Leaving cert maths important geometry

If the opposite angles of a quadrilateral add to 180, what is the quadrilateral?

It is a cyclic quadrilateral

What if two circles intersect at one point only?

The two centres and the point of contact are collinear

If angles at points on a circle are equal in size….

they lie on the same arc

Distinguish beween the diagonals in a parallelogram and rhombus.

What is a polygon?

A polygon is a closed shape (without gaps or openings) with straight sides.

What is a regular polygon?

A regular polygon has equal sides and angles.

How would you divide a hexagon (6-sided shape) into triangles?

At the centre.

360/6=60 degrees

What is the relationship between triangle sides and angles?

The largest angle is opposite the largest side. The smallest angle is opposite the smallest side.

What will the exterior angle of a triangle equal to?

The two opposite interior sides.

What is an ordinary angle?

An angle less than 180 degrees.

What are supplementary angles

angles that add to 180 degrees

What is the rule for calculating interior angles size?

If 3-sided= 180

If 4 sided= 360

If 5 sided= 540

Then divide by number of sides ie 180/3= 60, 540/5= 108

When dealing with cyclic quadrilaterals…

consider that there could be more than 1 in the circle

ALWAYS CONSIDER THIS WHEN DEALING WITH CIRCLE PROBLEMS

All the angles lying on the same arc

When given 2 sides and an angle, there may be what…

the ambiguous case where one triangle may satisfy the data. in this case, calculate as normal and then take 180 away from your first answer.

What should you consider when dealing with circles/ circular arcs.

All lines that make a radius will be equal!

How to identify the alternate segment theorem

Consider the angle formed with the tangent and chord. Trace this chord to the opposite point/ angle

How to prove a shape is a parallelogram

If the DIAGONALS BISECT each other

If 2 pairs of opposite sides are equal and parallel.

If both pairs of opposite sides are either parallel or equal.

If the diagonals create 2 congruent triangles.

What does is equivalent to mean?

This means that something has the same value or measure as something else. For example, 1 euro is equivalent to 100 cents.

“if and only if” meaning

This means that both the statement and its converse are true

Give examples of parallelograms.

Parallelograms Rectangles Squares Rhombuses

Is a square a rhombus?

Yes a square is a rhombus but not every rhombus is a square.

Is a square a rectangle?

Every square is a rectangle but not every rectangle is a square.

What is a theorem?

A statement that is proven to be true (eg a triangle’s angles add to 180 degrees)

What is a proof?

A proof is a logical explanation that shows that a theorem must be true.

What is an axiom?

A basic fact that we accept as being true.

What is a corollary?

A corollary is a small, easy fact that follows on from a theorem.

What is a converse?

The converse of a theorem is the opposite of it.

It is when you switch the “if” and “then” parts.

eg. If a figure is a square, then it has four equal sides.

converse= “if a figure has four equal sides, then it is a square.”

What does implies mean?

Implies means that one fact leads to another fact.

eg. a figure being a square implies that it has 4 equal sides and 4 equal angles.

What should you always look for even in squares, rectangles (ESPECIALLY WITH PARALLEL LINES)

NO.1 - PARALLEL LINES ARE EQUIDISTANT

NO 2- CORRESPONDING ANGLES OR ALTERNATE ANGLES

VERY IMPORTANT property of transversals

Parallel lines will always cut off equal segments in a transversal (as they are equidistant themselves)

Look at the base height. In congruent triangles, it will be the same.

What is an important aspect to consider in congruent triangles/ triangles in general?

Extend the transversal!

When you have a transversal that is halfway between the parallel lines, what should you do?

Right angled triangles.

What will all the triangles formed by the diagonals of a rhombus be?

It will split the angle in half

What will a rhombus do to each angle?

It will bisect each diagonal.

(Note however that all 4 bisected “lines” won’t be equal- only a pair of them)

What will the rhombus do to each diagonal?

0

Just imagine it folding over (despite there being two congruent triangles it won’t quite “reach” the other side)

How many lines of symmetry does a parallelogram have?

According to number of sides

eg. equilateral triangle=3

octagon=8

How many axis of symmetry will regular polygons have?

2

How many lines of symmetry does a rhombus have?