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Summary euclidean geometry theorem statements revision
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- Mathematics
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- 12th Grade
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EUCLIDEAN GEOMETRY:S
THEOREM STATEMENTS & ACCEPTABLE REASONS
LINES If three sides of one triangle are respectively equal to
SSS
three sides of another triangle, the triangles are congruent.
The adjacent angles on a straight line are supplementary. s
ø on a str line
A
If two sides and an included angle of one triangle are
If the adjacent angles are supplementary, the outer arms
adj øs supp respectively equal to two sides and an included angle of SAS OR SøS
of these angles form a straight line.
another triangle, the triangles are congruent.
EUCLIDEAN GEOMETRY: THEOREM STATEMENTS & ACCEPTABLE REASONS
The adjacent angles in a revolution add up to 360º. øs round a pt OR øs in a rev
If two angles and one side of one triangle are
Vertically opposite angles are equal. vert opp øs respectively equal to two angles and the corresponding AAS OR øøS
T
If AB || CD, then the alternate angles are equal. s
alt ø ; AB || CD side in another triangle, the triangles are congruent.
If AB || CD, then the corresponding angles are equal. corresp øs ; AB || CD If in two right angled triangles, the hypotenuse and one side
If AB || CD, then the co-interior angles are supplementary. co-int øs ; AB || CD of one triangle are respectively equal to the hypotenuse RHS OR 90ºHS
and one side of the other, the triangles are congruent.
If the alternate angles between two lines are equal, then
alt øs =
the lines are parallel. The line segment joining the midpoints of two sides of a
If the corresponding angles between two lines are equal, s triangle is parallel to the third side and equal to half the Midpt Theorem
corresp ø = length of the third side.
then the lines are parallel.
If the co-interior angles between two lines are
co-int øs supp The line drawn from the midpoint of one side of a line through midpt || to
supplementary, then the lines are parallel.
triangle, parallel to another side, bisects the third side. 2nd side
A line drawn parallel to one side of a triangle divides the line || one side of Δ OR
TRIANGLES
other two sides proportionally. prop theorem; name || lines
ø sum in Δ OR sum of øs in Δ
The interior angles of a triangle are supplementary. If a line divides two sides of a triangle in the same line divides two sides of Δ in
OR int øs in Δ
proportion, then the line is parallel to the third side. prop
The exterior angle of a triangle is equal to the sum of the
ext ø of Δ
interior opposite angles. If two triangles are equiangular, then the corresponding
The angles opposite the equal sides in an isosceles sides are in proportion (and consequently the triangles ||| Δs OR equiangular Δs
øs opp equal sides are similar).
triangle are equal.
The sides opposite the equal angles in an isosceles
sides opp equal øs If the corresponding sides of two triangles are
triangle are equal.
proportional, then the triangles are equiangular (and Sides of Δ in prop
In a right-angled triangle, the square of the hypotenuse is Pythagoras OR consequently the triangles are similar).
equal to the sum of the squares of the other two sides. Theorem of Pythagoras
If the square of the longest side in a triangle is equal to Converse Pythagoras If triangles (or parallelograms) are on the same base (or on
same base; same height OR
the sum of the squares of the other two sides then the OR Converse Theorem of bases of equal length) and between the same parallel lines,
equal bases; equal height
triangle is right-angled. Pythagoras then the triangles (or parallelograms) have equal areas.
xiv Copyright © The Answer Series: Photocopying of this material is illegal
, QUADRILATERALS CIRCLES
GROUP I
The interior angles of a quadrilateral add up to 360º. sum of øs in quad The tangent to a circle is perpendicular
S
O to the radius/diameter of the circle at tan ⊥ radius
The opposite sides of a parallelogram are parallel. opp sides of ||m
the point of contact. tan ⊥ diameter
If the opposite sides of a quadrilateral are parallel, then opp sides of quad are || OR
the quadrilateral is a parallelogram. converse opp sides of ||m
If a line is drawn perpendicular to a
line ⊥ radius OR
The opposite sides of a parallelogram are equal in length. opp sides of ||m O radius/diameter at the point where the
radius/diameter meets the circle, then converse tan ⊥ radius OR
A
If the opposite sides of a quadrilateral are equal, then the opp sides of quad are = OR the line is a tangent to the circle. converse tan ⊥ diameter
EUCLIDEAN GEOMETRY: THEOREM STATEMENTS & ACCEPTABLE REASONS
quadrilateral is a parallelogram. converse opp sides of a parm
The opposite angles of a parallelogram are equal. opp øs of ||m The line drawn from the centre of a
O
s
circle to the midpoint of a chord is line from centre to midpt of chord
opp ø of quad are = OR perpendicular to the chord.
If the opposite angles of a quadrilateral are equal then the
T
quadrilateral is a parallelogram. converse opp angles of a
parm
The line drawn from the centre of a
The diagonals of a parallelogram bisect each other. diag of ||m O
circle perpendicular to a chord bisects line from centre ⊥ to chord
diags of quad bisect each the chord.
If the diagonals of a quadrilateral bisect each other, then
other OR
the quadrilateral is a parallelogram.
converse diags of a parm
The perpendicular bisector of a
If one pair of opposite sides of a quadrilateral are equal O chord passes through the centre of perp bisector of chord
pair of opp sides = and ||
and parallel, then the quadrilateral is a parallelogram. the circle.
The diagonals of a parallelogram bisect its area. diag bisect area of ||m
The angle subtended by an arc at the
The diagonals of a rhombus bisect at right angles. diags of rhombus x
centre of a circle is double the size of
O øat centre
The diagonals of a rhombus bisect the interior angles. diags of rhombus the angle subtended by the same arc
2x = 2 % ø at circumference
at the circle (on the same side of the
All four sides of a rhombus are equal in length. sides of rhombus chord as the centre)
All four sides of a square are equal in length. sides of square øs in semi circle OR
The angle subtended by the diameter
diameter subtends right angle
at the circumference of the circle
The diagonals of a rectangle are equal in length. diags of rect O OR ø in ½ ?
is 90º.
The diagonals of a kite intersect at right-angles. diags of kite
If the angle subtended by a chord at chord subtends 90º OR
A diagonal of a kite bisects the other diagonal. diag of kite
the circumference of the circle is 90°,
O converse øs in semi circle
then the chord is a diameter.
A diagonal of a kite bisects the opposite angles. diag of kite
xv Copyright © The Answer Series: Photocopying of this material is illegal
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