Is it possible for all the eight angles to have different measurements? Why, why not?

Question

Homi · Accepted Answer

Understand the Question

When two straight lines intersect, the vertically opposite angles formed at the point of intersection are always equal.
When a transversal intersects two lines, it forms two sets of four angles (eight angles in total).
Within each set of four angles, there are two pairs of vertically opposite angles that must be equal in measure, regardless of whether the two lines are parallel or not.

Step 1 · Identify Vertically Opposite Angle Pairs

Diagram 1

At the intersection of line $l$ and transversal $t$ :

$\angle 1$ and $\angle 3$ are vertically opposite angles.
$\angle 2$ and $\angle 4$ are vertically opposite angles.

At the intersection of line $m$ and transversal $t$ :

$\angle 5$ and $\angle 7$ are vertically opposite angles.
$\angle 6$ and $\angle 8$ are vertically opposite angles.

Step 2 · Apply the Vertically Opposite Angles Property

Since vertically opposite angles are always equal: $\angle 1 = \angle 3$ $\angle 2 = \angle 4$ $\angle 5 = \angle 7$ $\angle 6 = \angle 8$

Because at least four pairs of angles are equal, it is impossible for all eight angles to have different measurements.

Answer

No, it is not possible because vertically opposite angles are always equal ( $\angle 1 = \angle 3$ , $\angle 2 = \angle 4$ , $\angle 5 = \angle 7$ , and $\angle 6 = \angle 8$ ).

Common Mistakes

Assuming Parallelism is Required: Thinking that angles are only equal if the lines are parallel. While corresponding and alternate interior angles require parallel lines to be equal, vertically opposite angles are always equal for any two intersecting lines.
Counting Distinct Values: Assuming eight distinct angle positions can yield eight independent values, ignoring the intersecting line theorem.

Question 11

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