Geometry Grade 9
Two angles are complementary if their sum is 90°, and they supplementary if their sum is 180°.
Two angles form one-position pair, if the corresponding arms are parallel rays with the same orientation.
Two angles form an alternate pair if the corresponding arms are parallel rays with opposite orientation.
Two angles form a vertical pair if their corresponding arms form straight lines.
A triangle is uniquely determined:
- the three sides
- two sides and the enclosed angle
- one side and the two angles on it
- two sides ant the angle opposite the longer side.
In a right triangle, the square of the hypotenuse is equal to the sum of the square of the legs.
If the sum of the square of two sides in a triangle is equal to the square of the third side, then the triangle is a right triangle.
The sum of the interior angles of a triangle is 180°.
Any exterior angle of a triangle is equal to the sum of the two interior angles which are not adjacent to the given exterior angle.
The sum of any two sides of a triangle is larger than the third side.
If two sides of a triangle are equal to each other, then the two opposite site angles are equal too.
If two angles of a triangle are equal, then the two opposite sides are equal too.
The angle opposite the longer side is larger than the angle opposite the shorter side.
The side opposite the larger angle is longer than the side opposite the smaller angle.
A polygon is said to be convex if it contains all the points of a segment determined ny any two points of the polygon.
A quadrilateral which has two parallel sides is called a trapezium.
A quadrilateral whose opposite sides are parallel is called parallelogram.
A quadrilateral which has two pairs of consecutive equal sides is called a deltoid.
The perpendicular bisector of a segment is the set of points in the plane which are at equal distances from the endpoints of the segment.
The bisector of a given angle is the set of points in the plane which are at equal distances from the arms of a given angle.
The point of intersection of the perpendicular bisectors of the sides of the triangle is the centre of the circumscribed circle of the triangle.
If we connect the two end points of a diameter of a circle with any other perimeter point of the circle, then we get a right angled triangle.
In any right triangle the centre point of the circumscribed circle is the midpoint of the hypotenuse.
An altitude of a triangle is a line which passes through a vertex of the triangle and is perpendicular to the side opposite this vertex.
The three altitudes of a triangle meet at a point. (orthocenter of the triangle)
The point of intersection of the bisectors of the angles of a triangle is the centre point of the inscribed circle.
The segment connecting the mid-point of two sides of a triangle is a mid-line.
Each mid-line is parallel and half as long as to the third side of the triangle.
The line joining a vertex of a triangle to the mid-point of the opposite side is called a median.
The three medians of any triangle meet at one point (centroid of the triangle), which divides the medians in the ratio 2:1 (the longer segments lie towards the vertices)
