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Lines and Angles - Comparing Angles

Grade 6CBSE

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

🔑Concepts

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Comparing angles by inspection: Sometimes we can tell which angle is larger just by looking at them. An angle with a wider opening between its two rays is the larger angle.

Comparison of two angles A and B where B has a wider opening than A.
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Comparison by superposition: To compare two angles exactly without a protractor, we can trace one angle on a piece of paper and place it over the other angle. If the rays of the traced angle fall inside the other, the traced angle is smaller.

One angle placed over another sharing a common vertex and base arm to compare size.
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Measuring with a Protractor: The most accurate way to compare angles is to measure their magnitude in degrees. A right angle is exactly 90∘90^\circ. If m∠P=60∘m\angle P = 60^\circ and m∠Q=110∘m\angle Q = 110^\circ, then ∠Q>∠P\angle Q > \angle P.

A protractor showing the 90 degree mark for measurement.
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Clock Hands as an Angle Comparator: The angle between the hands of a clock changes as time progresses. At 3 o'clock, the angle is 90∘90^\circ (Right Angle), while at 2 o'clock, the angle is smaller (60∘60^\circ, Acute Angle).

📐Formulae

Right Angle=90∘\text{Right Angle} = 90^\circ

Straight Angle=180∘\text{Straight Angle} = 180^\circ

Complete Angle=360∘\text{Complete Angle} = 360^\circ

Acute Angle<90∘\text{Acute Angle} < 90^\circ

90∘<Obtuse Angle<180∘90^\circ < \text{Obtuse Angle} < 180^\circ

💡Examples

Problem 1:

Compare ∠ABC\angle ABC and ∠XYZ\angle XYZ if m∠ABC=45∘m\angle ABC = 45^\circ and m∠XYZ=75∘m\angle XYZ = 75^\circ. Which one is larger?

Solution:

∠XYZ>∠ABC\angle XYZ > \angle ABC

Explanation:

By comparing the numerical values of the measures, we see that 75∘>45∘75^\circ > 45^\circ. Therefore, ∠XYZ\angle XYZ has a greater opening than ∠ABC\angle ABC.

Problem 2:

Identify which angle is larger: a right angle or an acute angle.

Solution:

Right Angle>Acute Angle\text{Right Angle} > \text{Acute Angle}

Explanation:

A right angle is exactly 90∘90^\circ, whereas an acute angle is always less than 90∘90^\circ. Thus, the right angle is always larger than any acute angle.

Problem 3:

Two angles are given. ∠1\angle 1 is part of a square's corner, and ∠2\angle 2 is 120∘120^\circ. Which is smaller?

Solution:

∠1<∠2\angle 1 < \angle 2

Explanation:

A corner of a square is a right angle, which measures 90∘90^\circ. Since 90∘<120∘90^\circ < 120^\circ, ∠1\angle 1 is the smaller angle.

Problem 4:

Compare the angles formed by the hands of a clock at 1:00 and 4:00. Which time shows a larger angle?

Two clocks showing 1:00 (acute angle) and 4:00 (obtuse angle).

Solution:

Angle at 1:00=30∘\text{Angle at 1:00} = 30^\circ Angle at 4:00=120∘\text{Angle at 4:00} = 120^\circ 120∘>30∘120^\circ > 30^\circ

Explanation:

On a clock, each hour mark represents an angle of 360∘÷12=30∘360^\circ \div 12 = 30^\circ. At 1:00, the hands are 1 hour mark apart (1×30∘=30∘1 \times 30^\circ = 30^\circ). At 4:00, the hands are 4 hour marks apart (4×30∘=120∘4 \times 30^\circ = 120^\circ). Therefore, the angle at 4:00 is larger.

Problem 5:

Given ∠LMN\angle LMN and ∠PQR\angle PQR, determine which is larger if ∠LMN\angle LMN is an obtuse angle and ∠PQR\angle PQR is a right angle.

Comparison of an obtuse angle LMN and a right angle PQR.

Solution:

Obtuse Angle>90∘\text{Obtuse Angle} > 90^\circ Right Angle=90∘\text{Right Angle} = 90^\circ Therefore, ∠LMN>∠PQR\text{Therefore, } \angle LMN > \angle PQR

Explanation:

By definition, a right angle measures exactly 90∘90^\circ, whereas an obtuse angle is any angle that measures more than 90∘90^\circ but less than 180∘180^\circ. Since any value greater than 90 is larger than 90, the obtuse angle is larger.