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Two points determine a
straight line.
AB
Is a segment that is extended
infinitely in one direction.
AB
Is a section of a straight line, of
finite length with two endpoints.
AB
Is the distance between its endpoints.
10 3
7
AB
103
If point Q is between point P and
point R, then
PQ QR PR
With a letter, a number, or
a group of three letters.
a
A
B C
Angle a Angle ABC
It is extremely unlikely that you’ll
see any fractional angles (23 ½o).
There are no negative angles (−30o).
Angles
greater than
zero but less
than 90o.
Angles with
measure
equal to 90o.
Angles
greater than
90o but less
than 180o.
An angle
that
measures
exactly 180o.
Two angles
that sum up
to 90o.
Two
angles
that sum
up to
180o.
Edwin Lapuerta, May 2014
LINES AND
ANGLES
Angles opposite each other
when two lines cross.
ac
bd
a b
c d
Angles that have a common vertex
and share a side.
ac
bd
,
180
180
180
180
a b c d
a c
a d
b c
b d
360a b c d
ABC CBD
A
B
C
D
If BC is the angle bisector
of angle ABD, then
L1
L2
Two lines which not meet.
L1
L2
L1 || L2
L1
L2
14
23
8 57 6
L1 || L2
1 3 5 7
2 4 6 8
small angles
big angles
L1
L2
14
23
8 57 6
L1 || L2
180small angle big angle
An angle is a right angle
only if:
You’re expressly told, “this
is a right angle”.
You see the
perpendicular symbol ().
You see the box in the
angle.
L1
L2
TRIANGLES
A triangle with
three equal sides
and three equal
angles (60o).
A triangle with
two equal sides
and two equal
angles.
A triangle with no
equal sides and no
equal angles.
If two sides of
a triangle are
unequal, the
angles
opposite these
sides are
unequal.
If two angles
of a triangle
are unequal,
the sides
opposite these
angles are
unequal.
The largest
angle is
opposite to the
largest side.
A triangle has 3
possible
midsegments. d
mED
B
CA
The midsegment is
always parallel to
the third side of
the triangle. d
mED
B
CA
The midsegment is
always half of the
length of the third
side. d
mED
B
CA
1
2m d
d
mED
B
CADBE ABC
DB EB DE
AB CB AC
The triangle formed
by the midsegment
is similar to the
original triangle.
DBE ABC
DB EB DE
AB CB AC
2a 2b
2c
a b
c
CongurentTriangles
have exactly the same three sides and exactly the same three angles.
Corresponding
sides and angles
are equal .
SAS
(Side-Angle-Side)
SSS
(Side-Side-Side)
ASA
(Angle-Side-Angle)
AAS
(Angle-Angle-Side)
The sum of the lengths of the sides.
a b
c
P a b c
The sum of the length of two sides
must be greater than the length of
the third side.
a b
c
a b c
b a c
c a b
The difference of two sides must be
less than the third side.
a b
c
b c a
a c b
a b c
ab
c
b c a b c
a c b a c
a b c a b
a b
c AB
C
The sum of the length of two sides must be
greater than the length of the third side.
+
In any triangle, the sum of the interior
angles is 180o.
a b
c AB
C
180A B C
The largest side is the hypotenuse and the
sides adjacent to the right angle (C = 90o) are
the legs.
C A
B
ac
b
180
90
A B C
A B
The measure of exterior angles of a triangle
is equal to the sum of the two remote
interior angles.
a
b
c
y
xz
360
x b c
y a c
z a b
x y z
The square of the length of the hypotenuse
is equal to the sum of the squares of the
lengths of the legs of the triangle.
a
b
c2 2 2a b c
It works only in right triangles.
3:4:5 (2 sides)
5
4
3
6:8:10 2 3: 4:5
15: 20: 25 5 3: 4:5
30: 40:50 10 3: 4:5
5:12:13 (2 sides)
13
12
5
10: 24: 26 2 5:12:13
25:60:65 5 5:12:13
50:120:130 10 5:12:13
Isosceles right triangle (1 side) 1:1:1 2
x
x 2x
45o
45o
: : 2 1:1:1 2
2: 2 : 2 2 2 1:1:1 2
5:5:5 2 5 1:1:1 2
x x x x
(1 side) 1: 3 : 2
2xx
3x
30o
60o
: 3 : 2 1: 3 : 2
2: 2 3 : 4 2 1: 3 : 2
5:5 3 :10 5 1: 3 : 2
x x x x
1
2A base height
The term "base" denotes any side, and
"height" denotes the length of a
perpendicular from the vertex opposite
the side onto the line containing the
side itself.
1
2A base height
h
b
1
2A base height
h
b
1
2A base height
h
b
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