Junior Cert Geometry Theorems

Theorem angles at centre and circumferenceA major part of the Geometry and Trigonometry sections of your Junior Cert maths course concerns the use of axioms, theorems and their corollaries and converses. Don’t worry, its not as bad as it might sound.

You could be asked to give a one line definition of these terms.

An axiom is a statement accepted without proof, as a basis for argument.

A theorem is a statement deduced from the axioms by logical argument.

A corollary is something that naturally follows or results from another thing already proved.

A converse is a switching of the order of the hypothesis and conclusion of a conditional statement. In other words in an “If ….then…” statement you swap around what comes between the “if” and the “then” with what comes after the “then”. For example the converse of “If a line is horizontal, then the line has slope 0″ would be “If a line has slope 0, then the line is horizontal.”

A proposition is a mathematical statement such as “3 is greater than 4,” “an infinite set exists,” or “7 is prime number.”

You should also know the meaning of “congruent“: Objects are congruent if they have the same dimensions and shape. Angles are congruent if they have the same measure in degrees. Congruent polygons (many sided shapes) have an equal number of sides, and all the corresponding sides and angles are congruent. On your exam papers it will usually be congruent triangles that you will be concerned with. These are triangles with all three sides and all three angles in one of them are the same length and measure as corresponding sides and angles in the other. If you could slide them around you could get one to sit exactly on top of the other.

Triangles are congruent in the following cases:

– Two angles and an opposite side. AAS

– Two angles and included side. ASA

– Two sides and included angle. SAS

– All three sides equal. SSS

Congruent triangles may share a common side.

.

But we can not assume that they are congruent in the following cases:

– SSA does not work

           – AAA does not work

Similar triangles have congruent (equal) corresponding angles but the sides are not congruent.

Remember also that every radius of a given circle is the same length. Questions often appear with triangles within circles and any sides of the triangles that are also radii of the circle are the same length.

It would be useful to have a summary of all the theorems on your course on a single page for easy reference. This is done for you on one of the pages of GMG’s latest publication, Junior Cert Maths – Geometry Theorems” which can be purchased as a PDF document .

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