a
Middle
School
Grade 6
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Ressource
Out Middle
High
School
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Grade 12 - Exp.
Ressource
Out High
4
3
2
1
Outside the middle school program
/ Inscribed and center angles
19 exercises (including 17 corrected)
a
Angles and trigonometry
(7 exercices)
Angles and isosceles triangles
(3 exercices)
Angles and right-angled triangles
(1 exercice)
Angles and thales
(2 exercices)
Small and large bows
(2 exercices)
A little of everything
(2 exercices)
O
A
B
C
D
C
60
o
A
H
M
T
C
A
B
M
O
C
A
B
C
H
O
E
C
A
B
C
O
H
C
O
C
A
B
H
37
o
C
1.
A
ngles
and
trigonometry
E.801
Let
C
b
e
a
circle
of
cen
ter
O
and
radius
5
cm
.
P
oin
ts
A
,
O
,
C
are
aligned.
D
is
a
p
oin
t
of
C
suc
h
that
B
D
C
=
60
o
.
Determine
the
measure
of
seg-
men
t
[
B
C
]
rounded
to
the
nearest
millimeter.
E.2493
On
the
figur
e
opp
osite
the
me
asur
es
ar
e
not
r
esp
e
cte
d.
Consider
a
circle
C
of
di-
ameter
H
A
=
9
cm
.
Let
M
b
e
a
p
oin
t
on
the
circle
C
suc
h
that
M
A
=
5.3
cm
and
T
another
p
oin
t
on
the
cir-
cle
C
.
1
Justify
that
M
AH
is
a
righ
t-angled
triangle.
2
Calculate
the
measure,
rounded
to
the
nearest
unit,
of
the
angle
M
H
A
.
3
Determine
the
measure,
rounded
to
the
nearest
unit,
of
the
angle
H
T
M
.
E.2647
The
figur
e
opp
osite
is
not
ful
l-
sc
ale.
It
is
not
r
e
quir
e
d
to
b
e
r
e-
pr
o
duc
e
d.
C
is
a
circle
of
cen
ter
O
and
diameter
[
AB
]
suc
h
that
AB
=
6
cm
.
M
is
a
p
oin
t
on
the
circle
suc
h
that
B
M
=
4.8
cm
.
1
Demonstrate
that
the
tri-
angle
AB
M
is
righ
t-angled
at
M
.
2
Calculate
the
measure
of
the
angle
AB
M
,
rounded
to
the
degree.
3
Deduce
the
measure
of
the
angle
AO
M
,
rounded
to
the
degree.
E.4051
The
figur
e
b
elow
is
not
to
b
e
r
e-
p
e
ate
d
on
the
c
opy.
It
is
not
given
in
ful
l
size.
A
,
B
and
C
are
three
p
oin
ts
on
a
circle
C
with
cen
ter
O
.
W
e
kno
w
that
AB
=
3
cm
.
The
heigh
t
AH
measures
2.5
cm
.
The
diameter
[
AE
]
is
plotted.
1
What
is
the
nature
of
the
triangle
AC
E
?
Justify
the
answ
er.
2
Explain
wh
y
the
angles
AB
C
and
AE
C
are
equal.
3
Using
the
triangle
AB
H
,
calculate
the
exact
v
alue
of
sin
AB
H
and
deduce
the
measure
of
the
angle
AE
C
rounded
to
the
degree.
E.798
The
figure
opp
osite
is
not
dra
wn
to
actual
dimensions.
The
circle
C
has
as
its
cen
ter
the
p
oin
t
O
and
its
radius
is
5
cm
.
The
triangle
AB
C
is
in-
scrib
ed
in
the
circle
C
and
suc
h
that
B
O
C
=
80
o
.
Note
H
the
p
oin
t
of
in
tersec-
tion
of
the
straigh
t
line
(
AO
)
with
the
circle
C
.
1
Sho
w
that
the
straigh
t
line
(
AO
)
is
the
p
erp
endicular
bisector
of
the
segmen
t
[
B
C
]
.
2
Calculate
the
v
alue
of
the
angle
O
AB
.
Justify
.
3
What
is
the
nature
of
the
triangle
AB
H
?
Justify
.
