Here are some basic definitions and properties of lines and angles in geometry. These concepts are tested in many competitive entrance exams like GMAT, GRE, CAT.
Line segment: A line segment has two end points with a definite length.
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||Angles|
|Interior Angles||∠3, ∠4, ∠5, ∠6|
|Exterior Angles||∠1, ∠2, ∠7, ∠8|
|Vertically opposite Angles||(∠1, ∠3), (∠2, ∠4), (∠5, ∠7), (∠6, ∠8)|
|Corresponding Angles||(∠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,
We can say that the lines are parallel if we can verify at least one of the aforementioned conditions.
Let us take a look at some examples.
Example 1. If the lines m and n are parallel to each other, then determine the angles ∠5 and ∠7.
Determining one pair can make it possible to find all the other angles. The following is one of the many ways to solve this question.
∠2 = 125°
∠2 = ∠4 since they are vertically opposite angles.
Therefore, ∠4 = 125°
∠4 is one of the interior angles on the same side of the transversal.
Therefore, ∠4 + ∠5 = 180°
125 + ∠5 = 180 → ∠5 = 180 – 125 = 55°
∠5 = ∠7 since vertically opposite angles.
Therefore, ∠5 = ∠7 = 55°
Note: Sometimes, the parallel property of the lines may not be mentioned in the problem statement and the lines may seem to be parallel to each other; but they may be not. It is important to determine whether two lines are parallel by verifying the angles and not by looks.
Example 2. If ∠A = 120° and ∠H = 60°. Determine if the lines are parallel.
Given ∠A = 120° and ∠H = 60°.
Since adjacent angles are supplementary, ∠A + ∠B = 180°
120 + ∠B = 180 → ∠B = 60°.
It is given that ∠H = 60°. We can see that ∠B and ∠H are exterior alternate angles.
When exterior alternate angles are equal, the lines are parallel.
Hence the lines p and q are parallel.
We can verify this using other angles.
If ∠H = 60°, ∠E = 120° since those two are on a straight line, they are supplementary.
Now, ∠A = ∠E = 120°. ∠A and ∠E are corresponding angles.
When corresponding angles are equal, the lines are parallel.
Likewise, we can prove using other angles too.
Example 3. If p and q are two lines parallel to each other and ∠E = 50°, find all the angles in the figure below.
It is given ∠E = 50°.
The two lines are parallel
→ The corresponding angles are equal.
Since ∠E and ∠A are corresponding angles, ∠A = 50° .
→ The vertically opposite angles are equal.
Since ∠A and ∠C are vertically opposite to each other, ∠C = 50°.
Since ∠E and ∠G are vertically opposite to each other, ∠G = 50°.
→ The interior angles on the same side of the transversal are supplementary.
∠E + ∠D = 180° → 50 + ∠D = 180° → ∠D = 130°
→ ∠D and ∠B are vertically opposite angles. So ∠B = 130°.
→ ∠B and ∠F are corresponding angles. So ∠F = 130°.
→ ∠F and ∠H are vertically opposite angles. So ∠H = 130°.
∠D = ∠O + 90° → 130 = ∠O + 90 → ∠O = 40°