Download MPM1DE Summary of Euclidean Geometry Theorems

MPM1DE
CA
SA
CAT
SAT
OAT
PLT-Z
PLT-F
PLT-C
AST
ITT
EAT
ETT
Prop. Tri.
Prop. Quad.
Prop. Poly.
IAT
Summary of Euclidean Geometry Theorems and Properties
Complementary Angles sum to 90o.
Supplementary Angles sum to 180o.
Complementary Angle Theorem – if a and b are complementary and c and d are
complementary then if a = c, b = d.
Supplementary Angle Theorem – if a and b are supplementary and c and d are
supplementary then if a = c, b = d.
Opposite Angle Theorem – opposite angles are equal
Parallel Line Transversal Theorem – alternate angles are equal
Parallel Line Transversal Theorem – corresponding angles are equal
Parallel Line Transversal Theorem – co-interior angles are supplementary
Angle Sum Theorem – angles in a triangle sum to 180 degrees (aka SATT)
Isosceles Triangle Theorem – angles opposite the equal sides are equal
Exterior Angle Theorem
Equilateral Triangle Theorem – all angles in an equilateral triangle 60o.
The sum of the exterior angles of a triangle sum to 360 degrees.
The midsegment of a triangle is parallel to the third side and half its length.
The ratio of the area of the four triangles formed by the midsegments and the original
triangle is 1:4.
The sum of the interior angles of a quadrilateral is 360 degrees.
The sum of the exterior angles of a quadrilateral is 360 degrees.
Opposite angles of a parallelogram are equal.
The four midsegments of a quadrilateral always form a parallelogram.
The sum of the interior angles of an n sided polygon is 180(n-2) degrees.
The sum of the exterior angles of a convex polygon is 360 degrees.
Inscribed Angle Theorem – the two inscribed angles
subtended by the same arc are equal
SIAT
SIAT sector Inscribed Angle Theorem – an inscribed angle is
half of the central angle subtended by the same arc
DIAT
DIAT Diameter Inscribed Angle Theorem – an angle inscribed
in a semi-circle is always a right angle (half of the central
angle which is 180 degrees
CQT
CQT Cyclic Quadrilateral Theorem – opposite angles of a cyclic
quadrilateral sum to 180 degrees
Triangle Centres
The orthocenter is the point where the three altitudes of a triangle intersect. Properties: if the triangle is acute the
orthocenter lies inside the triangle, if it is obtuse it lies outside the triangle and if it is right it lies on a vertex of the
triangle.
The centroid is the point where the three medians of a triangle intersect. Properties: it always lies inside the triangle, it
is the centre of balance of the triangle, and the medians are divided in the ratio of 2:1.
The circumcentre is the point where the three perpendicular bisectors of a triangle intersect. Properties: if the triangle
is acute the circumcentre lies inside the triangle, if the triangle is obtuse it lies outside the triangle and if it is a right
triangle it lies on a side of the triangle. The circumcentre is the centre of a circle that passes through the vertices of the
triangle.
The incentre is the point where the three angle bisectors of a triangle intersect. Properties: the incentre is always inside
the triangle. The incentre is a centre of a circle inside the triangle that meets each side at exactly one point. The
inscribed circle touches the triangle sides at the perpendicular distance from the incentre to the side and not where the
angle bisectors meets the sides.
Diagonals of Quadrilaterals
The diagonals of a parallelogram bisect each other.
The diagonals of a rhombus bisect each other at right angles and bisect the angles at the vertices.
The diagonals of a rectangle are equal in length.
The diagonals of a kite are perpendicular to each other. The major diagonal bisects the minor one.