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Practical Geometry - Bisector of an Angle

Grade 6CBSE

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

🔑Concepts

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Definition of an Angle Bisector: An angle bisector is a ray that originates from the vertex of an angle and divides it into two smaller angles of equal measure. Visually, if you have an angle shaped like a 'V', the bisector is a straight line that cuts exactly through the center of that 'V'.

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Symmetry Property: The bisector of an angle acts as its line of symmetry. If you were to fold a piece of paper along the bisector ray, the two arms of the angle would lie perfectly on top of each other. This shows that every point on the bisector is equidistant from the two arms of the angle.

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Initial Construction Step: To construct a bisector for an angle ∠AOB\angle AOB, place the compass pointer at the vertex OO and draw an arc that intersects both rays OAOA and OBOB. Label these intersection points PP and QQ. This ensures that both points are the same distance from the vertex.

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Finding the Mid-Point Intersection: From points PP and QQ, use a compass to draw two arcs with the same radius inside the opening of the angle. These arcs should cross each other to form an 'X' shape. Let this intersection point be RR. This point RR is visually positioned exactly halfway between the two arms.

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The Resulting Ray: Draw a straight line using a ruler from the vertex OO through the intersection point RR. The ray OROR is the bisector. This ray effectively splits the original wide opening into two identical narrower openings.

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Mathematical Relationship: If a ray OCOC bisects ∠AOB\angle AOB, then the measures of the resulting angles are equal: ∠AOC=∠BOC\angle AOC = \angle BOC. The total measure of the original angle is always twice the measure of either of the two bisected parts.

📐Formulae

Measure of each bisected angle=12×(Measure of the original angle)\text{Measure of each bisected angle} = \frac{1}{2} \times (\text{Measure of the original angle})

∠AOB=∠AOC+∠COB\angle AOB = \angle AOC + \angle COB

If OC is the bisector, then ∠AOC=∠COB=∠AOB2\text{If } OC \text{ is the bisector, then } \angle AOC = \angle COB = \frac{\angle AOB}{2}

∠AOB=2×∠AOC\angle AOB = 2 \times \angle AOC

💡Examples

Problem 1:

If ∠XYZ=90∘\angle XYZ = 90^{\circ} and ray YMYM is its angle bisector, find the measure of ∠XYM\angle XYM.

Solution:

  1. Identify the given total angle: ∠XYZ=90∘\angle XYZ = 90^{\circ}.
  2. Since YMYM is the bisector, it divides the angle into two equal parts.
  3. Use the formula: ∠XYM=12×∠XYZ\angle XYM = \frac{1}{2} \times \angle XYZ.
  4. Substitute the value: ∠XYM=12×90∘=45∘\angle XYM = \frac{1}{2} \times 90^{\circ} = 45^{\circ}.
  5. Therefore, ∠XYM=45∘\angle XYM = 45^{\circ}.

Explanation:

The problem applies the fundamental property that an angle bisector divides the total angle into two equal halves.

Problem 2:

A student constructs a bisector ADAD for ∠BAC\angle BAC. If the measure of ∠BAD\angle BAD is found to be 22.5∘22.5^{\circ}, what was the original measure of ∠BAC\angle BAC?

Solution:

  1. We are given the measure of one of the bisected parts: ∠BAD=22.5∘\angle BAD = 22.5^{\circ}.
  2. Since ADAD is the bisector, the whole angle is twice the measure of one part.
  3. Use the formula: ∠BAC=2×∠BAD\angle BAC = 2 \times \angle BAD.
  4. Calculate: ∠BAC=2×22.5∘=45∘\angle BAC = 2 \times 22.5^{\circ} = 45^{\circ}.
  5. The original angle ∠BAC\angle BAC measured 45∘45^{\circ}.

Explanation:

This example demonstrates how to find the total angle when the measure of a bisected part is known by multiplying by 22.