4
Calculate
the
length
of
[
B
H
]
.
E.799
The
figure
opp
osite
do
es
not
sho
w
the
actual
dimensions.
Let
C
b
e
a
circle
of
cen
ter
O
and
radius
5
k
m
.
A
,
B
and
C
are
three
p
oin
ts
on
the
cir-
cle
C
suc
h
that
B
AC
=
37
o
.
H
is
the
fo
ot
of
the
heigh
t
from
O
in
the
triangle
AB
C
.
1
a
Determine
the
measure
of
the
angle
B
O
C
.
b
Calculate
the
v
alue
of
the
angle
O
B
C
.
2
Calculate
the
length
of
segmen
t
[
B
H
]
rounded
to
the
nearest
decameter.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/801
sacados/801
O
A
B
C
D
C
60
o
chapExoCorrec/2493
sacados/2493
A
H
M
T
C
chapExoCorrec/2647
sacados/2647
Brevet 2007
A
B
M
O
C
chapExoCorrec/4051
sacados/4051
Reims Septembre 2002
A
B
C
H
O
E
C
chapExoCorrec/798
sacados/798
A
B
C
O
H
C
chapExoCorrec/799
sacados/799
O
C
A
B
H
37
o
C
C
A
B
C
O
H
E
O
C
N
B
A
M
80
o
25
o
C
O
M
B
A
I
20
o
C
O
I
J
K
C
E.5406
The
figure
b
elo
w
is
not
to
b
e
rep
eated
on
the
cop
y
.
It
is
not
giv
en
in
full
size.
A
,
B
and
C
are
three
p
oin
ts
on
a
circle
C
(se
e
fig-
ur
e)
.
W
e
kno
w
that
AB
=
3
cm
,
the
heigh
t
[
AH
]
measures
2.5
cm
.
W
e
trace
the
diameter
[
AE
]
.
1
What
is
the
nature
of
the
triangle
AC
E
?
Justify
the
answ
er.
2
Explain
wh
y
the
angles
AB
C
and
AE
C
are
equal.
3
Using
the
triangle
AB
H
,
calculate
the
exact
v
alue
of
sin
AB
H
and
deduce
the
measure
of
the
angle
AE
C
rounded
to
the
nearest
degree.
2.
A
ngles
and
isosc
eles
triangles
E.800
The
p
oin
ts
A
,
B
,
C
,
M
and
N
b
elong
to
the
cir-
cle
C
of
cen
ter
O
v
erify-
ing
the
measures
:
AO
B
=80
o
;
B
N
C
=25
o
1
Calculate
the
mea-
sure
of
AM
B
.
2
Calculate
the
mea-
sure
of
eac
h
angle
of
triangle
B
O
C
.
3
Deduce
the
measure
of
eac
h
of
the
angles
AO
C
and
AM
C
.
E.802
Let
C
b
e
a
circle
of
cen
ter
O
.
A
,
B
,
M
three
p
oin
ts
of
the
circle
suc
h
that
AB
M
=
20
o
.
Let
I
b
e
the
middle
of
seg-
men
t
[
AB
]
.
Calculate
the
v
alue
of
the
an-
gle
AO
B
.
Justify
.
E.5405
The
figure
opp
osite
is
not
full-
scale
;
w
e
do
not
ask
y
ou
to
re-
pro
duce
it.
Consider
a
circle
C
with
cen
ter
O
and
diameter
8
cm
.
I
and
J
are
t
w
o
diametrically
opp
osed
p
oin
ts.
K
is
a
p
oin
t
of
C
suc
h
that
J
K
=
4
cm
.
1
a
Sp
ecify
the
nature
of
the
triangle
O
J
K
.
Justify
.
b
Deduce
the
measure
of
the
angle
J
I
K
.
Justify
y
our
answ
er.
2
a
Sp
ecify
the
nature
of
the
triangle
I
J
K
.
Justify
.
b
Giv
e
the
measuremen
t,
to
the
nearest
millimetre,
of
segmen
t
[
I
K
]
.
3
W
e
call
R
the
image
of
K
b
y
symmetry
of
axis
(
I
J
)
.
