An Angle is something that we use throughout Geometry, we talk about it all the way to the very end when we're talking about Trigonometry. Well an angle is formed by two rays that share common end point. It's measured in degrees and is between 0 and 180 degrees, if it's over 180 then you're going to subtract that number. So let's say you had 220 degrees you're going to subtract 180 from that so it's actually a 40 degree angle.
If we look at an example where we have angle a, b, c, there's two ways that you could label this. You can write this as angle abc or since there are no other adjacent angles that is an adjacent angle would be something like this where it would share that vertex that common end point. Since there are no other adjacent angles you could also just label this based on the vertex which is b.
Now there's something very specific about the way that I wrote angle abc, whenever you write the angle its vertex must be the middle letter. But what is the vertex? The vertex is this point that is the common end point of your rays. So I'm going to label this as the vertex, so the rays form what are called the sides. So bc, ray bc is one side of this angle and ray ba is another side. So again you can label an angle two different ways, one using three letters that make up the two sides and the vertex making sure that your vertex is the middle letter or if there aren't any other adjacent angles you can just label it based on its vertex.
There are four key types of angles first one is a acute, so if I drew this angle and I said that's angle x if it is acute that means that is less than 90 degrees but also greater than 0 degrees. So it has to be somewhere in between them, it cannot be exactly 90 degrees it cannot be exactly 0 degrees.
A right angle if this is x is equal to exactly 90 degrees a right angle. So we're going going to label all of our right angles in Geometry using these two segments which will tell you the student that this is a 90 degree angle.
The third type is an obtuse angle. So here if we measured x, x is going to be less than 180 degrees but more than 90 degrees. Because if this angle was able to be exactly 90 degrees it would be a right angle if it was less than 90 it would be acute.
The last one which is key to a lot of proofs is a straight angle. If you have a straight angle it is the equal to exactly 180 degrees. Which means if we had the rotation about point x that was form the full rotation around any given point is 360 degrees. So 2x=360 degrees.
So keep this in mind and remember obtuse is going to be in between 180 and 90, right is going to be exactly 90, acute is going to be between 90 and 0. And that there are two different ways of labeling your angles. And the way that will always work is using three letters.
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.
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||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,
- 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.
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°
– Properties and formulas of Circles
– Types of Triangles and Properties
– Properties of Quadrilaterals (parallelograms, trapezium, rhombus)