Why do We Need to Learn to Construct Angles?
One of the most significant components of geometry is the creation of angles, which is the "perfect" type of geometric construction. In geometry, the phrase construction refers to the precise sketching of forms, lines, or angles using mathematical instruments. In this tutorial, we will learn how to make angles with a protractor, compass, ruler, and pencil.
What is an Angle?
Before talking about constructing angles with a protractor, let us quickly reminisce about angles and their types.
An angle is made when two rays start from the same point or common point. This common point is known as the vertex of the angle and the two rays forming the angle are known as sides or arms.
On the basis of the inclination between its two arms, an angle can be obtuse (i.e. more than 90 degrees), acute (i.e. less than 90 degrees) or right-angled (i.e. exactly 90 degrees).
The construction of angles is a very crucial part of geometry because its knowledge is extended for the construction of other geometrical figures as well, for example, the triangles.
The Protractor and Its Uses
A protractor is a semi-circular disc that you can use to draw or measure angles. It has marks from 0 to 180-degree angles and can be used directly to measure any angle within the range. It has two sets of markings which are 0 to 180 degrees from left to right and vice versa.
How To Use a Protractor for Constructing Angles
For constructing angles of any measure whether it's acute, obtuse or a right-angle the easiest method is to use a protractor. Let us assume that you need to construct an angle of 160 degrees. Here are the steps.
Draw a line and name it BC.
Place the protractor with its point O on point B of the line segment BC.
Align OQ along the edge BC.
As we said earlier, the protractor has two-way markings. We will examine the scale which has 0 degrees near point C for this construction. Mark point A next to the 160 degrees mark on the scale.
Join points A and B. ∠ABC = 160 degrees is the required angle.
Constructing Angles of Unknown Measure
Constructing angles of unknown measures is quite fun and easy for it is like copying a given angle with unknown measurements. We achieve this task using compasses. Let us assume that you are given an acute angle ∠BAC that you are supposed to copy. Here are the steps:
Draw a line PQ and point P is the vertex of the duplicate angle.
Put your compass pointer at point A and develop a circular segment or arc that cuts arms AC and AB at points K and J separately.
Don't change the radius of the compass. Make an arc at PQ at point M.
Control the compass so that the pointer is set down at K and the pencil head is placed at J.
Keep the equal radius and then form an arc on the first arc while holding the compass pointer at M and mark the intersecting point as L.
Join the points P and L using a scale and extend the line up to R.
∠RPQ is the needed angle.
How to Construct a 90-Degree Angle
Here we are explaining how to make a 90-degree angle with a compass and we have added the steps of construction of a 90-degree angle for your convenience:
Step 1: Take any ray OA.
Step 2: Let O be the centre and take any radius, draw an arc cutting OA at B.
Step 3: Take B as the centre and the same radius, and draw an arc cutting the first arc at C.
Step 4: Taking C as the centre and the same radius, make an arc intersecting again the first arc at D.
Step 5: Take C and D as the centre and keep radius as more than half of CD.
Step 6: Draw two arcs cutting each other at E then join OE.
Step 7: Here you have your desired angle ∠EOA = 90 degrees.
How to Construct a 75 Degree Angle
Here we are explaining how to make a 75-degree angle with a compass and we have added the steps of construction of 75-degree angle for your convenience.
Step 1: Take a look at OA.
Step 2: Using O as the centre and any appropriate radius, draw an arc that cuts OA at C.
Step 3: Using C as the centre and the same radius, cut the first arc at M.
Step 4: Using M as the centre and the same radius, cut off an arc that intersects the first arc at L.
Step 5: Using L and M as the centre and radius of more than half of LM, draw two arcs intersecting at B and joining OB at 90°.
Step 6: Now, with N and M as the centres, draw two arcs that intersect at P.
Step 7: Lastly, join OP.
Step 8: Angle POA is your required angle and this is how to draw a 75-degree angle.
Practice Question MCQs
1. The angle that is less than 360° but more than 180° is referred to as ________.
Reflex Angle
Acute Angle
Right Angle
Obtuse Angle
Answer: A) Reflex Angle
2. What do we call an angle that is exactly equal to 180°?
ObtuseAngle
Right Angle
Straight Angle
Acute Angle
Answer: C) Straight Angle
Conclusion
A geometric shape generated by the intersection of two line segments, lines, or rays is known as an angle. Angles, as opposed to linear distance, are a measure of rotational distance. An angle can alternatively be considered as a part of a circle. The angle between two line segments is the distance (in degrees or radians) that one segment must be turned about the crossing point in order for the two segments to overlap. Angles are essential in the definition and study of polygons such as triangles and quadrilaterals. They are utilized in a number of fields, including animation, woodworking, and physics.
FAQs on Constructing Angles
1. What are the essential tools required for constructing angles in geometry?
To construct angles accurately as per geometric principles, you primarily need the following tools:
- Compass: Used for drawing arcs and circles of a specific radius, which is essential for creating intersections.
- Straightedge or Ruler: Used for drawing straight line segments. While a ruler has markings for measurement, a straightedge is used purely for drawing lines.
- Protractor: A semi-circular tool used for directly measuring or drawing angles of a specific degree.
