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2. SSS SAS ASA Two triangles are congruent if… All 3 sides are equal
2 sides and the contained angle are equal
SAS
1 side and the 2 adjacent angles are equal
ASA
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4. 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
MC is common
RAMC = RCMB = 90° A B M
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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6. 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
RBDC = RABD
as alternate angles D C A B
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7. 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
RBDC = RABD
as alternate angles D C
RDCA = RCAB
as alternate angles A B
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8. RBDC = RABD RDCA = RCAB rDCA = rCAB
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
RBDC = RABD
as alternate angles D C
RDCA = RCAB
as alternate angles
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
So RAOD = RDOC = RCOB = RAOB
Since all four add up to 360°, each must be 90°
S S S
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9. Exam Question
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10. In the diagram below ABCD and DEFG are squares.
Prove that the triangles ADE and CDG are congruent. F E G A D B C
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11. In the diagram below ABCD and DEFG are squares.
Prove that the triangles ADE and CDG are congruent. F E G A D = + B C
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12. In the diagram below ABCD and DEFG are squares.
Prove that the triangles ADE and CDG are congruent. F E G A D = + = + B C
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13. SAS In the diagram below ABCD and DEFG are squares.
Prove that the triangles ADE and CDG are congruent. F
SAS E G A 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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14. Exam
Question
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15. Triangle congruency SSS OM = ON (given) (circle radii)
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
AB = CD
AO = CO
BO = DO
Triangle congruency SSS
OM = ON M O A
(given)
(circle radii) C D N
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17. In the diagram below ABD and BCE are equilateral triangles.
Prove that the triangles ABE and DBC are congruent. D AB = DB
[ABD is equilateral] BC = BE
[CBE is equilateral] A
RDBC =
RABE
[both angles are 60° + θ ]
60° B 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.
60°
SAS C E
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