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MATHEMATICS 406
GEOMETRY
ANGLES
Angles are classified according to their measures, x, in degrees.
 acute angle : 0 0  x 0  90 0
 right angle : x = 90 0
 obtuse angle: 90 0  x 0  180 0
 straight angle : x = 180 0
 reflex angle : 180 0  x 0  360 0
Complementary angles : a pair of angles whose measures have a sum of 90 0 .
Ex. 1
A
 ABC and  CBD are complementary angles
C
B
D
Ex. 2
E
 EFG and  HIJ are
complementary angles
H
570
330
F
G
I
J
Supplementary angles : a pair of angles whose measures have a sum of 180 0 .
Ex. 1
P
 NQP is the supplement
of  PQT.
N
Q
T
Ex. 2
Z
 ZYX and  LMR are
supplementary angles
L
350
X
1450
Y
M
R
Vertically opposite angles (V.O.) : Pairs of V.O. angles are formed when two lines or
two line segments intersect. V.O. angles have the same vertex.
Ex.
a and b are V.O. angles
c and d are V.O. angles
c
a
b
d
Note: Vertically opposite angles are always congruent (  ).
Adjacent angles : pairs of angles that share a common vertex and a common interior
side.
They may be congruent, they may be complementary or they may be supplementary, but
they need not be any of these.
Ex.
A
 ABC and  CBD are adjacent angles
BUT
 ABC and  ABD are not.
C
B
D
t
When a transversal line, t, intersects two other lines,
as illustrated in the adjacent figure, the following
pairs of angles are formed.
a b
c d
e f
g h

corresponding angles :
two angles at different vertices, located on
the same side of the transversal line, one
inside the other two lines and the other
outside them.
In the given figure, the following pairs of angles are corresponding angles.
a and e ; b and f ; c and g ; d and h

alternate-interior angles :
two angles at different vertices, located on opposite sides of the transversal line,
and inside the other two lines.
In the given figure, the following pairs of angles are alternate-interior angles.
c and f ; d and e

alternate-exterior angles :
two angles at different vertices, located on opposite sides of the transversal line,
and outside the other two lines.
In the given figure, the following pairs of angles are alternate-exterior angles.
a and h ; b and g
When the two lines intersected by a transversal line are parallel, then all pairs of
corresponding, alternate-interior, and alternate-exterior angles are congruent. The
converse of this statement is also always true. Namely, if any two corresponding,
alternate-interior, or alternate-exterior angles are congruent, then they are formed by
parallel lines.
An angle bisector is a segment or line that cuts an angle into 2 congruent angles.
TRIANGLES
Triangles can be classified according to the measures of their angles or according to
the measures of their sides.
According to angle measures:



an acute triangle has three acute angles
a right triangle has one right angle
an obtuse triangle has one obtuse angle
A
R
O
According to side measures:



