krit.club logo

Practical Geometry - Constructing a copy of an angle

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

Definition of Copying an Angle: Constructing a copy of an angle means creating a new angle that has the exact same measure as a given angle without using a protractor to measure the degrees. Visually, this results in two angles that look identical in their 'opening' or 'spread'.

•

The Role of the Compass: A compass is the primary tool used to 'measure' distances between points on the arms of the angle. When we draw an arc, we are ensuring that the distance from the vertex to the points on the arms is constant, creating a circular reference line.

•

Vertex and Rays: Every angle consists of a vertex (the meeting point) and two rays (the arms). To copy an angle, we first draw a base ray for the new angle, which serves as the starting line where the vertex will be placed.

•

Initial Arc Placement: In the original angle ∠AOB\angle AOB, we place the compass pointer at vertex OO and draw an arc that intersects both rays OAOA and OBOB at points PP and QQ. This arc represents the path of points equidistant from the vertex.

•

Transferring the Radius: Using the exact same compass setting (radius) used on the original angle, we place the pointer on the new vertex O′O' and draw an arc that intersects the new base ray. This ensures the 'scale' of our reference points remains identical to the original.

•

Measuring the Width (Arc Length): To capture the specific 'opening' of the angle, we place the compass pointer at point PP and adjust the pencil to point QQ. This physical distance between the two points on the arms represents the angular width.

•

Finding the Second Ray: We move the compass to the intersection point on our new base ray and draw a small arc that crosses our first arc. The point where these two arcs intersect (the 'X' mark) defines the path for the second ray of our copied angle.

•

Verification: Once the construction is complete, the resulting angle ∠A′O′B′\angle A'O'B' is congruent to ∠AOB\angle AOB. This means if you were to place one over the other, they would match perfectly, which can be verified using a protractor to see that m∠AOB=m∠A′O′B′m\angle AOB = m\angle A'O'B'.

📐Formulae

Measure of Original Angle = Measure of Copied Angle

m∠ABC=m∠PQRm\angle ABC = m\angle PQR

RadiusofArc1(Original)=RadiusofArc1(Copy)Radius of Arc_{1} (Original) = Radius of Arc_{1} (Copy)

Distance between intersection points P and Q = Distance between intersection points X and Y

ΔOPQ≅ΔO′X′Y′\Delta OPQ \cong \Delta O'X'Y' (The underlying geometric principle of SSS congruence)

💡Examples

Problem 1:

Given an angle ∠PQR\angle PQR of unknown measure, construct a copy of this angle named ∠ABC\angle ABC using a ruler and compass.

Solution:

  1. Draw a ray BCBC which will serve as the base of the new angle. 2. Place the compass pointer at vertex QQ of the given angle and draw an arc of any radius that cuts the arms QPQP and QRQR at points XX and YY respectively. 3. Without changing the compass radius, place the pointer at point BB (the new vertex) and draw an arc that cuts the ray BCBC at a point ZZ. 4. Now, place the compass pointer at point YY and adjust the width so the pencil touches point XX. 5. With this width, move the compass pointer to point ZZ on your new drawing and draw an arc that intersects the first arc at a point DD. 6. Use a ruler to draw a ray starting from BB and passing through DD. The resulting ∠DBC\angle DBC (or ∠ABC\angle ABC) is the copy of ∠PQR\angle PQR.

Explanation:

The solution uses the compass to transfer the distance from the vertex to the arms and the distance between the arms themselves. By maintaining the same radius and the same arc-width, we ensure the two angles are congruent.

Problem 2:

If ∠XYZ=75∘\angle XYZ = 75^{\circ}, describe the steps to construct ∠LMN=75∘\angle LMN = 75^{\circ} without using a protractor for the construction.

Solution:

  1. Draw ray MNMN. 2. At vertex YY, draw an arc intersecting YXYX and YZYZ at AA and BB. 3. At vertex MM, draw the same arc intersecting MNMN at PP. 4. Measure the distance ABAB with the compass. 5. From point PP, draw an arc with radius ABAB to intersect the previous arc at point QQ. 6. Draw ray MLML through QQ. Since the construction replicates the exact arc length between the arms at a fixed distance from the vertex, ∠LMN\angle LMN will measure exactly 75∘75^{\circ}.

Explanation:

This demonstrates that even when the degree measure is known, the compass-and-ruler method relies on transferring geometric lengths (arcs and chords) rather than numerical degree values.