Lines and Angles:
Line segment: A line segment has two end points with a definite length.
Ray: A ray has one end point and infinitely extends in one direction.
Straight line: A straight line has neither starting nor end point and is of infinite length.
Acute angle: The angle that is between 0° and 90° is an acute angle, ∠A in the figure below.
Obtuse angle: The angle that is between 90° and 180° is an obtuse angle, ∠B as shown below.
Right angle: The angle that is 90° is a Right angle, ∠C as shown below.
Straight angle: The angle that is 180° is a straight angle, ∠AOB in the figure below.
In the figure above, ∠AOC + ∠COB = ∠AOB = 180°
If the sum of two angles is 180° then the angles are called supplementary angles.
Two right angles always supplement each other.
The pair of adjacent angles whose sum is a straight angle is called a linear pair.
∠COA + ∠AOB = 90°
If the sum of two angles is 90° then the two angles are called complementary angles.
The angles that have a common arm and a common vertex are called adjacent angles.
In the figure above, ∠BOA and ∠AOC are adjacent angles. Their common arm is OA and common vertex is ‘O’.
Vertically opposite angles:
When two lines intersect, the angles formed opposite to each other at the point of intersection (vertex) are called vertically opposite angles.
In the figure above,
x and y are two intersecting lines.
∠A and ∠C make one pair of vertically opposite angles and
∠B and ∠D make another pair of vertically opposite angles.
Perpendicular lines: When there is a right angle between two lines, the lines are said to be perpendicular to each other.
Here, the lines OA and OB are said to be perpendicular to each other.
Here, A and B are two parallel lines, intersected by a line p.
The line p is called a transversal, that which intersects two or more lines (not necessarily parallel lines) at distinct points.
As seen in the figure above, when a transversal intersects two lines, 8 angles are formed.
Let us consider the details in a tabular form for easy reference.
Types of Angles
∠3, ∠4, ∠5, ∠6
∠1, ∠2, ∠7, ∠8
Vertically opposite Angles
(∠1, ∠3), (∠2, ∠4), (∠5, ∠7), (∠6, ∠8)
(∠1, ∠5), (∠2, ∠6), (∠3, ∠7), (∠4, ∠8)
Interior Alternate Angles
(∠3, ∠5), (∠4, ∠6)
Exterior Alternate Angles
(∠1, ∠7), (∠2, ∠8)
Interior Angles on the same side of transversal
(∠3, ∠6), (∠4, ∠5)
When a transversal intersects two parallel lines,
- The corresponding angles are equal.
- The vertically opposite angles are equal.
- The alternate interior angles are equal.
- The alternate exterior angles are equal.
- The pair of interior angles on the same side of the transversal is supplementary.
We can say that the lines are parallel if we can verify at least one of the aforementioned conditions.
When two lines intersect they form two pairs of opposite angles, A + C and B + D. Another word for opposite angles are vertical angles.
Vertical angles are always congruent, which means that they are equal.
Adjacent angles are angles that come out of the same vertex. Adjacent angles share a common ray and do not overlap.
The size of the angle xzy in the picture above is the sum of the angles A and B.
Two angles are said to be complementary when the sum of the two angles is 90°.
Two angles are said to be supplementary when the sum of the two angles is 180°.
If we have two parallel lines and have a third line that crosses them as in the ficture below - the crossing line is called a transversal
When a transversal intersects with two parallel lines eight angles are produced.
The eight angles will together form four pairs of corresponding angles. Angles 1 and 5 constitutes one of the pairs. Corresponding angles are congruent. All angles that have the same position with regards to the parallel lines and the transversal are corresponding pairs e.g. 3 + 7, 4 + 8 and 2 + 6.
Angles that are in the area between the parallel lines like angle 2 and 8 above are called interior angles whereas the angles that are on the outside of the two parallel lines like 1 and 6 are called exterior angles.
Angles that are on the opposite sides of the transversal are called alternate angles e.g. 1 + 8.
All angles that are either exterior angles, interior angles, alternate angles or corresponding angles are all congruent.
The picture above shows two parallel lines with a transversal. The angle 6 is 65°. Is there any other angle that also measures 65°?
6 and 8 are vertical angles and are thus congruent which means angle 8 is also 65°.
6 and 2 are corresponding angles and are thus congruent which means angle 2 is 65°.
6 and 4 are alternate exterior angles and thus congruent which means angle 4 is 65°.
- Identifying Lines and Parts of Lines http://web.cerritos.edu/dford/SitePages/Math_70_F13/Worksheet-pts-lines-planes-angles.pdf
- Lines and Angles https://www.superteacherworksheets.com/lines-segments-rays.html
- Parallel Lines and Angles http://worksheets.tutorvista.com/parallel-lines-and-angles-worksheet.html