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Vertical angles

 

The intersection of lines forms vertical angles. The intersecting line can be a straight line crossing a parallel line or a set of two perpendicular lines. Parallel lines are those lines that never intersect, and the distance between them remains the same forever. Perpendicular lines are those lines that cross each other precisely at 90 degrees.

Properties of vertical angles

  • Adjacent angles are said to be supplementary if their sum is equal to 180°.
  • Opposite angles are said to be complementary if their sum is equal to 90°.
  • The sum of all angles of a vertical angle is equal to 360°.
  • Vertical angles are always congruent, i.e., the opposite angles measure equal. It is commonly known as the Vertical Angle Theorem.

Example 1: - Find the rest of the angles in the given figure.

Solution 1 

Solution: - We know that the opposite angles measure equal. 

Therefore, angle b is equal to 40°. 

Angle c and angle a must be equal by the same property. 

Therefore, angle c = angle a. 

We know, the sum of angles of a vertical angle is equal to 360°. 

This implies, angle a + angle b + angle c + 40° = 360° 

Also, angle c = angle a 

And, angle b = 40°

Therefore, angle a + 40° + 40° angle a = 360°

This gives angle a = 140° 

Angle c is also equal to 140° because it is a vertically opposite angle to angle a. 

Example 2: - Find angles x, y and z as shown in the figure given that angle x is right angled. 

Solution 2

Solution: - We know that the angle formed on a straight line is equal to 180°. 

Therefore, angle z + 140° = 180°

This gives, angle z = 40°. 

Also, angle x + angle y + angle z = 180° (straight line property) 

Angle x is equal to 90° (right angled) 

This implies, 90° + 40° + angle y = 180° 

Therefore, angle y = 50°. 

 

Applications of vertical angles

  • Vertical angles help design railway tracks, crossovers, and roller coaster rides. The maximum angle allowed for these crossovers is 112° to prevent accidents and reduce wear and tear of railway carriages.
  • The kite is made to not tear off at vast heights, even at high wind pressures. The two wooden sticks form vertical angles.
  • The bull's eye on a dartboard is a vertical vertex of ten vertical angles.
  • Scissors and pliers are made concerning vertical angles so that they cut equally.

Fun facts:

Vertical angles were used by ancient astronomers to plot their telescopes at correct angles so that they can have a full view of the entire sky. The telescope's rotation used to be in such a way that the angle remained equal if turned on any side of the sky.

Today's lighthouses also have the same concept to guide the ships and sailors that come near the seashore.

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