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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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An angle bisector is a ray that originates from the vertex of an angle and divides it into two equal parts. If ray OCOC bisects ∠AOB\angle AOB, then ∠AOC=∠COB\angle AOC = \angle COB.

An angle AOB divided into two equal parts by ray OC.
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To construct an angle bisector using a compass, we draw an arc from the vertex OO to intersect both arms at points PP and QQ. Then, from PP and QQ, we draw arcs with the same radius that intersect at a point RR in the interior of the angle.

Geometry construction showing intersection of arcs for angle bisector.
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The bisector is always the axis of symmetry for the angle, meaning if you fold the paper along the bisector, the two arms of the angle will overlap perfectly.

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If we bisect an angle and then bisect the resulting angles again, we divide the original angle into four equal 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.

Problem 3:

Given ∠PQR=120∘\angle PQR = 120^{\circ}, if ray QSQS is the bisector of ∠PQR\angle PQR, find the measure of ∠SQR\angle SQR.

Angle PQR of 120 degrees bisected into two 60 degree angles by ray QS.

Solution:

  1. Measure of the original angle ∠PQR=120∘\angle PQR = 120^{\circ}.
  2. Since QSQS is the bisector, it divides the angle into two equal parts.
  3. ∠SQR=12×∠PQR\angle SQR = \frac{1}{2} \times \angle PQR
  4. ∠SQR=12×120∘=60∘\angle SQR = \frac{1}{2} \times 120^{\circ} = 60^{\circ}

Explanation:

The angle bisector property states that each resulting angle is exactly half of the original angle's measure.

Problem 4:

In the figure, ∠AOB\angle AOB is a straight angle (180∘180^{\circ}). Ray OCOC is perpendicular to ABAB, and ray ODOD bisects ∠AOC\angle AOC. Find the measure of ∠DOB\angle DOB.

Straight angle AOB with perpendicular OC and bisector OD of angle AOC.

Solution:

  1. ∠AOC=90∘\angle AOC = 90^{\circ} (since OC⊥ABOC \perp AB).
  2. ∠COB=90∘\angle COB = 90^{\circ} (linear pair with ∠AOC\angle AOC).
  3. Since ODOD bisects ∠AOC\angle AOC, ∠DOC=90∘2=45∘\angle DOC = \frac{90^{\circ}}{2} = 45^{\circ}.
  4. ∠DOB=∠DOC+∠COB\angle DOB = \angle DOC + \angle COB
  5. ∠DOB=45∘+90∘=135∘\angle DOB = 45^{\circ} + 90^{\circ} = 135^{\circ}

Explanation:

We first find the half-measure of the right angle and then add it to the adjacent right angle to find the total measure.