SAGS Acceptable geometry reasons

[LINES] The adjacent angles on a straight line are supplementary.

∠s on a str line

[LINES] If the adjacent angles are supplementary, the outer arms of these angles form a straight line.

adj ∠s supp

[LINES] The adjacent angles in a revolution add up to 360°.

∠s round a pt OR ∠s in a rev

[LINES] Vertically opposite angles are equal.

vert opp ∠s =

[LINES] If AB || CD, then the alternate angles are equal.

alt ∠s; AB || CD

[LINES] If AB || CD, then the corresponding angles are equal.

corresp ∠s; AB || CD

[LINES] If AB || CD, then the co-interior angles are supplementary.

co-int ∠s; AB || CD

[LINES] If the alternate angles between two lines are equal, then the lines are parallel.

alt ∠s =

[LINES] If the corresponding angles between two lines are equal, then the lines are parallel.

corresp ∠s =

[LINES] If the co-interior angles between two lines are supplementary, then the lines are parallel.

coint ∠s supp

[TRIANGLES] The interior angles of a triangle are supplementary.

∠ sum in Δ OR sum of ∠s in Δ OR Int ∠s Δ

[TRIANGLES] The exterior angle of a triangle is equal to the sum of the interior opposite angles.

ext ∠ of Δ

[TRIANGLES] The angles opposite the equal sides in an isosceles triangle are equal. (i.e. Given image - Prove B and C equal)

∠s opp equal sides

[TRIANGLES] The sides opposite the equal angles in an isosceles triangle are equal. (i.e. Given image - prove c and b equal)

sides opp equal ∠s

[TRIANGLES] In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Pythagoras OR Theorem of Pythagoras

[TRIANGLES] If the square of the longest side in a triangle is equal to the sum of the squares of the other two sides then the triangle is right-angled.

Converse Pythagoras OR Converse Theorem of Pythagoras

[TRIANGLES] If three sides of one triangle are respectively equal to three sides of another triangle, the triangles are congruent.

SSS

[TRIANGLES] If two sides and an included angle of one triangle are respectively equal to two sides and an included angle of another triangle, the triangles are congruent.

SAS OR S∠S

[TRIANGLES] If two angles and one side of one triangle are respectively equal to two angles and the corresponding side in another triangle, the triangles are congruent.

AAS OR ∠∠S

[TRIANGLES] If in two right-angled triangles, the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and one side of the other, the triangles are congruent.

RHS OR 90°HS

[TRIANGLES] The line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half the length of the third side. (i.e. Given image - prove DE parallel to BC AND BC = 2DE)

Midpt Theorem

[TRIANGLES] The line drawn from the midpoint of one side of a triangle, parallel to another side, bisects the third side. (i.e. Given image - Given AD = AB and DE is parallel to BC, prove AE = EC)

line through midpt || to 2nd side

[TRIANGLES] A line drawn parallel to one side of a triangle divides the other two sides proportionally. (i.e. Given image - Given that HF is parallel to BC, Prove AH/HB = AF/FC

line || one side of Δ OR prop theorem; name || lines (i.e. - In ΔABC, HF || BC)

[TRIANGLES] If a line divides two sides of a triangle in the same proportion, then the line is parallel to the third side. (i.e. Given image - Prove that HF is parallel to BC)

line divides two sides of Δ in prop

[TRIANGLES] If two triangles are equiangular, then the corresponding sides are in proportion (and consequently the triangles are similar).

||| Δs OR equiangular Δs

[TRIANGLES] If the corresponding sides of two triangles are proportional, then the triangles are equiangular (and consequently the triangles are similar).

Sides of Δ in prop

[TRIANGLES] If triangles (or parallelograms) are on the same base (or on bases of equal length) and between the same parallel lines, then the triangles (or parallelograms) have equal areas.

same base; same height OR equal bases; equal height

[CIRCLES] The tangent to a circle is perpendicular to the radius/diameter of the circle at the point of contact.

tan ⊥ radius OR tan ⊥ diameter

[CIRCLES] If a line is drawn perpendicular to a radius/diameter at the point where the radius/diameter meets the circle, then the line is a tangent to the circle.

line ⊥ radius OR converse tan ⊥ radius OR converse tan ⊥ diameter

[CIRCLES] The line drawn from the centre of a circle to the midpoint of a chord is perpendicular to the chord. (i.e. Given Image - Prove Angle ACE is perpendicular)

line from centre to midpt of chord

[CIRCLES] The line drawn from the centre of a circle perpendicular to a chord bisects the chord. (i.e. Given image - Prove that DC = EC)

line from centre ⊥ to chord

[CIRCLES] The perpendicular bisector of a chord passes through the centre of the circle.

perp bisector of chord

[CIRCLES] The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle (on the same side of the chord as the centre).

