Question

How many different angles would be formed by a transversal intersecting three parallel lines? How many different angle measures would there be?

Answer 1

Hint: In geometry, a transversal is a line that intersects two or more parallel lines.

Here we need to find the different angles that would be formed by a transversal intersecting three parallel lines. We get three intersecting points if the transversal passes through three parallel lines.

Complete step by step solution:

At each intersection four angles will be formed.

From the given condition we get three intersecting points. So the total number of angles formed will be:

\[\Rightarrow \]3\[\times \]4 angles.

\[\Rightarrow \]12 angles.

So, the total number of different angles that are formed by a transversal intersecting three parallel lines is 12.

The diagram of a transversal line intersecting three parallel lines:

Here a, b, c are parallel lines and d is the transversal line.

Angles are \[\angle 1,\angle 2,\angle 3,\angle 4,\angle 5,\angle 6,\angle 7,\angle 8,\angle 9,\angle 10,\angle 11\] and \[\angle 12\].

At the first intersection point(intersection of parallel line a and transversal d) four angles are formed. 

We know that from the concept of geometry, the sum of adjacent angles is \[{{180}^{\circ }}\] and the adjacent angles are equal.

\[\Rightarrow \]\[\angle 1\]+\[\angle 2\]=\[{{180}^{\circ }}\] , \[\angle 3\]+\[\angle 4\]=\[{{180}^{\circ }}\],\[\angle 1\]=\[\angle 4\] and \[\angle 2\]=\[\angle 3\].

Similarly, at second intersection point we get:

\[\Rightarrow \]\[\angle 5\]+\[\angle 6\]=\[{{180}^{\circ }}\] , \[\angle 7\]+\[\angle 8\]=\[{{180}^{\circ }}\],\[\angle 5\] =\[\angle 8\] and \[\angle 6\]=\[\angle 7\].

Similarly, at third intersection point we get:

\[\Rightarrow \]\[\angle 9\]+\[\angle 10\]=\[{{180}^{\circ }}\] , \[\angle 11\]+\[\angle 12\]=\[{{180}^{\circ }}\], \[\angle 9\]=\[\angle 12\] and \[\angle 10\]=\[\angle 11\].

From the concept of geometry, 

If the transversal line is passed through the parallel lines their corresponding angles are equal. So,

\[\Rightarrow \]\[\angle 1\]=\[\angle 5\]=\[\angle 9\], \[\angle 2\]=\[\angle 6\]=\[\angle 10\], \[\angle 3\]=\[\angle 7\]=\[\angle 11\], \[\angle 4\]=\[\angle 8\]=\[\angle 12\].

From the above all equations we get,

\[\Rightarrow \]\[\angle 1\]=\[\angle 4\]=\[\angle 5\]=\[\angle 8\]=\[\angle 9\]=\[\angle 12\].

\[\Rightarrow \]\[\angle 2\]=\[\angle 3\]=\[\angle 6\]=\[\angle 7\]=\[\angle 10\]=\[\angle 11\].

Hence it is concluded that 12 different angles are formed and only 2 different angle measures will be there by a transversal intersecting three parallel lines.

Note: Students must know the basic concepts of geometry like the sum of adjacent angles is \[{{180}^{\circ }}\] and the opposite angles are equal. If a transversal intersects the two or more parallel lines then their corresponding angles are equal.