Geometry 1

Basic concepts

• A plane is a flat 2-D surface it has length and width but no thickness
• A point is a position on a plane it has no dimensions
• If points are on the same plane they are coplanar
• A line is a straight, infinitely thin 1-D figure that continues forever in both directions, it has no end points
• Points that lie on the same line are called collinear points
• Perpendicular lines are lines that are at 90° to each other
• Parallel lines are the same distance apart and never meet
• A line segment is part of a straight line which has endpoints and can be measured using a ruler. Ex; |ab| * if has a measurement |ab|=5cm
• A ray is part of a line that starts at a point and goes on forever in only one direction

Angles

An angle is formed when two rays meet at a point called a vertex ex; ∠B

• Supplementary angles do not need to be beside or adjacent to each other

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Axioms

• An Axiom is a statement we accept without any proof

*To Note:

• Axiom 1 - There is exactly one line through any two given points
• Axiom 2 - The properties of the distance between points
• Axiom 3 - The properties of the degree measure of and angle
• Axiom 5 - Given line L and a point P, there is exactly one line through P that is parallel to L.

Properties, Rules or theorems of Angles and Lines

• Given two intersecting lines vertically opposite angles are angles that have the same vertex and are not adjacent to each other (look for X)

• Theorem 1: Vertically opposite angles are equal in measure

• A line that cuts two or more lines (usually parallel) is a transversal

• Alternate angles are on opposite sides of the transversal that cuts two lines but are between the two lines (look for Z)

• Theorem 3: If transversal makes equal alternate angles on two lines then the lines are parallel ( and converse)

• The converse of a theorem is formed by swapping hypothesis and conclusion and may or may not be true

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Geometry 1

• Corresponding angles are on the same side of the transversal that cuts two lines. One angle is between the lines, the other outside the lines (F shape)

• Theorem 5: Two lines are parallel if, and only if, for any transversal, the corresponding angles are equal.

• Interior angles between two parallel lines add up to 180°

Triangles

Equilateral triangle	Isosceles triangle	Scalene triangle
All sides equal length	At least two sides same length	No sides same length
All angles the same size (60°)	At least two angles same size	No angles same size

• An equilateral triangle is also an isosceles triangle
• Theorem 2: In an isosceles triangle the angles opposite the equal sides are equal conversely if two angles in a triangle are equal in measure, then the triangle is isosceles.
• Theorem 4: Angles in any triangle add to 180°
• An external angle of a triangle is the angle between one side of the triangle and the extension of an adjacent side

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Geometry 1

Triangle Theorems

Theorem 6: Each exterior angle of a triangle is equal to the sum of the interior opposite angles

Theorem 7: The angle opposite the greater of two sides is greater than the angle opposite the lesser side converse is true

Theorem 8: Two sides of a triangle are together greater than the third. This is called the triangle inequality theorem

Theorem 16: For a triangle, base times height does not depend on the choice of base.

Theorem 11: If three parallel lines cut off equal segments on same transversal line, they will cut off equal segments on any other transversal

Theorem 12: Let ABC be a triangle. If a line l is parallel to BC and cuts [AB] in the ratio m

then it also cuts [AC] in the same ratio.

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Geometry 1

• The converse of that theorem also stands true and all these ratios can be inverted or turned upside down

• In similar or equiangular triangles, all three angles in one triangle have the same measurement as the corresponding three angles in the other.

• Theorem 13: if two triangles are similar, their sides are proportional in order. The converse also stands true.

• If a triangle is cut by a line parallel to one of its sides, this line divides the triangle into two similar triangles

• Theorem 14: (Pythagoras) In a right angled triangle the square of the hypotenuse is the sum of the squares of the other two sides

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Geometry 1

• Theorem 15: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Quadrilaterals

• A polygon is a closed shape with straight sides.
  • At least three sides
• A regular polygon has equal sides and equal angles
• A parallelogram is a quadrilateral for which both pairs of opposite sides are parallel

PARALLELOGRAM	RHOMBUS	RECTANGLE	SQUARE
Opposite sides equal	Four equal sides	Opposite sides equal	Four equal sides
Opposite sides parallel	Opposite sides parallel	Opposite sides parallel	Opposite sides parallel
Diagonals bisect each other	Diagonals bisect each other & form 90° angle	Diagonals bisect each other	Diagonals bisect each other & form 90° angle

• Theorem 9: In a parallelogram, opposite sides are equal and opposite angles are equal. Converse is also true.
  • Corollary 1: A diagonal divides a parallelogram into congruent triangles[SAS]
• Theorem 10: The diagonals of a parallelogram bisect each other

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Geometry 1

• Theorem 17: A diagonal of a parallelogram bisects the area. From theorem 17 a parallelogram can be cut into two triangles of equal area

• Theorem 18: The area of a parallelogram is the base times the height

Circles

• A circle is set of points in a plane that are all equidistant from a fixed point. It's centre

• Radius is the line segment from centre of the circle to any point on the circle

• Chord is any segment that joins two points on a circle.

• Diameter is a chord that passes through the centre of a circle. The diameter is twice the radius in length. It is the longest chord of a circle.

