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CONGRUENCE OF SIMPLE POLYGON


C:thlbcrtzcongruence of simple polygon_filesimage001.png
The triangles above are drawn such that
CC:thlbcrtzcongruence of simple polygon_filesimage002.pngB= ZC:thlbcrtzcongruence of simple polygon_filesimage003.pngY
AC:thlbcrtzcongruence of simple polygon_filesimage004.pngC=XC:thlbcrtzcongruence of simple polygon_filesimage005.pngZ
BC:thlbcrtzcongruence of simple polygon_filesimage006.png=YC:thlbcrtzcongruence of simple polygon_filesimage007.pngX
Corresponding sides in the triangles are those sides which are opposite to the equal angles i.e.
C:thlbcrtzcongruence of simple polygon_filesimage008.png
If the corresponding sides are equal i.e.
C:thlbcrtzcongruence of simple polygon_filesimage009.png
In general, polygons are congruent if corresponding sides and corresponding angles are equal.
The symbol for congruence is C:thlbcrtzcongruence of simple polygon_filesimage010.png
Congruence of triangles
Case 1: Given three sides
Two triangles are congruent if the three pairs of corresponding sides are such that the sides in each pair are equal.
Consider the triangles below:
C:thlbcrtz__i__images__i__aa221.png
C:thlbcrtzcongruence of simple polygon_filesimage012.png
C:thlbcrtzcongruence of simple polygon_filesimage013.png
Note: SSS- is an abbreviation of side- side- side
Examples :
C:thlbcrtz__i__images__i__aa233.png
Solution

C:thlbcrtz__i__images__i__aa29.png
Construction of A is joined C
C:thlbcrtzcongruence of simple polygon_filesimage015.jpg
C:thlbcrtz__i__images__i__AA37.png
C:thlbcrtzcongruence of simple polygon_filesimage017.png
C:thlbcrtz__i__images__i__s24.png
C:thlbcrtz__i__images__i__aa332.png

C:thlbcrtz__i__images__i__A332.png
Construction; A joined to D

C:thlbcrtz__i__images__i__aa333.png
C:thlbcrtz__i__images__i__ss.png
Case 2; Given two sides and the included angle (SAS)
Two triangles are congruent if two pairs of corresponding sides are such that the sides in each pair are equal and the angles included between the given sides in each triangle are equal.
Examples
C:thlbcrtz__i__images__i__s125.png
C:thlbcrtz__i__images__i__a333.png

C:thlbcrtz__i__images__i__a1220.png
C:thlbcrtz__i__images__i__aaa10.png
C:thlbcrtzcongruence of simple polygon_filesimage025.jpg

C:thlbcrtz__i__images__i__a1316.png
Case 3; Given two angles and a corresponding side
Two triangles are congruent if two pairs of corresponding angles are such that the angles in each triangle are equal.
C:thlbcrtzcongruence of simple polygon_filesimage030.png
Example
C:thlbcrtz__i__images__i__a2211.png

Solution
C:thlbcrtz__i__images__i__a2311.png
Case 4: Given that a right angle hypotenuse and one side (RHS)
The right angled triangles are congruent if the hypotenuse and a side of one triangle are respectively equal to the hypotenuse and side of another triangle
C:thlbcrtzcongruence of simple polygon_filesimage032.png
Example:
Use the figure below to prove that
C:thlbcrtzcongruence of simple polygon_filesimage033.png
Solution
C:thlbcrtzcongruence of simple polygon_filesimage034.jpg
AC:thlbcrtzcongruence of simple polygon_filesimage035.pngC= AC:thlbcrtzcongruence of simple polygon_filesimage035.pngB -right angles
Therefore
C:thlbcrtzcongruence of simple polygon_filesimage036.jpg
Note:
R.H.S – Right angle hypotenuse side
Isosceles triangle theorem
The base angles of an isosceles triangle are equal
C:thlbcrtzcongruence of simple polygon_filesimage037.png
C:thlbcrtzcongruence of simple polygon_filesimage038.jpg
Construction:-
C:thlbcrtzcongruence of simple polygon_filesimage039.jpg
Exercise 1.
C:thlbcrtz__i__images__i__a3110.png
C:thlbcrtzcongruence of simple polygon_filesimage042.png
C:thlbcrtzcongruence of simple polygon_filesimage043.jpg
C:thlbcrtzcongruence of simple polygon_filesimage044.png
C:thlbcrtzcongruence of simple polygon_filesimage045.png
Solution
C:thlbcrtzcongruence of simple polygon_filesimage046.png
ABCD = Common line
C:thlbcrtzcongruence of simple polygon_filesimage047.png
C:thlbcrtzcongruence of simple polygon_filesimage048.png
They are alternate interior angle
C:thlbcrtzcongruence of simple polygon_filesimage049.png
C:thlbcrtzcongruence of simple polygon_filesimage050.jpg
AB=CD given
BC =AD given
C:thlbcrtzcongruence of simple polygon_filesimage051.jpg
C:thlbcrtzcongruence of simple polygon_filesimage052.png
SOLUTION
C:thlbcrtzcongruence of simple polygon_filesimage053.png
C:thlbcrtzcongruence of simple polygon_filesimage054.jpg
6. O is the center of the circle ABCD, if AC and BD and diameter of the circle and the line segments AD, AB and CB are drawn prove thatC:thlbcrtzcongruence of simple polygon_filesimage055.png
C:thlbcrtzcongruence of simple polygon_filesimage056.png

