Introduction to Angles
An angle is a figure in Plane Geometry generated by two rays or lines that share a common endpoint. "Angle" comes from the Latin word "angulus," which means "corner." The angle in the plane is present in two forms: a positive angle and a negative angle. A positive angle is formed when the angle is rotated counterclockwise. A negative angle is one that is measured in a clockwise direction.
Types of Angle
What are Angles?
When two rays arise from the same spot, an angle is generated. These rays form an angle known as the arms of the angle, and their origin is known as the vertex of the angle.
Angle Formed by Two Rays
An angle is denoted by the symbol \[\angle \]. The created angle is indicated by \[\angle \]PQR in the diagram above. \[\angle \]RQP is another way to represent the same angle. The degree is used to measure angles.
The above two rays may be combined in a variety of ways to generate the many sorts of angles in geometry.
Types of Angles
Based on the degree of the angle formed there are different types of angles.
Zero Angle – The angle formed is zero degrees.
Acute Angle – The angle formed is greater than zero and less than \[90^\circ \].
Right Angle – The angle formed is \[90^\circ \].
Obtuse Angle – The angle formed is greater than \[90^\circ \]and less than \[180^\circ \].
Straight Angle – The angle is at \[180^\circ \]and appears like a straight line.
Reflex Angle – The angle is greater than \[180^\circ \]but less than \[360^\circ \].
Complete Angle – The angle which is of \[360^\circ \]is the complete angle.
Types of Angles Based on the Degree of Angle Formed
Angles Based on Their Pairing
Based on the pairing of two angles there are different types of angles.
Complementary Angles – When two angles add up to form \[90^\circ \]then it is called complementary angles. For example, if we add \[30^\circ \]and \[60^\circ \] we get \[90^\circ \].
Supplementary Angles – When two angles add up to form \[180^\circ \]then we call it a supplementary angle. For example, if we add \[110^\circ \]and \[70^\circ \]we will get \[180^\circ \].
Linear Angles – Two angles are termed to be linear if they are produced by two intersecting lines. Their total is always \[180^\circ \].
Adjacent Angles – Adjacent angles are two angles that share a vertex and a side but do not intersect.
Complementary and Supplementary Angles
Construction of Angles
To construct the angle you need to follow these simple steps.
Step 1: Draw an OA line segment.
Step 2: Set the protractor's center at point O.
Step 3: Working clockwise from point A, mark a point at 50 degrees by glancing at the protractor's outer circle. Make this point B.
The \[\angle \]BOA represents the needed \[50^\circ \] angle.
For the construction of equal angles, you need to use a compass. The method of its construction is bit complex and would be taught to you in higher classes.
Construction of Angle
Conclusion
Angle is constructed by the two line segments who have a common end-point. Angles can be formed by two-line segments using a mathematical tool named a protractor. Two equal supplementary angles will result to form a whole circle or whole angle of \[360^\circ \]. The different types of angles formed can be calculated using the simple operations of addition and subtraction.
Sample Questions
1. The angle complementary to \[40^\circ \] is
a. \[30^\circ \]
b. \[40^\circ \]
c. \[50^\circ \]
d. \[60^\circ \]
Ans: \[50^\circ \]
Explanation: For angles to be complementary their sum should be equal to 90.
\[\begin{array}{l}40 + x = 90\\x = 90 - 40\\x = 50^\circ \end{array}\]
2. The angle supplementary to \[60^\circ \] is
a. \[160^\circ \]
b. \[140^\circ \]
c. \[120^\circ \]
d. \[100^\circ \]
Ans: \[120^\circ \]
Explanation: For angles to be supplementary their sum should be equal to 180.
\[\begin{array}{l}60 + x = 180^\circ \\x = 180^\circ \ - 60^\circ \\x = 120^\circ \end{array}\]
3. Find the measure of all angles of an equilateral triangle?
a. \[60^\circ \]
b. \[10^\circ \]
c. \[120^\circ \]
d. \[100^\circ \]
Ans: \[60^\circ \]
Explanation: The angles in the equilateral triangle are also equal. The sum of all the angles in a triangle is 180.
\[\begin{array}{l}3x = 180\\x = \dfrac{{180}}{3} = 60^\circ \end{array}\]
So, in an equilateral triangle, each angle is 60 degrees.
FAQs on Equal Angles
1. What does it mean for two angles to be equal in geometry?
In geometry, two angles are considered equal when they have the exact same measure in degrees or radians. For instance, if ∠A measures 45° and ∠B also measures 45°, they are equal angles. This concept is crucial for proving properties of shapes and lines. Angles with equal measures are also referred to as congruent angles.
2. Which common geometric figures have equal angles?
Many common geometric figures contain equal angles, which define their properties. Key examples include:
- Squares and Rectangles: All four interior angles are equal, each measuring 90° (right angles).
- Equilateral Triangles: All three interior angles are equal, with each angle measuring 60°.
- Isosceles Triangles: The two angles opposite the equal-length sides, known as base angles, are always equal.
- Regular Polygons: In any regular polygon, such as a regular pentagon or hexagon, all interior angles are equal to one another.
3. Which angles are equal when a transversal intersects two parallel lines?
When a transversal line cuts across two parallel lines, several pairs of equal angles are created. According to the CBSE syllabus for 2025-26, students should identify:
- Corresponding angles: Angles in the same relative position at each intersection are equal.
- Alternate interior angles: Angles on opposite sides of the transversal and between the parallel lines are equal.
- Alternate exterior angles: Angles on opposite sides of the transversal and outside the parallel lines are equal.
4. How are 'equal angles' and 'congruent angles' different?
While often used interchangeably, 'equal angles' and 'congruent angles' have a subtle technical difference. 'Equal' refers to the numerical measurement of the angles (e.g., m∠A = m∠B). 'Congruent' refers to the angles as geometric shapes that are identical in form (e.g., ∠A ≅ ∠B). In practice, if two angles have equal measures, they are congruent, and if they are congruent, their measures are equal.
5. Why are vertically opposite angles always equal to each other?
Vertically opposite angles are formed by two intersecting lines and are always equal because of their relationship with adjacent angles. An angle and its adjacent angle form a linear pair, meaning their sum is 180°. Since each of the vertically opposite angles shares an adjacent angle with the other, they are both supplementary to the same angle. This forces their measures to be equal.
6. If two angles are both equal and supplementary, what must be their measure?
If two angles are supplementary, their measures add up to 180°. If they are also equal, each angle must be half of the total sum. By setting up the equation x + x = 180°, we find that 2x = 180°, which solves to x = 90°. Therefore, each angle must be a right angle, measuring exactly 90°.
7. In an isosceles triangle, which angles are equal?
In an isosceles triangle, the two angles that are opposite the two sides of equal length are always equal to each other. These are commonly referred to as the base angles of the triangle. For example, in triangle ABC, if side AB is equal to side AC, then the angle opposite AB (which is ∠C) will be equal to the angle opposite AC (which is ∠B).
8. How can you construct an angle equal to a given angle using only a compass and a straightedge?
As per the NCERT practical geometry syllabus, you can copy an angle by following these steps:
- Draw a ray which will form one side of the new angle.
- On the original angle, place your compass point on the vertex and draw an arc that crosses both sides.
- Without changing the compass width, draw a similar arc on your new ray.
- Measure the distance between the two intersection points on the original angle's arc with your compass.
- Use that measurement to mark a point on the new arc from its intersection with the ray.
- Draw a line from the endpoint of your ray through this new point. This creates an angle congruent to the original.
