all sides are congruent
@@ Line 614: / Line 611: @@
 * Construct congruent triangles and
 * prove their congruency.
 |-
@@ Line 659: / Line 657: @@
 * Choose the minute and second where you have the question.</nowiki>
-*Explain your question briefly</nowiki>
+* Explain your question briefly</nowiki>
 * Someone from our team will answer them.