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

Question

Homi · Accepted Answer

Vertically opposite angles are always equal.

Step 1 — Identify angle relationships

Let us look at the diagram. Line $l$ and transversal $t$ cross. They form angles 1, 2, 3, and 4. Angles 1 and 3 are vertically opposite. Angles 2 and 4 are vertically opposite.

Line $m$ and transversal $t$ cross. They form angles 5, 6, 7, and 8. Angles 5 and 7 are vertically opposite. Angles 6 and 8 are vertically opposite.

Diagram 1

Step 2 — Explain why angles cannot be different

Vertically opposite angles are always equal. This is a basic geometry rule. So, angle 1 must be equal to angle 3. This means $\angle 1$ and $\angle 3$ cannot be different. Similarly, $\angle 2$ must be equal to $\angle 4$ . Also, $\angle 5$ must be equal to $\angle 7$ . And $\angle 6$ must be equal to $\angle 8$ . Not all eight angles can have different measurements.

Answer

(i) No. (ii) Vertically opposite angles are always equal. For example, angle 1 and angle 3 are vertically opposite. So, they must have the same measure. This means they cannot be different. The same is true for other pairs. These pairs are angle 2 and angle 4. Also angle 5 and angle 7. And angle 6 and angle 8.

Question 11

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