Leaving Cert Geometry

Leaving Cert Geometry
Propositions Propositions are useful ideas which deserve mention. Proofs are not required on the syllabus.		F (B)	O (G)	H (A)
	Propositions
1	Two lines perpendicular to the same line are parallel to one another.
2	There is a unique line perpendicular to a given line and passing through a given point. This applies to a point on or off the line.
3	Each rectangle is a parallelogram.
4	A quadrilateral in which one pair of opposite sides is equal and parallel, is a parallelogram.
5	Each rhombus is a parallelogram.
6	(Converse of Theorem 10) If the diagonals of a quadrilateral bisect one another, then the quadrilateral is a parallelogram.
7	If two triangles have an angle in one equal to an angle in the other, and the lengths of sides surrounding that angle proportional, then the triangles are similar (words supplied here for notation).
8	(Converse of Theorem 13) If the sides of two triangles are proportional, in order, then the two triangles are similar.
9	(RHS) If two right angled triangles have hypotenuse and another side equal in length, respectively, then they are congruent.
10	Each point on the perpendicular bisector of a segment [AB] is equidistant from the ends.
11	The perpendicular from a point on an angle bisector to the arms of the angle have equal length.
12	Congruent triangles have equal areas. (Remark: The converse of this Proposition is false).
13	If a triangle ABC is cut into two by a line AD from A to a point D on the segment [BC], then the areas add up properly: Area Triangle ABC = area of Triangle ABD + area of Triangle ADC.
14	The area of a rectangle having sides of length a and b is ab.
DEF39	Let the side AB of a parallelogram ABCD be chosen as a base. Then the height of the parallelogram corresponding to that base is the height of the Triangle ABC.
15	This height is the same as the height of the triangle ABD, and as the length of the perpendicular segment from D onto AB.
16	If l is a line and S a circle, then l meets S in zero, one, or two points.
17	If a circle passes through three non-collinear points A, B and C, then its centre lies on the perpendicular bisector of each side of the Triangle ABC.
18	If a circle lies inside the Triangle ABC and is tangent ot each of its sides, then its centre lies on the bisector of each of the angles ÐA, ÐB, and ÐC.
19	The lines joining the vertices of a triangle to the centre of the opposite sides meet in one point. (Medians meet at the centriod)
20	The perpendicular from the vertices of a triangle to the opposite sides meet in one point. (Orthocentre)

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Leaving Cert Geometry

Propositions

Let the side AB of a parallelogram ABCD be chosen as a base. Then the height of the parallelogram corresponding to that base is the height of the Triangle ABC.