Transcript triangle_congruency

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Two triangles are congruent if…
All 3 sides are equal
SSS
2 sides and the contained
angle are equal
SAS
1 side and the 2 adjacent
angles are equal
ASA
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© T Madas
Prove that the any point that lies on the perpendicular
bisector of a line segment is equidistant from the
endpoints of the segment
C
Let AB be a line segment and M its
midpoint
Let C be a point on the perpendicular
bisector
Two right angled triangles are formed
AM = MB
A
M
B
MC is common
RAMC = RCMB = 90°
The two triangles have two sides and the contained angle of
those sides, correspondingly equal (SAS)
Therefore the triangles are congruent
AC = CB
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© T Madas
Given that a parallelogram has four equal sides, prove that
its diagonals are perpendicular to each other.
A parallelogram with 4 equal sides is in general a rhombus
C
D
A
RBDC = RABD
as alternate angles
B
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Given that a parallelogram has four equal sides, prove that
its diagonals are perpendicular to each other.
A parallelogram with 4 equal sides is in general a rhombus
C
D
RBDC = RABD
as alternate angles
RDCA = RCAB
as alternate angles
A
B
© T Madas
Given that a parallelogram has four equal sides, prove that
its diagonals are perpendicular to each other.
A parallelogram with 4 equal sides is in general a rhombus
C
D
RBDC = RABD
as alternate angles
RDCA = RCAB
as alternate angles
0
rDCA = rCAB A S A
hence all their sides are equal
A
B
but all four sides of a rhombus are equal
thus all four triangles are congruent S S S
So RAOD = RDOC = RCOB = RAOB
Since all four add up to 360°, each must be 90°
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© T Madas
In the diagram below ABCD and DEFG are squares.
Prove that the triangles ADE and CDG are congruent.
F
E
G
A
B
D
C
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In the diagram below ABCD and DEFG are squares.
Prove that the triangles ADE and CDG are congruent.
AD = DC
F
GD = DE
E
G
A
D
=
B
+
C
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In the diagram below ABCD and DEFG are squares.
Prove that the triangles ADE and CDG are congruent.
AD = DC
F
E
G
A
B
GD = DE
R GDC = R ADE
D
=
+
R GDC = R GDA + 90°
=
+
R ADE = R GDA + 90°
C
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In the diagram below ABCD and DEFG are squares.
Prove that the triangles ADE and CDG are congruent.
AD = DC
F
E
G
A
GD = DE
R GDC = R ADE
SAS
D
ADE and CDG are congruent because
2 sides and the contained angle of
ADE are equal to 2 sides and the
contained angle of ADE.
B
C
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© T Madas
In a circle, centre O, two chords AB and CD are marked, so
that AB = CD
Prove that the chords are equidistant from the centre O
B
Need to prove OM = ON
If we prove that AOB and
COD are congruent then
their corresponding heights
OM and ON will be equal
M
O
A
C
N
D
AB = CD (given)
AO = CO (circle radii)
BO = DO (circle radii)
Triangle congruency SSS
OM = ON
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© T Madas
In the diagram below ABD and BCE are equilateral triangles.
Prove that the triangles ABE and DBC are congruent.
D
A
60°
θ
B
60°
AB = DB
BC = BE
RDBC = RABE
[ABD is equilateral]
[CBE is equilateral]
[both angles are 60° + θ ]
VABE
and VDBC
are congruent
because 2 sides and the contained
angle of VABE are equal to 2 sides
and the contained angle of VDBC.
SAS
C
E
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© T Madas