Demonstrate
that
the
quadrilateral
R
O
K
J
is
a
rhom
bus.
3.
A
ngles
and
right-angle
d
triangles
E.4338
Consider
the
figure
b
elo
w,
whic
h
is
not
full-scale.
It
is
not
necessary
to
redo
the
figure.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5406
sacados/5406
Brevet de Reims Septembre 2002
C
A
B
C
O
H
E
chapExoCorrec/800
sacados/800
O
C
N
B
A
M
80
o
25
o
C
chapExoCorrec/802
sacados/802
O
M
B
A
I
20
o
C
chapExoCorrec/5405
sacados/5405
O
I
J
K
C
chapExoCorrec/4338
sacados/4338
Pondichery Avril 2011
A
B
D
O
M
C
75
o
A
B
C
O
H
C
(
d
A
B
D
E
C
B
D
A
C
C
AB
D
is
an
isosceles
tri-
angle
at
A
suc
h
that
:
AB
D
=
75
o
;
C
is
the
circumscrib
ed
circle
of
the
triangle
AB
D
;
O
is
the
cen
ter
of
the
cir-
cle
C
;
[
B
M
]
is
a
diameter
of
C
.
1
What
is
the
nature
of
the
triangle
B
M
D
?
Justify
the
answ
er.
2
a
Calculate
the
measure
of
the
angle
B
AD
.
b
Name
an
inscrib
ed
angle
that
in
tercepts
the
same
arc
of
the
circle
C
as
the
angle
B
M
D
.
c
Justify
that
the
angle
B
M
D
measures
30
o
.
3
W
e
giv
e
:
B
D
=
5.6
cm
;
B
M
=
11.2
cm
Calculate
D
M
.
Round
the
result
to
the
nearest
ten
th.
4.
A
ngles
and
thales
E.805
The
figure
opp
osite
is
not
made
to
actual
dimensions.
Let
C
b
e
a
circle
of
cen
ter
O
radius
6
cm
.
P
oin
ts
A
and
C
are
diametri-
cally
opp
osed.
B
is
a
p
oin
t
on
the
circle
suc
h
that
B
C
=
3
cm
.
The
segmen
t
[
B
C
]
measures
3
cm
in
length.
The
line
(
d
)
is
p
erp
endicular
to
(
B
C
)
passing
through
the
p
oin
t
O
.
It
in
tersects
segmen
t
[
B
C
]
at
H
.
1
What
is
the
nature
of
the
triangle
AB
C
?
Justify
.
2
Determine
the
measure
of
the
angle
B
AC
.
3
a
Determine
the
measure
of
the
angle
B
O
C
.
Justify
b
What
is
the
nature
of
the
triangle
O
B
C
?
Justify
c
Deduce
the
measure
of
the
angle
B
O
H
d
Justify
the
fact
that
:
B
H
=
1.5
cm
.
4
Calculate
the
v
alue
of
O
H
to
the
nearest
millimetre.
E.3928
On
the
figure
opp
osite,
whic
h
is
not
full-scale,
w
e
kno
w
that
:
(
C
)
est
a
circle
with
cen
ter
E
don
t
the
diameter
[
AD
]
mesure
9
cm
.
B
is
a
p
oin
t
of
circle
C
suc
h
that
:
AE
B
=
46
o
1
Mak
e
the
figure
resp
ecting
the
giv
en
dimensions.
2
Sho
w
that
the
triangle
AB
D
is
a
righ
t-angled
triangle.
3
Justify
that
:
AD
B
=
23
o
.
4
Calculate
the
length
AB
and
sp
ecify
its
v
alue
rounded
to
the
h
undredth
of
cm
.
5
Dra
w
the
line
parallel
to
the
line
(
AB
)
passing
through
E
.
It
in
tersects
segmen
t
[
B
D
]
at
p
oin
t
F
.
Place
the
p
oin
t
F
.
6
Calculate
the
length
E
F
and
sp
ecify
its
v
alue
rounded
to
the
ten
th
of
cm
.
5.
Smal
l
and
lar
ge
b
ows
E.2938
Consider
a
quadrilateral
AB
C
D
inscrib
ed
in
a
circle
C
.
