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Practical Geometry - Constructing a copy of an angle

Grade 6CBSE

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

🔑Concepts

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The objective is to construct an angle equal to a given angle ∠AOB\angle AOB using only a ruler and a compass, without knowing the actual numerical degree measure.

A given angle AOB with an arc drawn across its arms.
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The first step involves drawing a ray LL and then drawing an arc on the original angle ∠AOB\angle AOB with center OO. This same radius is used to draw an arc on the new ray starting from its endpoint PP.

A ray with a circular arc drawn from the endpoint P.
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The compass is then adjusted to the width between the points where the arc intersects the arms of the original angle. This width is transferred to the new construction to locate the second point on the copy.

The points X and Y on the arms of the angle used to measure the width with a compass.
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Finally, the intersection point is joined to the origin of the ray. This ensures the two angles are congruent based on the SSS (Side-Side-Side) criterion of the triangles formed by the arcs.

📐Formulae

Measure of Original Angle = Measure of Copied Angle

m∠ABC=m∠PQRm\angle ABC = m\angle PQR

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.

Problem 3:

Construct a copy of a given obtuse angle ∠DEF\angle DEF using only a compass and a straightedge.

An obtuse angle DEF with a construction arc.

Solution:

  1. Draw a ray QRQR.
  2. On the original ∠DEF\angle DEF, place the compass pointer at EE and draw an arc cutting EDED at XX and EFEF at YY.
  3. Without changing the compass width, place the pointer at QQ and draw an arc cutting QRQR at SS.
  4. Adjust the compass to the distance XYXY.
  5. With the pointer at SS, draw an arc intersecting the previous arc at point TT.
  6. Draw ray QTQT. ∠TQS\angle TQS is the required copy of ∠DEF\angle DEF.

Explanation:

By keeping the radii of the arcs and the chord lengths (XY=ST)(XY = ST) identical, we create two congruent triangles, ensuring ∠DEF=∠TQS\angle DEF = \angle TQS.

Problem 4:

If you are given an angle of 45∘45^{\circ}, describe the construction of another angle that is exactly twice the size of the given angle.

Two 45 degree angles constructed adjacently to form a 90 degree angle.

Solution:

  1. Construct a copy of the 45∘45^{\circ} angle on a base ray OXOX to get ∠YOX=45∘\angle YOX = 45^{\circ}.
  2. Treat the new ray OYOY as the base for a second copy.
  3. Using the same compass width used for the first arc, draw another arc from OO starting from ray OYOY.
  4. Measure the chord length of the original 45∘45^{\circ} arc and mark it off starting from the point on ray OYOY.
  5. Draw the final ray OZOZ. ∠ZOX\angle ZOX will be 45∘+45∘=90∘45^{\circ} + 45^{\circ} = 90^{\circ}.

Explanation:

This method uses the 'copying an angle' technique twice consecutively (adjacent to each other) to perform angle addition.