an equilateral triangle has three congruent sides (and as a result, three congruent
angles, all measuring 60o)
a isosceles triangle has two congruent sides (and as a result, two congruent
angles)
a scalene triangle has no congruent sides (and as a result, no congruent angles)
E
I
I
S
The median of a triangle is a line segment that joins one vertex of the triangle to the
midpoint of the side opposite this vertex. Every triangle has 3 medians.
The altitude or height of a triangle is a line segment that joins one vertex of the triangle
to the opposite side, or to an extension of the opposite side, forming a right angle. Every
triangle has 3 altitudes.
The sum of the measures of the interior angles of a triangle is always 180 0 .
In any right triangle, the acute angles are complementary.
In a triangle in which the angles measure 30o, 60o and 90o, the measure of the side
opposite the 30o angle is half that of the hypotenuse.
60o
x
2x
30o
To find the measure of the third side, use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the
hypotenuse is equal to the sum of the squares of the lengths of the other two sides, called
the legs of the triangle.
a2  b2  c2
c
a
b
In a triangle in which the angles measure 45o, 45o and 90o, the legs of the right triangle
are congruent and the hypotenuse is 2 times longer than one of these legs.
45o
x
2x
45o
In any triangle, the side opposite the largest angle is the longest side and the side
opposite the smallest angle is the shortest side.
A
c
b
B
C
a
If mC  mA  mB , then c  a  c.
The formula for calculating the area of a triangle is A 
bh
.
2
The right bisector of a segment is a line that divides a segment into 2 congruent
segments forming a right angle.
ex.
l
A
M
Line l is the right bisector
of AB if AM  MB
B
QUADRILATERALS
A quadrilateral is a four-sided polygon.
TYPES OF QUADRILATERALS
Parallelogram - a quadrilateral having two pairs of opposite sides parallel
Square – a quadrilateral having four congruent sides and one right angle
Rectangle – a parallelogram having one right angle
Rhombus – a parallelogram having four congruent sides
Trapezoid – a quadrilateral having a pair of opposite sides parallel
Isosceles trapezoid – a trapezoid having a pair of opposite sides congruent
PROPERTIES OF QUADRILATERALS
Type
opposite
sides
are //
square
rectangle
parallelogram
rhombus
trapezoid
isosceles
trapezoid
x
x
x
x
opposite
sides
are 
opposite
angles
are 
diagonals
are 
x
x
x
x
x
x
x
x
x
x
diagonals
bisect
each
other
x
x
x
x
diagonals are
perpendicular
x
x
x
Properties of isosceles trapezoids:
- Consecutive angles belonging to different bases are supplementary.
mA  mB  180 0 and mC  mD  180 0
- Angles belonging to the same base are congruent.
A  D and B  C
B
A
C
D
consecutive
angles are
supplementary
x
x
x
x
FORMULAS FOR CALCULATING AREAS OF QUADRILATERALS
Square : A  s 2
Rectangle : A  bh or A  l  w
Parallelogram : A  bh
Rhombus : A  bh or A 
Trapezoid : A 
( B  b) h
2
Dd
2
POLYGONS
A polygon is a closed figure with at least three sides.
Types of polygons:
Number
of sides
3
4
5
6
7
8
9
10
.
n
Name
triangle
quadrilateral
pentagon
hexagon
hepta or septagon
octagon
nonagon
decagon
n-gon
Sum of the measures of
the interior angles
180 0
360 0
540 0
720 0
900 0
1080 0
1260 0
1440 0
Measure of one angle in a
regular n-sided polygon
60 0
90 0
108 0
120 0
128.6 0
135 0
140 0
144 0
180(n  2) 0
180( n  2)
n
0
A regular polygon is a polygon in which all sides and all angles are congruent.
To calculate the area of a regular polygon, we use the following formula:
pa
A
2
where p is the perimeter of the polygon and a is the measure of the apothem.
The apothem of a regular polygon is a segment that joins the center of the polygon to the
midpoint of one of its sides, forming a right angle.
ISOMETRIES
Isometric figures are congruent figures.
The transformations that produce isometric figures are the following:
 translations – slides
defined by a translation arrow of a specific length and direction
 reflections – flips
defined by an axis of symmetry
 rotations – turns
defined by a center, a direction (clockwise or counter-clockwise) and an angle
measure of rotation
 glide-reflections – slide-turns
defined by a translation arrow and an axis of symmetry
The composite of any two or more of the above isometries is also an isometry.
Isometry type
translation
reflection
rotation
glide-reflection
Preserves orientation
yes
no
yes
no
Parallel paths
yes
yes
no
no
EQUALITY RELATIONS
The following are properties of the equality relation.

The reflexive property :
mABC  mABC or m AB  m AB

The symmetric property :
mABC  mCBA or m AB  mBA

The transitive property :
If mA  mB and mB  mC , then mA  mC .
Or
If m AB  mCD and mCD  mEF then m AB  mEF .