Practical Geometry - Class 7 Maths Chapter 8 Question Answer - Free Printable
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Step-by-step solution for: Practical Geometry - Class 7 Maths Chapter 8 Question Answer
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Show Answer Key & Explanations
Step-by-step solution for: Practical Geometry - Class 7 Maths Chapter 8 Question Answer
The image you provided illustrates the construction of various important angles using geometric tools such as a compass and straightedge. Below, I will explain how each angle is constructed step by step.
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- Construction:
- Draw a ray \( OA \).
- Place the compass at point \( O \) and draw an arc that intersects \( OA \) at a point, say \( P \).
- Without changing the compass width, place the compass at \( P \) and draw another arc that intersects the first arc at a point, say \( Q \).
- Draw a ray \( OB \) passing through \( Q \). The angle \( \angle AOB \) is now \( 60^\circ \).
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- Construction:
- Start with \( \angle AOB = 60^\circ \) (constructed as above).
- Bisect \( \angle AOB \):
- Place the compass at \( O \) and draw an arc intersecting \( OA \) and \( OB \) at points \( P \) and \( Q \), respectively.
- Place the compass at \( P \) and draw an arc.
- Place the compass at \( Q \) and draw another arc intersecting the previous arc at a point, say \( R \).
- Draw a ray \( OC \) passing through \( R \). The angle \( \angle AOC \) is now \( 30^\circ \).
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- Construction:
- Start with \( \angle AOC = 30^\circ \) (constructed as above).
- Bisect \( \angle AOC \):
- Place the compass at \( O \) and draw an arc intersecting \( OA \) and \( OC \) at points \( P \) and \( Q \), respectively.
- Place the compass at \( P \) and draw an arc.
- Place the compass at \( Q \) and draw another arc intersecting the previous arc at a point, say \( S \).
- Draw a ray \( OD \) passing through \( S \). The angle \( \angle AOD \) is now \( 15^\circ \).
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- Construction:
- Draw a ray \( OA \).
- Construct \( \angle AOB = 60^\circ \) as described earlier.
- Construct another \( \angle BOF = 60^\circ \) adjacent to \( \angle AOB \).
- The ray \( OE \) coincides with \( OF \). The angle \( \angle AOE \) is now \( 120^\circ \).
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- Construction:
- Draw a ray \( OA \).
- Extend the ray \( OA \) in the opposite direction to form ray \( OF \). The angle \( \angle AOF \) is now \( 180^\circ \).
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- Construction:
- Draw a ray \( OA \).
- Place the compass at \( O \) and draw an arc intersecting \( OA \) at a point, say \( P \).
- Place the compass at \( P \) and draw two arcs intersecting the first arc at points \( Q \) and \( R \).
- Place the compass at \( Q \) and draw an arc.
- Place the compass at \( R \) and draw another arc intersecting the previous arc at a point, say \( S \).
- Draw a ray \( OG \) passing through \( S \). The angle \( \angle AOG \) is now \( 90^\circ \).
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- Construction:
- Start with \( \angle AOG = 90^\circ \) (constructed as above).
- Bisect \( \angle AOG \):
- Place the compass at \( O \) and draw an arc intersecting \( OA \) and \( OG \) at points \( P \) and \( Q \), respectively.
- Place the compass at \( P \) and draw an arc.
- Place the compass at \( Q \) and draw another arc intersecting the previous arc at a point, say \( R \).
- Draw a ray \( OH \) passing through \( R \). The angle \( \angle AOH \) is now \( 45^\circ \).
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- Construction:
- Start with \( \angle AOH = 45^\circ \) (constructed as above).
- Bisect \( \angle AOH \):
- Place the compass at \( O \) and draw an arc intersecting \( OA \) and \( OH \) at points \( P \) and \( Q \), respectively.
- Place the compass at \( P \) and draw an arc.
- Place the compass at \( Q \) and draw another arc intersecting the previous arc at a point, say \( S \).
- Draw a ray \( OI \) passing through \( S \). The angle \( \angle AOI \) is now \( 22.5^\circ \).
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- Construction:
- Start with \( \angle AOB = 60^\circ \) (constructed as above).
- Construct \( \angle BOG = 15^\circ \) adjacent to \( \angle AOB \) (using the \( 15^\circ \) angle construction).
- The ray \( OJ \) coincides with \( OG \). The angle \( \angle AOJ \) is now \( 75^\circ \).
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- Construction:
- Start with \( \angle AOG = 90^\circ \) (constructed as above).
- Bisect \( \angle AOG \) to get \( \angle AOH = 45^\circ \).
- Bisect \( \angle HOG \) to get \( \angle AOK = 37.5^\circ \).
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- Construction:
- Start with \( \angle AOE = 120^\circ \) (constructed as above).
- Construct \( \angle EOL = 30^\circ \) adjacent to \( \angle AOE \).
- The ray \( OL \) is now such that \( \angle AOL = 150^\circ \).
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- Construction:
- Start with \( \angle AOG = 90^\circ \) (constructed as above).
- Construct \( \angle GOM = 45^\circ \) adjacent to \( \angle AOG \).
- The ray \( OM \) is now such that \( \angle AOM = 135^\circ \).
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Each angle is constructed using basic geometric principles such as bisecting angles and combining known angles. The constructions are systematic and rely on the properties of circles and arcs.
\[
\boxed{\text{All angles are constructed as explained above.}}
\]
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1. ∠AOB = 60°
- Construction:
- Draw a ray \( OA \).