Sho
w
that
the
opp
osite
angles,
in
the
quadrilateral
AB
C
D
,
are
angles
of
supplemen
tary
mea-
sures.
Hint
:
use
the
eight
angles
forme
d
by
the
interse
ction
of
the
two
diagonals
of
the
quadrilat-
er
al.
h
ttps:/
/c
hingmath.fr
A
B
D
O
M
C
75
o
chapExoCorrec/805
sacados/805
A
B
C
O
H
C
(
d
chapExoCorrec/3928
sacados/3928
Asie Juin 2009
A
B
D
E
C
chapExoCorrec/2938
sacados/2938
B
D
A
C
C
C
A
B
C
I
J
K
M
30
40
50
A
B
C
Figure
n
o
1
A
B
C
D
E
(
DE
=
=
(
AC
Figure
n
o
2
A
B
C
E
Figure
n
o
3
40
o
50
o
Figure 1
Figure 2
A
B
C
E
60
o
20
o
A
B
C
D
E
(
AC
=
=
(
DE
Figure 3
Figure 4
A
B
C
5
;
5
cm
4
;
5
cm
7
;
5
cm
BCDE
est un losange
de centre
A
A
B
C
D
E
E.2939
In
the
plane,
consider
an
y
triangle
AB
C
and
its
circumscribing
circle
C
;
the
p
oin
t
M
is
an
y
p
oin
t
on
the
circle
C
;
let
I
,
J
,
K
b
e
the
orthogonal
pro
jects
of
the
p
oin
t
M
on
to
the
straigh
t
lines
(
AC
)
,
(
B
C
)
,
(
AB
)
,
resp
ec-
tiv
ely
.
The
aim
of
this
exercise
is
to
sho
w
that
the
p
oin
ts
I
,
J
,
K
are
aligned.
1
a
Establish
that
the
p
oin
ts
I
and
J
b
elong
to
the
circle
of
diameter
[
M
C
]
.
b
Justify
that
the
angles
I
J
M
and
I
C
M
ha
v
e
the
same
measure.
c
Deduce
the
relationship
:
AC
M
=
180
−
I
J
M
2
a
Justify
that
the
p
oin
ts
B
,
J
,
K
,
M
are
co
cyclic
;
Sp
ecify
the
nature
of
the
circle
to
whic
h
they
b
elong.
b
Establish
the
equalit
y
:
AB
M
=
180
−
M
J
K
T
o
demonstr
ate
this
r
elationship,
we’l
l
use
the
diago-
nals
of
the
quadrilater
al
J
M
B
K
3
The
p
oin
ts
A
,
B
,
C
,
M
are
co
cyclic
(as
in
question
2
b
)
;
w
e
accept
the
follo
wing
relationship
:
AB
M
=
180
−
AC
M
Deduce
that
the
p
oin
ts
I
,
J
,
K
are
aligned
The
straigh
t
line
passing
through
these
three
p
oin
ts
is
called
Simson’s
straigh
t
line.
6.
A
little
of
everything
E.2492
Demonstrate,
for
eac
h
of
the
three
figures
b
elo
w,
that
the
triangle
AB
C
is
a
righ
t-angled
triangle
using
the
information
pro
vided.
E.3274
Complete
the
table
b
elo
w
:
h
ttps:/
/c
hingmath.fr
chapExoCorrec/2939
sacados/2939
C
A
B
C
I
J
K
M
chapExoCorrec/2492
sacados/2492
Groupe est - juin 2002 - 4 points
30
40
50
A
B
C
Figure
n
o
1
A
B
C
D
E
(
DE
=
=
(
AC
Figure
n
o
2
A
B
C
E
Figure
n
o
3
40
o
50
o
chapExoCorrec/3274
sacados/3274
Brevet juin 2010 - 6 points
Figure 1
Figure 2
A
B
C
E
60
o
20
o
A
B
C
D
E
(
AC
=
=
(
DE
Figure 3
Figure 4
A
B
C
5
;
5
cm
4
;
5
cm
7
;
5
cm
BCDE
est un losange
de centre
A
A
B
C
D
E
C
C
O
P
A
B
Q
6
cm
4
cm
A
B
C
O
M
5
cm
Figure
1
Figure
2
Figure
3
Figure
4
Is
triangle
AB
C
a
righ
t
triangle
in
A
?
