GEOMETRY FACTS AND THEOREMS
Appendix B
Appendix B: Geometry Facts and Theorems
Alternate Angle Theorem:
When a transversal intersects two parallel lines,
the alternate angles are equal.
a = b
a
c
b
c = d
d
Corresponding Angle Theorem:
When a transversal intersects two parallel lines,
the corresponding angles are equal.
a = b
e
a
c
c = d
f
e = g
g
b
d
h
f = h
Interior Angle Theorem:
When a transversal intersects two parallel lines,
the interior angles add to 180° (are supplementary.)
a
a + b = 180°
x
b
x + y = 180°
y
Appendix B - 1
Appendix B
GEOMETRY FACTS AND THEOREMS
Triangle Angle Sum Theorem:
The sum of the interior angles of any triangle is 180°.
a
a + b + c = 180°
b
c
Isosceles Triangle Theorem:
In an isosceles triangle, the base angles are equal.
a = b
a
b
Exterior Angle Sum Theorem
The sum of the exterior angles of any triangle is 360°.
a
a + b + c = 360°
c
b
Appendix B - 2
GEOMETRY FACTS AND THEOREMS
Appendix B
Pythagorean Theorem:
In a right triangle, the area of the square on the hypotenuse
is equal to the sum of the areas of the squares on the other
two sides.
area 1
+
area 2
area 1
area 2
Converse Theorems
If the opposite sides of a quadrilateral are equal, then they are also parallel. That is, the
quadrilateral is a parallelogram.
If two alternate angles are equal, then the lines intersected by the transversal are parallel.
If two corresponding angles are equal, then the lines intersected by the transversal are parallel.
If two interior angles add to 180°, then the lines intersected by the transversal are parallel.
If the areas of the squares on two sides of a triangle add to the area of the square on the third
side, then the triangle is a right triangle.
Appendix B - 3
Appendix B
GEOMETRY FACTS AND THEOREMS
Other Theorems and Facts
60°
Each angle in an equilateral triangle is 60°.
60°
60°
The sum of the interior angles of any quadrilateral is 360°.
c
d
a + b + c + d = 360°
a
b
For any polygon, the number of sides = the number of interior angles (or vertices).
3
2
2
3
1
1
4
4
7
5
7
5
6
6
The sum of the exterior angles of any polygon is 360°.
f
e
a
d
b
a + b + c + d + e + f = 360°
c
Appendix B - 4
GEOMETRY FACTS AND THEOREMS
Appendix B
The sum of the interior angles of any polygon can be found by dividing the polygon into nonoverlapping triangles. Each triangle contributes 180°.
The sum of interior angles = the number of triangles x 180°.
a
f
1
b
2
3
c
e
4
d
a + b + c + d + e + f
=
(4 triangles) x 180° = 720°
The number of non-overlapping triangles = number of sides ! 2
For a regular polygon,
Each interior angle = the sum of the interior angles ÷ number of angles (or sides).
The perpendicular distance between two parallel lines is the same no matter where you are on the
lines.
In similar triangles, the angles in one triangle equal matching angles in the other.
However, the sides of one triangle are multiples of matching sides of the other.
If the three sides of one triangle equal the three sides of another triangle then they are congruent
triangles. The angles of one triangle will equal the corresponding angles of the other triangle.
Appendix B - 5