- Place the compass at point \( O \) and draw an arc that intersects \( OA \) at a point, say \( P \).
- Without changing the compass width, place the compass at \( P \) and draw another arc that intersects the first arc at a point, say \( Q \).
- Draw a ray \( OB \) passing through \( Q \). The angle \( \angle AOB \) is now \( 60^\circ \).
---
2. ∠AOC = 30°
- Construction:
- Start with \( \angle AOB = 60^\circ \) (constructed as above).
- Bisect \( \angle AOB \):
- Place the compass at \( O \) and draw an arc intersecting \( OA \) and \( OB \) at points \( P \) and \( Q \), respectively.
- Place the compass at \( P \) and draw an arc.
- Place the compass at \( Q \) and draw another arc intersecting the previous arc at a point, say \( R \).
- Draw a ray \( OC \) passing through \( R \). The angle \( \angle AOC \) is now \( 30^\circ \).
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3. ∠AOD = 15°
- Construction:
- Start with \( \angle AOC = 30^\circ \) (constructed as above).
- Bisect \( \angle AOC \):
- Place the compass at \( O \) and draw an arc intersecting \( OA \) and \( OC \) at points \( P \) and \( Q \), respectively.
- Place the compass at \( P \) and draw an arc.
- Place the compass at \( Q \) and draw another arc intersecting the previous arc at a point, say \( S \).
- Draw a ray \( OD \) passing through \( S \). The angle \( \angle AOD \) is now \( 15^\circ \).
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4. ∠AOE = 120°
- Construction:
- Draw a ray \( OA \).
- Construct \( \angle AOB = 60^\circ \) as described earlier.
- Construct another \( \angle BOF = 60^\circ \) adjacent to \( \angle AOB \).
- The ray \( OE \) coincides with \( OF \). The angle \( \angle AOE \) is now \( 120^\circ \).
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5. ∠AOF = 180°
- Construction:
- Draw a ray \( OA \).
- Extend the ray \( OA \) in the opposite direction to form ray \( OF \). The angle \( \angle AOF \) is now \( 180^\circ \).
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6. ∠AOG = 90°
- Construction:
- Draw a ray \( OA \).
- Place the compass at \( O \) and draw an arc intersecting \( OA \) at a point, say \( P \).
- Place the compass at \( P \) and draw two arcs intersecting the first arc at points \( Q \) and \( R \).
- Place the compass at \( Q \) and draw an arc.
- Place the compass at \( R \) and draw another arc intersecting the previous arc at a point, say \( S \).
- Draw a ray \( OG \) passing through \( S \). The angle \( \angle AOG \) is now \( 90^\circ \).
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7. ∠AOH = 45°
- Construction:
- Start with \( \angle AOG = 90^\circ \) (constructed as above).
- Bisect \( \angle AOG \):
- Place the compass at \( O \) and draw an arc intersecting \( OA \) and \( OG \) at points \( P \) and \( Q \), respectively.
- Place the compass at \( P \) and draw an arc.
- Place the compass at \( Q \) and draw another arc intersecting the previous arc at a point, say \( R \).
- Draw a ray \( OH \) passing through \( R \). The angle \( \angle AOH \) is now \( 45^\circ \).
---
8. ∠AOI = 22½°
- Construction:
- Start with \( \angle AOH = 45^\circ \) (constructed as above).
- Bisect \( \angle AOH \):
- Place the compass at \( O \) and draw an arc intersecting \( OA \) and \( OH \) at points \( P \) and \( Q \), respectively.
- Place the compass at \( P \) and draw an arc.
- Place the compass at \( Q \) and draw another arc intersecting the previous arc at a point, say \( S \).
- Draw a ray \( OI \) passing through \( S \). The angle \( \angle AOI \) is now \( 22.5^\circ \).
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9. ∠AOJ = 75°
- Construction:
- Start with \( \angle AOB = 60^\circ \) (constructed as above).
- Construct \( \angle BOG = 15^\circ \) adjacent to \( \angle AOB \) (using the \( 15^\circ \) angle construction).
- The ray \( OJ \) coincides with \( OG \). The angle \( \angle AOJ \) is now \( 75^\circ \).
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10. ∠AOK = 37½°
- Construction:
- Start with \( \angle AOG = 90^\circ \) (constructed as above).
- Bisect \( \angle AOG \) to get \( \angle AOH = 45^\circ \).
- Bisect \( \angle HOG \) to get \( \angle AOK = 37.5^\circ \).
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11. ∠AOL = 150°
- Construction:
- Start with \( \angle AOE = 120^\circ \) (constructed as above).
- Construct \( \angle EOL = 30^\circ \) adjacent to \( \angle AOE \).
- The ray \( OL \) is now such that \( \angle AOL = 150^\circ \).
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12. ∠AOM = 135°
- Construction:
- Start with \( \angle AOG = 90^\circ \) (constructed as above).
- Construct \( \angle GOM = 45^\circ \) adjacent to \( \angle AOG \).
- The ray \( OM \) is now such that \( \angle AOM = 135^\circ \).
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Final Answer:
Each angle is constructed using basic geometric principles such as bisecting angles and combining known angles. The constructions are systematic and rely on the properties of circles and arcs.
\[
\boxed{\text{All angles are constructed as explained above.}}
\]
Parent Tip: Review the logic above to help your child master the concept of geometry construction practice worksheet.