• Circumference is the perimeter or length of the circle

• Arc is any part of the circumference of the circle

• Tangent is a line that touches the circle at only one point. Where the tangent touches the circle is called the point of tangency.

• Sector is the region of a circle enclosed by two radii and the area between those two radii.

• Theorem 19: The angle at the centre of a circle standing on a given area is twice the angle at any point of the circle standing on the same arc.

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Geometry 1

• Corollary 2 All angles at a point of a circle, standing on the same arc are equal in measure. Converse is also true

• A cyclic quadrilateral is a quadrilateral in which all four vertices are points of a circle

• Corollary 5- if ABCD is a cyclic quadrilateral, then opposite angles sum to 180° The converse of this will prove if vertices of quadrilateral stand on a circle

• Corollary 3- Each angle in a semicircle is a right angle

• Corollary 4- if the angle standing on a chord [BC] at some point of a circle is a right angle, then [BC] is a diameter.

• Theorem 20: Each tangent is perpendicular to the radius that goes to the point of contact the converse is also true→ if P lies on s, and line L is perpendicular to the radius at P then L is a tangent

• Corollary 6- If two circles intersect at only one point, the two centres and the point of contact are collinear.

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Geometry 1

Theorem 21 (i): The perpendicular from the centre to a chord bisects the chord

Theorem 21 (ii): The perpendicular from the centre to a chord bisects the chord

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Geometry 2&3

Definitions for proofs

• A proposition is a mathematical statement or theorem it may be true or false
• A proof is a sequence of logical steps to prove a statement or theorem.
• A theorem is a statement which has been proved to be true, deduced from axioms by logical arguments
• An axiom is a rule or statement that we accept without any proof, assumed true
• A corollary is a statement which must be true based on a previous theorem, follows after a theorem
• The converse of a theorem is formed by swapping the order of the hypothesis and conclusion. Conclusion is taken as starting point and starting point is conclusion
• Implies indicates a logical relationship between two statements. A fact we have proved from previous statements. Symbol =>
• If and only if (iff) - one statement is true if and only the second statement is true. Both must be true or both must be false.
• Two things are congruent if they are identical in size and shape.
• Two things are equivalent if they have the same value but different forms

Types of proofs

• A direct proof is proving a statement by a series of logical steps using known facts, theorems or axioms
• Proof by induction is used to establish if a given statement is true for one step of the process, it is also true for the next step.
• Proof by contradiction is used to establish the truth of a proposition showing that the proposition being false leads to contradiction. Proving its falsely to be impossible proces it to be true

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Congruent triangles

• The symbol ≡ is shorthand to show congruency △ABC≡△DEF
• Congruent triangles have all corresponding sides and interior angles equal
• Two triangles are congruent if:

Writing a proof

• Heading 1 : Theorem: state
• Heading 2 : Given: All information/diagram given to be listed.
• Heading 3 : Statement to prove: State
• Heading 4 : Construction : Extra lines / labels
• Heading 5 : Proof : Logical progression of statements with reasons
• Heading 6 : Q.E.D to conclude.

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Geometry 2&3

Proofs to learn

Junior Cert:

• Theorem 4 - Angles add to 180°
• Theorem 6 - Exterior opposite
• Theorem 9 - Parallelogram
• Theorem 14 - Pythagoras
• Theorem 19 - Arc + angles in a circle

Leaving Cert:

• Theorem 11 - Transversals split
• Theorem 12 - Equal segments ratio in triangle
• Theorem 13 - Proportional Sides in similar triangles

Congruent triangles

• In a triangle the angle opposite the greater of two sides is greater than the angle opposite the lesser side.
• Inequality : Lengths of any two sides of a triangle added together are always greater than the third sides length.

In ABC if |AC| > |AB|
then |∠B| > |∠C|
converse also applies

a+b > c
a+c > b
b+c > a

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Triangle centres

Centre	Intersecting lines	Acute△ (angles less than 90°)	Right-angled △	Obtuse△ (one angle > 90°)
Circumcentre (equidistant from all vertices)	Perpendicular bisectors of sides	Inside	On midpoint of hypotenuse	Outside
Incentre (equidistant from all sides)	Intersection of angle bisectors	Inside	Inside	Inside
Centroid (centre of mass balancing point)	Medians (centroid divides median in ratio 2:1)	Inside	Inside	Inside
Orthocentre	Altitudes	Inside	On right angle	Outside

• The circumcentre is the point where a triangles three perpendicular bisectors meet
• The circumcircle of a triangle is a circle that passes through all three vertices
• The incentre is the point where a triangles three angle bisectors meet
• The incircle of a triangle is the largest circle that will fit inside the triangle (△ sides will be tangents).
• A median of a triangle is a segment that goes from one of the triangles vertices to the midpoint of the opposite side

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Geometry 2&3

• The centroid is the triangles centre of gravity ( point where medians meet)
• An altitude is a segment drawn from a vertex of a triangle to its opposite side such that it forms 90° with the side
• The orthocentre is the point where the attitudes of a triangle meet.
• In a non-equilateral triangle a line will pass through the circumcentre, the centroid and the orthocentre of a triangle. It is called the Euler line.
• In an isosceles triangle the incentre also lies on the line.