Activity-construction of angles

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Construction of an angle with measure 22.5∘

Objectives

Understand construction of an angle bisector,

Problem Class 9. Construction of an angle with measure 22.5∘

Estimated Time

One period

Prerequisites/Instructions, prior preparations, if any

Introduction to Angles

Materials/ Resources needed

Geogebra file - Construction of angle with measure 22.5∘

From Geogebra.org

Process (How to do the activity)

  1. Ask students which angle could be drawn with a scale? (Straight Angle)
  2. If we can 'halve' any angle, then we can derive angles from the straight angle - in succession, this would be as follows - ∡180∘ -> ∡90∘, ∡90∘ -> ∡45∘, ∡45∘ - > ∡22.5∘ and so on.
  3. Explain the process of constructing an Angle bisector using only compass (without using a protractor). Students can follow and do the construction in their books
    1. Draw a line segment or a line. Identify a point on this segment. We can treat this point as the vertex of a straight angle
    2. Construct two arcs from the vertex of the straight angle, such that each cuts the line segment on either side of the vertex.
    3. Plot the intersection point of each arc and the arms of the straight angle.
    4. From each intersection, draw an arc of same measure, such that the two arcs intersect.
    5. Plot the intersection point of these two arcs
    6. Construct a line segment (or line or ray) from the vertex to this intersection point.
    7. This process will create two angles. Measure both angles. They will be 90 each
    8. Why does this process work? (Explanation - locus of points equidistant from the two points which are equidistant from the vertex)
    9. Following the same process. construct the bisector of one of these two right angles. We will get two angles of 45 each
    10. Following the same process. construct the bisector of one of these angles, which measure 45. We will get two angles of 22.5 each
  4. You can use the 'Play - Construction protocol' to show the above steps one by one. But do this after constructing the bisectors using Geogebra.

Evaluation at the end of the activity

  1. Ask students, what are the other angles that can drawn in the same construction. (Hint, there are new angles created, which are adjacent/ cocomplementary/plementary to the angles we have discussed. For eg. we get an angle with measure 135 (by considering one 90 and one 45 angles adjacent to each other).

Construction of an angle with measure 60∘

Objectives

Understand construction of an an angle with measure 60∘. Problem Class 9.

Estimated Time

One period

Prerequisites/Instructions, prior preparations, if any

Introduction to Angles

Materials/ Resources needed

Geogebra file - Construction of angle with measure 60∘

From Geogebra.org

Process (How to do the activity)

  1. Explain the process of constructing an angle with measure 60∘ (without using a protractor). Students can follow and do the construction in their books
    1. Draw a circle with centre as point A, and with any given radius. Select a point B on the circle, and with the same radius, construct another circle.
    2. Mark the intersection points of the two circles, C and D
    3. Draw line segment AC.
    4. Measure ∡BAC, it will be 60∘
    5. Draw line segments AB and BC.
    6. Measure ∡ABC, ∡ACB.
    7. This process will create two more angles. Measure both angles. They will be 60∘ each
    8. Why does this process work? (Explanation - We have constructed a triangle whose all sides are congruent to one another. This is an equilateral triangle. Equilateral triangle also is an 'equiangular' triangle, where all angles are congruent to one another. The sum of measures of angles of a triangle is 180∘, and each angle is 60∘)
  2. You can use the 'Play - Construction protocol' to show the above steps one by one. But do this after constructing the equilateral triangle using Geogebra.

Evaluation at the end of the activity

  1. Ask students, what are the other angles that can drawn in the same construction. (Hint, there are new angles created, which are adjacent/ complementary / supplementary to the angles we have discussed. For eg. we get an angle with measure 120 (supplement of ∡ABC) if we extend segment AB beyond point B.