∠ at centre = 2 × ∠ at circumference

[CIRCLES] The angle subtended by the diameter at the circumference of the circle is 90°.

∠s in semi-circle OR diameter subtends right angle OR ∠ in ½

[CIRCLES] If the angle subtended by a chord at the circumference of the circle is 90°, then the chord is a diameter.

chord subtends 90° OR converse ∠s in semi-circle

[CIRCLES] Angles subtended by a chord of the circle, on the same side of the chord, are equal.

∠s in the same seg

[CIRCLES] If a line segment joining two points subtends equal angles at two points on the same side of the line segment, then the four points are concyclic.

line subtends equal ∠s OR converse ∠s in the same seg

[CIRCLES] Equal chords subtend equal angles at the circumference of the circle.

equal chords; equal ∠s (Angle A = Angle D)

[CIRCLES] Equal chords subtend equal angles at the centre of the circle. (i.e. Given image, prove that angle FAC is = angle EAD)

equal chords; equal ∠s

[CIRCLES] Equal chords in equal circles subtend equal angles at the circumference of the circles.

equal circles; equal chords; equal ∠s

[CIRCLES] Equal chords in equal circles subtend equal angles at the centre of the circles.

equal circles; equal chords; equal ∠s

[CIRCLES] The opposite angles of a cyclic quadrilateral are supplementary.

opp ∠s of cyclic quad

[CIRCLES] If the opposite angles of a quadrilateral are supplementary then the quadrilateral is cyclic.

opp ∠s quad supp OR converse opp ∠s of cyclic quad

[CIRCLES] The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

ext ∠ of cyclic quad

[CIRCLES] If the exterior angle of a quadrilateral is equal to the interior opposite angle of the quadrilateral, then the quadrilateral is cyclic.

ext ∠ = int opp ∠ OR converse ext ∠ of cyclic quad

[CIRCLES] Two tangents drawn to a circle from the same point outside the circle are equal in length.

Tans from common pt OR Tans from same pt

[CIRCLES] The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment.

tan chord theorem

[CIRCLES] If a line is drawn through the end-point of a chord, making with the chord an angle equal to an angle in the alternate segment, then the line is a tangent to the circle.

converse tan chord theorem OR ∠ between line and chord

[QUADRILATERALS] The interior angles of a quadrilateral add up to 360°.

sum of ∠s in quad

[QUADRILATERALS] The opposite sides of a parallelogram are parallel.

opp sides of ||m

[QUADRILATERALS] If the opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram.

opp sides of quad are ||

[QUADRILATERALS] The opposite sides of a parallelogram are equal in length.

opp sides of ||m

[QUADRILATERALS] If the opposite sides of a quadrilateral are equal, then the quadrilateral is a parallelogram.

opp sides of quad are = OR converse opp sides of a parallelogram

[QUADRILATERALS] The opposite angles of a parallelogram are equal.

opp ∠s of ||m

[QUADRILATERALS] If the opposite angles of a quadrilateral are equal then the quadrilateral is a parallelogram.

opp ∠s of quad are = OR converse opp angles of a parm

[QUADRILATERALS] The diagonals of a parallelogram bisect each other.

diag of ||m

[QUADRILATERALS] If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

diags of quad bisect each other OR converse diags of a parm

[QUADRILATERALS] If one pair of opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram.

pair of opp sides = and ||

[QUADRILATERALS] The diagonals of a parallelogram bisect its area.

diag bisect area of ||m

[QUADRILATERALS] The diagonals of a rhombus bisect at right angles.

diags of rhombus

[QUADRILATERALS] The diagonals of a rhombus bisect the interior angles.

diags of rhombus

[QUADRILATERALS] All four sides of a rhombus are equal in length.

sides of rhombus

[QUADRILATERALS] All four sides of a square are equal in length.

sides of square

[QUADRILATERALS] The diagonals of a rectangle are equal in length.

diags of rect

[QUADRILATERALS] The diagonals of a kite intersect at right-angles.

diags of kite

[QUADRILATERALS] A diagonal of a kite bisects the other diagonal.

diag of kite

[QUADRILATERALS] A diagonal of a kite bisects the opposite angles.

diag of kite

[TRIANGLES] What is the reason for the statement: XY² = (y1 + z1) . y1 ?

Line from 90° ∠ ⟂ to hypotenuse

[TRIANGLES] What is the reason for the statement: XZ² = (y1 + z1) . z1 ?

Line from 90° ∠ ⟂ to hypotenuse

[TRIANGLES] What is the reason for the statement: XA² = y1 . z1 ?

Line from 90° ∠ ⟂ to hypotenuse