MATH
Key Geometric Ideas
B
45°
45°
O
C
B
90°
90°
O
C
There is a direct relationship between the
central angle and the arc subtended by that
central angle.
Even if you don’t remember the rule, you can
tell the rule just by looking at simple
examples.
180°
B
180°
O
C
This rule goes both ways …
B
70°
?
C
70°
O
This rule goes both ways …
B
70°
C
?
70°
O
a b
a  b  180
B
110°?
70°
C
70°
110°
O
B
These are both the same length, because
they’re both radii of the circle.
Now form a triangle.
O
C
If sides of a triangle are equal lengths, then the
opposite angles are equal.
B
70°
C
70°
110°
?
O
110  x  x  180
2 x  70
x  35
Q
R
P
?
T
30°
S
Q
R
P
30°
T
S
?
What fraction of the circle is shaded?
If Line1 is parallel to Line2 , then:
ae
a
line 1
e
line 2
a
c
line 1
e
g
a  c  180
line 2
ae
c  e  180
In quadrilateral PQRS below, sides PS and
QR are parallel for what value of x ?
Mark the parallel lines. This is the first – and
most important – step.
To make clear where the parallel lines are cut
by a diagonal line, extend them.
Q
P
70°
x°
112°
S
R
In quadrilateral PQRS below, sides PS and
QR are parallel for what value of x ?
Q
P
70°
x°
112°
S
R
In quadrilateral PQRS below, sides PS and
QR are parallel for what value of x ?
Q
P
70°
x°
112°
S
R
In quadrilateral PQRS below, sides PS and
QR are parallel for what value of x ?
P
70°
Q
x° 70°
112°
S
R
x  110
In the figure below, A, B, C, and D are
collinear, FC is parallel to ED, BE is
perpendicular to ED, and the measures of
FAB and EBA are as marked. What is the
measure of FCB ?
E
63°
A
147°
B
?
C
D
In the figure below, A, B, C, and D are
collinear, FC is parallel to ED, BE is
perpendicular to ED, and the measures of
FAB and EBA are as marked. What is the
measure of FCB ?
E
63°
A
147°
B
33
90
57
?
C
D
33  90  x  180
123  x  180
x  57
E
63°
A
147°
B
33
90
57
?
C
D
47. In the figure below, AB || CD, AE
bisects ∠BAC, and CE bisects ∠ACD. If the
measure of ∠BAC is 82°, what is the
measure of ∠AEC ?
47. In the figure below, AB || CD, AE
bisects ∠BAC, and CE bisects ∠ACD. If the
measure of ∠BAC is 82°, what is the
measure of ∠AEC ?
ABCD is a trapezoid. What is angle BDC?
B
C
30°
?°
60°
A
D
Lines 1 and 2 are parallel. Find x.
110°
65°
line 1
line 2
x