Solution
C:thlbcrtzcongruence of simple polygon_filesimage057.png
CONVERSE THE ISOSCELES TRIANGLE THEO
REM
If two angles of a triangle are equal then sides opposite those angles are equal
C:thlbcrtzcongruence of simple polygon_filesimage058.png
Given that C:thlbcrtzcongruence of simple polygon_filesimage059.pngC=C:thlbcrtzcongruence of simple polygon_filesimage060.png
Required to prove C:thlbcrtzcongruence of simple polygon_filesimage061.png= C:thlbcrtzcongruence of simple polygon_filesimage062.png
Construction A and D are joined such that
C:thlbcrtzcongruence of simple polygon_filesimage063.png
THEOREMS OF PARALLELOGRAMS
1) The opposite sides of the parallelogram are equal
C:thlbcrtzcongruence of simple polygon_filesimage064.png
Given a parallelogram ABCD
Required to prove
C:thlbcrtzcongruence of simple polygon_filesimage065.png
Construction:D is formed to B
AC:thlbcrtzcongruence of simple polygon_filesimage035.pngB= CC:thlbcrtzcongruence of simple polygon_filesimage066.pngD -is interior angles AB//DC
AC:thlbcrtzcongruence of simple polygon_filesimage066.pngD= BC:thlbcrtzcongruence of simple polygon_filesimage035.pngC -is interior angles AB//DC
C:thlbcrtzcongruence of simple polygon_filesimage067.png
Therefore
C:thlbcrtzcongruence of simple polygon_filesimage068.jpg
2. The opposite angles of the parallelogram are equal
C:thlbcrtzcongruence of simple polygon_filesimage069.png
DC:thlbcrtzcongruence of simple polygon_filesimage070.pngB= DC:thlbcrtzcongruence of simple polygon_filesimage071.pngB
AC:thlbcrtzcongruence of simple polygon_filesimage035.pngC + DC:thlbcrtzcongruence of simple polygon_filesimage070.pngB=180C:thlbcrtzcongruence of simple polygon_filesimage072.pngInterior angle of the same side of C:thlbcrtzcongruence of simple polygon_filesimage073.png// C:thlbcrtzcongruence of simple polygon_filesimage074.png
AC:thlbcrtzcongruence of simple polygon_filesimage075.pngC + DC:thlbcrtzcongruence of simple polygon_filesimage070.pngB=180º interior angles on side of C:thlbcrtzcongruence of simple polygon_filesimage076.pngC:thlbcrtzcongruence of simple polygon_filesimage077.png//C:thlbcrtzcongruence of simple polygon_filesimage078.png
Therefore
C:thlbcrtzcongruence of simple polygon_filesimage079.png
Similarly
DC:thlbcrtzcongruence of simple polygon_filesimage070.pngB + AC:thlbcrtzcongruence of simple polygon_filesimage075.pngC=180º interior angles the same side of C:thlbcrtzcongruence of simple polygon_filesimage076.pngC:thlbcrtzcongruence of simple polygon_filesimage077.png//C:thlbcrtzcongruence of simple polygon_filesimage078.png
BC:thlbcrtzcongruence of simple polygon_filesimage071.pngD + AC:thlbcrtzcongruence of simple polygon_filesimage075.pngC=180º interior angles the same side of C:thlbcrtzcongruence of simple polygon_filesimage078.png,C:thlbcrtzcongruence of simple polygon_filesimage080.png// C:thlbcrtzcongruence of simple polygon_filesimage078.png
Therefore
DC:thlbcrtzcongruence of simple polygon_filesimage070.pngB + ABC= BC:thlbcrtzcongruence of simple polygon_filesimage071.pngD + AC:thlbcrtzcongruence of simple polygon_filesimage075.pngC
DC:thlbcrtzcongruence of simple polygon_filesimage070.pngB= BC:thlbcrtzcongruence of simple polygon_filesimage071.pngD
Hence opposite angles of a parallelogram are equal.
3.The diagonals of a parallelogram bisect each other
C:thlbcrtzcongruence of simple polygon_filesimage081.png
4. The diagonals of a parallelogram intersect each other
If one pair of the opposite sides of a quadrilateral are equal and parallel then the other pair of the opposite side are equal and parallel.
C:thlbcrtzcongruence of simple polygon_filesimage082.png
C:thlbcrtz__i__images__i__aaaa5.png
Example
C:thlbcrtzcongruence of simple polygon_filesimage084.jpg
C:thlbcrtzcongruence of simple polygon_filesimage085.png
C:thlbcrtzcongruence of simple polygon_filesimage086.png
C:thlbcrtzcongruence of simple polygon_filesimage087.jpg
C:thlbcrtzcongruence of simple polygon_filesimage088.png




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EcoleBooks | MATHEMATICS O LEVEL(FORM TWO) NOTES - CONGRUENCE OF SIMPLE POLYGON

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1 Comment

  • EcoleBooks | MATHEMATICS O LEVEL(FORM TWO) NOTES - CONGRUENCE OF SIMPLE POLYGON

    DJ cholo, September 19, 2023 @ 6:41 pm Reply

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