Download Properties of Triangles Results to be Discussed

Properties of Triangles
You should be familiar with the following properties of triangles:
1. If two sides and the included angle of one triangle are equal to two sides and the
included angle of another triangle, the triangles are congruent. (S.A.S.)
2. If two angles and the included side of one triangle are equal to two angles and the
included side of another triangle, the triangles are congruent. (A.S.A.)
3. If the three sides of one triangle are equal to the three sides of another triangle, the
triangles are congruent. (S.S.S.)
4. If two sides of a triangle are equal, the angles opposite them are equal.
5. If two angles of a triangle are equal, the sides opposite them are equal.
6. If a triangle is equilateral, it is also equiangular. If a triangle is equiangular, it is also
equilateral.
7. If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar; i.e. corresponding sides are proportional. (A.A. Similarity Theorem)
8. If an angle of one triangle is equal to an angle of another triangle and the sides including these angles are proportional, then the triangles are similar. (S.A.S. Similarity
Theorem)
9. An angle bisector in a triangle divides the opposite side into segments that have the
same ratio as the other two sides.
10. In a right-angled triangle, the square of the hypotenuse is equal to the sum of the
swuares of the other two sides. (Pythagoras)
11. The angle bisectors of a triangle meet in a point, called the incentre of the triangle.
12. The perpendicular bisectors the sides of a triangle meet in a point, called the circumcentre of the triangle.
13. The three altitudes (lines through a vertex, perpendicular to the opposite side) of a
triangle meet in a point called the orthocentre of the triangle.
14. The three medians (lines through a vertex to the midpint of the opposite side) of a
triangle meet in a point called the centroid of the triangle.
15. Let ABC be a triangle, and let D,E be the midpoints of AB and AC respectively.
Then the line DE is parallel to the ase BC and equal to one-half of it.
Results to be Discussed
16. The Euler Line In a triangle ABC, let O be the circumcentre, let G be the centroi,
and let H be the orthocentre. Then O, G, H lie on a line called the Euler line of the
triangle and GH ∼
= OG.
17. In any triangle, the midpoints of the three sides, the feet of the three altitudes, and
the midpoints of the segments joining the three vertices to the orthocentre all lie on
a circle called the nine-point circle.
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