Some Important concept for Lines & Angles
Concepts (Angles, Lines)
1. Angles: In geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.
2. Types of Angles: There are six types of angles, namely:
· Acute: <90°
· Obtuse: >90° (but less than 180°)
· Right Angle: =90°
· Straight: =180°
· Reflex: >180° (but less than 360°)
· Perigon: =360°
3. Sum of all the angles of a quadrilateral is 360°
4. Two lines are said to be perpendicular if the angle between them is 90°
5. Two angles are said to be complementary if there sum is 90°
∠ A° + ∠B° = 90°
6. The sum of all the angles made on one side of a straight line is 180°
7. Two angles are said to be supplementary if there is sum is 180°
∠ A° + ∠B° = 180°
8. When two straight lines intersect each other, then the vertically opposite angles are equal.
9. Perpendicular Angle is 90°
10. Two lines are said to be perpendicular if the angle between is 90°
11. If one line divides another line equally into two parts, then the first line is said to be the Bisector of the second line.
12. Angular Bisector divides an angle equally.
13. Perpendicular Bisector is the line which is perpendicular as well as divided the line equally.
14. Two lines are said to be parallel if the angle between them is Zero (even if we extend them to infinity)
Triangles:
15. A polygon with three corners or vertices and three sides or edges which are line segments is known triangle.
16. There are 3 types of triangles (When considered according to the lengths of the sides):
· In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°
· In an isosceles triangle, two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length.
· In a scalene triangle, all sides are unequal and equivalently all angles are unequal.
17. There are 4 types of triangles (When considered according to the internal angles):
· A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one of its interior angles measuring 90° (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side of the right triangle.
· Triangles that do not have an angle that measures 90° are called oblique triangles.
· A triangle that has all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If the greatest side length is c, then a2 + b2 > c2
· A triangle that has one interior angle that measures more than 90° is an obtuse triangle or obtuse-angled triangle. If the greatest side length is c, then a2 + b2 < c2.
· A "triangle" with an interior angle of 180° (and collinear vertices) is degenerate.
18. A triangle has three corners, called vertices.
19. Sum of all the angles of a triangle is 180°
20. The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.
21. Perpendicular bisector of a line in triangle doesn’t mandatorily passes through the opposite vertex.
22. Altitude: An altitude is the perpendicular segment from a vertex to its opposite side.
23. Median:A median of a triangle is a line segment joining a vertex to the midpoint of the opposing side.
24. Every triangle has exactly three medians: one running from each vertex to the opposite side. In the case of isosceles and equilateral triangles, a median bisects any angle at a vertex whose two adjacent sides are equal in length.
25. In an isosceles triangle, the perpendicular drawn to the base from the vertex opposite the base, bisects the base as well as the vertical angle.
26. The altitude of an equilateral triangle is √3*a/2 (a = side of the triangle).
27. Sum of any two sides of the triangle is greater than the third side.
28. In right angled triangle, Hypotenuse is the largest side of the triangle.
29. In obtuse angled triangle, the angle opposite to the largest side is the obtuse angle.
30. Pythogoras Theorem:
(Hypotenuse)2 = (Base)2 + (Perpendicular)2
Concepts (Angles, Lines)
1. Angles: In geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.
2. Types of Angles: There are six types of angles, namely:
· Acute: <90°
· Obtuse: >90° (but less than 180°)
· Right Angle: =90°
· Straight: =180°
· Reflex: >180° (but less than 360°)
· Perigon: =360°
3. Sum of all the angles of a quadrilateral is 360°
4. Two lines are said to be perpendicular if the angle between them is 90°
5. Two angles are said to be complementary if there sum is 90°
∠ A° + ∠B° = 90°
6. The sum of all the angles made on one side of a straight line is 180°
7. Two angles are said to be supplementary if there is sum is 180°
∠ A° + ∠B° = 180°
8. When two straight lines intersect each other, then the vertically opposite angles are equal.
9. Perpendicular Angle is 90°
10. Two lines are said to be perpendicular if the angle between is 90°
11. If one line divides another line equally into two parts, then the first line is said to be the Bisector of the second line.
12. Angular Bisector divides an angle equally.
13. Perpendicular Bisector is the line which is perpendicular as well as divided the line equally.
14. Two lines are said to be parallel if the angle between them is Zero (even if we extend them to infinity)
Triangles:
15. A polygon with three corners or vertices and three sides or edges which are line segments is known triangle.
16. There are 3 types of triangles (When considered according to the lengths of the sides):
· In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°
· In an isosceles triangle, two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length.
· In a scalene triangle, all sides are unequal and equivalently all angles are unequal.
17. There are 4 types of triangles (When considered according to the internal angles):
· A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one of its interior angles measuring 90° (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side of the right triangle.
· Triangles that do not have an angle that measures 90° are called oblique triangles.
· A triangle that has all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If the greatest side length is c, then a2 + b2 > c2
· A triangle that has one interior angle that measures more than 90° is an obtuse triangle or obtuse-angled triangle. If the greatest side length is c, then a2 + b2 < c2.
· A "triangle" with an interior angle of 180° (and collinear vertices) is degenerate.
18. A triangle has three corners, called vertices.
19. Sum of all the angles of a triangle is 180°
20. The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.
21. Perpendicular bisector of a line in triangle doesn’t mandatorily passes through the opposite vertex.
22. Altitude: An altitude is the perpendicular segment from a vertex to its opposite side.
23. Median:A median of a triangle is a line segment joining a vertex to the midpoint of the opposing side.
24. Every triangle has exactly three medians: one running from each vertex to the opposite side. In the case of isosceles and equilateral triangles, a median bisects any angle at a vertex whose two adjacent sides are equal in length.
25. In an isosceles triangle, the perpendicular drawn to the base from the vertex opposite the base, bisects the base as well as the vertical angle.
26. The altitude of an equilateral triangle is √3*a/2 (a = side of the triangle).
27. Sum of any two sides of the triangle is greater than the third side.
28. In right angled triangle, Hypotenuse is the largest side of the triangle.
29. In obtuse angled triangle, the angle opposite to the largest side is the obtuse angle.
30. Pythogoras Theorem:
(Hypotenuse)2 = (Base)2 + (Perpendicular)2
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