Y
es
No
Y
es
No
Y
es
No
Y
es
No
Num
b
er(s)
of
the
prop
ert
y
or
prop-
erties
that
pro
v
e
this
List
of
prop
erties
:
1
If
a
quadrilateral
is
a
rhom
bus,
then
its
diagonals
ha
v
e
the
same
midp
oin
t
and
are
p
erp
endicular.
2
If
t
w
o
lines
are
p
erp
endicular
to
the
same
third
line,
then
they
are
parallel
to
eac
h
other.
3
If,
in
a
triangle,
the
square
of
the
length
of
the
longest
side
is
not
equal
to
the
sum
of
the
squares
of
the
lengths
of
the
other
t
w
o
sides,
then
the
triangle
is
not
a
righ
t
triangle.
4
In
a
triangle,
the
sum
of
the
measures
of
the
three
angles
is
equal
to
180
o
.
5
If
t
w
o
lines
are
parallel
and
a
third
line
is
p
erp
endicular
to
one
of
them,
then
it
is
p
erp
endicular
to
the
other.
6
If
a
quadrilateral
has
four
sides
of
equal
length,
then
it
is
a
rhom
bus.
7
If
t
w
o
angles
inscrib
ed
in
a
circle
in
tercept
the
same
arc,
then
they
ha
v
e
the
same
measure.
8
If
in
a
triangle,
the
square
of
the
length
of
the
longest
side
is
equal
to
the
sum
of
the
squares
of
the
lengths
of
the
other
sides,
then
this
triangle
is
a
righ
t
triangle
and
the
righ
t
angle
is
the
angle
opp
osite
the
longest
side.
7.
Unclassifie
d
financial
ye
ars
E.5989
The
circle
C
has
cen
ter
O
and
radius
4
cm
.
Note
[
P
Q
]
a
diameter
of
the
circle
C
.
W
e
construct
the
circle
C
of
cen
ter
P
and
radius
6
cm
.
Note
A
the
p
oin
t
of
in
tersec-
tion
of
the
circle
C
and
the
segmen
t
[
P
Q
]
and
B
one
of
the
of
the
in
tersection
p
oin
ts
of
the
t
w
o
circles
C
and
C
.
The
aim
of
the
exercise
is
to
determine
an
appro
ximate
v
alue
for
the
area
of
the
shaded
part.
W
e
will
w
ork
with
rounded
measuremen
ts
:
1
a
Determine
the
area
of
the
triangle
B
P
Q
.
b
Deduce
the
area
of
triangle
P
O
B
.
2
a
Determine
the
measure,
rounded
to
a
ten
th
of
a
de-
gree,
of
the
angle
P
QB
.
b
Deduce
the
area,
rounded
to
the
ten
th
of
cm
2
,
of
the
angular
sector
of
the
circle
C
defined
b
y
the
arc
P
B
.
3
Deduce
a
measure,
rounded
to
the
ten
th
of
cm
2
,
of
the
shaded
part.
E.6058
Consider
an
isosceles
triangle
AB
C
at
A
suc
h
that
angle
B
AC
measures
50
o
and
AB
is
equal
to
5
cm
.
Let
O
b
e
the
cen
ter
of
the
cir-
cle
circumscrib
ed
around
trian-
gle
AB
C
.
The
line
(
O
A
)
in
ter-
sects
this
circle,
denoted
(
C
)
,
at
another
p
oin
t
M
.
1
What
is
the
measure
of
an-
gle
B
AM
?
No
justification
is
required.
2
What
is
the
nature
of
triangle
B
AM
?
Justify
y
our
answ
er.
3
Calculate
the
length
AM
and
round
it
to
the
nearest
ten
th.
4
The
line
(
B
O
)
in
tersects
the
circle
C
at
another
p
oin
t
K
.
What
is
the
measure
of
angle
B
K
C
?
Justify
y
our
answ
er.
h
ttps:/
/c
hingmath.fr
sacados/5989
C
C
O
P
A
B
Q
6
cm
4
cm
sacados/6058
A
B
C
O
M
5
cm
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