Download Congruent Triangles - Mr. Garrett's Learning Center

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To Prove that AG is the Angle Bisector of CAB
To Prove that CD bisects AB at M.
Congruent Triangles
Remember: Two shapes are congruent if all sides and angles in one, are
equal to the sides and angles in the other.
Congruent triangles are of particular importance in mathematics because
they enable us to determine/prove many geometrical properties/theorems.
In his book “The Elements” Euclid
proved four theorems concerning the
conditions under which triangles are
guaranteed to be congruent. He used
some of these theorems to help
establish proofs of other important
theorems such as the Theorem of
Pythagoras and the bisection of a
chord.
Euclid of Alexandria
O
The Windmill
Conditions for Congruency of Triangles
Three sides equal. (SSS)
Two sides and the included angle equal. (SAS)
Two angles and a corresponding side equal . (AAS)
Right angle, hypotenuse and side (RHS)
SSS
SAS
Decide which of the triangles
are congruent to the yellow
triangle, giving reasons.
10 cm
SAS
1

8 cm
25o
35o
35o
120o

SSS
2

120o
3

4 cm
10 cm
10 cm
4 cm
AAS
RHS
10 cm
8 cm
4 cm
AAS
4
120o
8 cm
8 cm
35o
4 cm
120o
SAS
5
Not to Scale!

25o
8 cm
35o

120o
6
SSS
SAS
AAS
Decide which of the triangles
are congruent to the yellow
triangle, giving reasons.
1
2
5 cm

13 cm
13 cm
13 cm

4
5 cm
20o
20o
5
3
70o
70o
AAS

12 cm

12 cm
RHS
12 cm
SSS
RHS
20o
13 cm
RHS
13 cm
5 cm
Not to Scale!

13 cm
70o
13 cm
SAS

5 cm
6
Proving Relationships
A
In the diagram AB is parallel to
DC and M is the midpoint of DB.
Prove that AM = MC
D
Angle ABM = angle CDM (Alternate angles)
Angle BAM = angle DCM (Alternate angles)
Triangles ABM and CDM are congruent (AAS)
 AM = MC
It would be wrong in this example to say
that triangles ABM and DCM are congruent.
B
M
C
The order of the lettering is
important when naming congruent
triangles. Corresponding sides are
identified by ordered letter pairs.
AB CD
AM CM
BM DM
Proving Relationships
Q
P
In the diagram, PQRS is a
quadrilateral with opposite
sides parallel.
Prove that PQ = SR
and that
S
PS = QR
Angle PRS = angle RPQ (Alternate angles)
Angle PRQ = angle RPS (Alternate angles)
Triangles PQR and RSP are congruent (AAS)
 PQ = SR and PS = QR
R
Proving Relationships
In the diagram, TP is a line
perpendicular to the chord SU
that passes through the
centre of the circle at O.
P
O
Prove that the chord SU is
bisected by line TP.
S
T
U
In triangles OST and OUT, OS = OU (radii of the same circle)
Also, OT is common to both triangles
Angle OTS = angle OTU (angles on a straight line)
Triangles OST and OUT are congruent (RHS)
 ST = TU
Draw the locus
of the point that remains
Perp
Bisect
equidistant from points A and B.
A
B
Congruent triangles can be used
to prove results from some of our
earlier work on loci.
An example of this would be
proving the construction of a line
bisector.
1. Join both points with a straight line.
2. Place compass at A, set over halfway and
draw two arcs.
3. Place compass at B, with same distance set
and draw two arcs to intersect first two.
4. Draw the perpendicular bisector through the
points of intersection.
To prove that CD bisects AB at M.
Perp Bisec Proof
C
A
M
B
D
Arcs lie on the circumference of circles of equal radii.
AC = AD = BC = BD (radii of the same circle).
Triangles ACD and BCD are congruent with CD common to both (SSS).
So Angle ACD = BCD
Triangles CAM and CBM are congruent (SAS) Therefore AM = BM QED
Angle Bisect
Draw the locus of the point that remains
equidistant from lines AC and AB.
Congruent triangles can be used
to prove results from some of our
earlier work on loci.
C
Another example of this would
be proving the construction of an
angular bisector.
A
B
1. Place compass at A and draw an arc
crossing both arms.
2. Place compass on each intersection
and set at a fixed distance. Then draw two
arcs that intersect.
3. Draw straight line from A through
point of intersection for angle bisector.
To prove that AG is the Angle Bisector of CAB
Ang Bisect Proof
C
D
G
F
A
E
B
AD = AE (radii of the same circle)
DG = EG (both equal to radius of circle DE)
Triangle ADG is congruent to AEG (AG common to both) SSS
So angle EAG = DAG
Therefore AG is the angle bisector of CAB
QED
SSS
SAS
Decide which of the triangles
are congruent to the yellow
triangle, giving reasons.
10 cm
AAS
RHS
4 cm
8 cm
1
25o
2
10 cm
8 cm
4 cm
35o
35o
120o
10 cm
10 cm
4 cm
120o
120o
3
4
8 cm
8 cm
35o
4 cm
120o
25o
8 cm
5
Not to Scale!
6
Worksheets
120
35o
o
SSS
SAS
AAS
Decide which of the triangles
are congruent to the yellow
triangle, giving reasons.
1
RHS
2
12 cm
12 cm
13 cm
13 cm
13 cm
4
5 cm
5 cm
70o
20o
13 cm
12 cm
20o
3
5
20o
13 cm
70o
13 cm
5 cm
13 cm
5 cm
70o
Not to Scale!
6
Proving Relationships
To Prove that CD bisects AB at M.
C
A
M
B
D
To Prove that AG is the Angle Bisector of CAB
C
D
G
F
A
E
B