- Pencil: A sharp pencil is crucial for drawing precise lines and points.
2. What is an angle, and what are its main types based on measurement?
An angle is a geometric figure formed when two rays originate from a common endpoint, called the vertex. The two rays are known as the arms or sides of the angle. Based on their measure, angles are commonly classified as:
- Acute Angle: An angle that measures less than 90°.
- Right Angle: An angle that measures exactly 90°.
- Obtuse Angle: An angle that measures more than 90° but less than 180°.
- Straight Angle: An angle that measures exactly 180°.
- Reflex Angle: An angle that measures more than 180° but less than 360°.
3. What is the main difference between constructing an angle with a protractor versus a compass?
The key difference lies in the method and principle. A protractor is a measuring tool; you simply align it and mark a pre-existing degree. It is fast and easy for any angle. In contrast, a compass and a straightedge are used for pure geometric construction. This method doesn't rely on pre-marked degrees but on creating arcs and points based on geometric properties. Constructions with a compass are fundamental for proving theorems and understanding geometric relationships.
4. Why is it important to learn how to construct angles using only a compass and ruler?
Learning to construct angles with a compass and ruler is crucial because it develops a deeper understanding of geometric principles beyond simple measurement. It teaches precision, logical reasoning, and the relationships between different angles and shapes. This skill is foundational for constructing more complex figures like triangles, quadrilaterals, and polygons, and it is a core concept in fields like architecture, engineering, and design.
5. How can you construct a 60-degree angle using only a compass and a straightedge?
A 60-degree angle is one of the most basic and fundamental constructions. The steps are as follows:
- Step 1: Draw a ray, let's call it OA.
- Step 2: Place the compass point at the vertex O and draw an arc that intersects the ray OA at a point, let's call it B.
- Step 3: Without changing the compass width, place the compass point at B and draw another arc that cuts the first arc at a new point, C.
- Step 4: Join the vertex O to point C with a straightedge. The resulting angle, ∠AOC, is exactly 60 degrees.
6. What are the steps to construct a 90-degree angle with a compass?
A 90-degree angle, or a right angle, is constructed by creating a perpendicular bisector or by bisecting the angle between 60° and 120°.
- Step 1: Draw a ray OA.
- Step 2: With O as the centre, draw a large arc that intersects OA at B.
- Step 3: With the same radius and B as the centre, cut the arc at C (this marks 60°).
- Step 4: With C as the centre and the same radius, cut the arc again at D (this marks 120°).
- Step 5: Now, with C and D as centres and a radius more than half of the distance CD, draw two arcs that intersect each other at a point, say E.
- Step 6: Join O and E. The angle ∠AOE is a 90-degree angle.
7. How is a 45-degree angle constructed using a compass?
A 45-degree angle is constructed by bisecting a 90-degree angle. First, you must construct a 90-degree angle.
- Step 1: Follow all the steps to construct a 90° angle (∠AOE = 90°), where the angle's arms intersect the initial arc at points B and F.
- Step 2: With B and F as centres and a radius more than half the distance between them, draw two arcs that intersect at a point, G.
- Step 3: Join the vertex O to point G. The line OG is the angle bisector of ∠AOE.
- Step 4: Both ∠AOG and ∠GOE will be exactly 45 degrees.
8. How do you construct a 75-degree angle, and which angles is it related to?
A 75-degree angle is constructed by combining the constructions of 60° and 90°. It is the angle created by bisecting the 30° difference between 90° and 60° (75 = 60 + 15).
- Step 1: Construct a 90-degree angle. Let the points on the initial arc corresponding to 60° and 90° be C and F, respectively.
- Step 2: The angle between OC and OF is 30°. Now, bisect this 30° angle.
- Step 3: With C and F as centres and a radius more than half their distance, draw two arcs that intersect at a point, H.
- Step 4: Join the vertex O to H. The angle ∠AOH will be 75 degrees (60° + 15°).
9. Is it possible to construct any angle, for example, 20 degrees, using only a compass and a straightedge?
No, it is not possible to construct every angle using only a compass and a straightedge. This is a famous problem in geometry known as Angle Trisection. You can only construct angles that can be derived from a 60-degree angle through bisection (halving) or addition/subtraction of constructible angles. For example, you can make 60°, 30°, 15°, 90°, 45°, 75°, etc. However, an angle like 20° cannot be constructed because it would require trisecting a 60° angle, which is proven to be impossible with these tools.
10. How can you copy an angle of an unknown measure using just a compass and a straightedge?
Copying an angle is a fundamental construction skill that allows you to replicate an angle without knowing its degree measure. Here are the steps:
- Step 1: Let's say you want to copy ∠BAC. First, draw a new ray, P'Q'.
- Step 2: On the original angle, place the compass point at vertex A and draw an arc that cuts both arms, AB and AC, at points J and K.
- Step 3: Without changing the compass width, place the compass point at P' and draw a similar arc that intersects the ray P'Q' at a point M.
- Step 4: Now, adjust your compass to measure the distance between J and K on the original angle.
- Step 5: With that same width, place the compass point at M on the new ray and draw an arc that intersects the first arc at a point L.
- Step 6: Join P' and L with a straightedge. The new angle, ∠QP'L, is an exact copy of ∠BAC.
