What each fact hides
Is "corresponding angles are equal" always true?
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The rule
No. It is true when the two lines are parallel, and not otherwise.
Ask what a fact needs before you use it.
- Always true. The fact needs no extra condition. Vertically opposite angles are equal. A linear pair totals 180°.
- True on parallel lines. Corresponding and alternate angles are equal. Co-interior angles total 180°.
- False as stated. One drawing can show it is false. Two angles that total 180° need not be a linear pair.
| Claim | Verdict | What decides it |
|---|---|---|
| Vertically opposite angles are equal. | Always true | Two linear pairs. |
| Corresponding angles are equal. | On parallel lines | The axiom needs parallel lines. |
| Equal alternate angles make lines parallel. | Always true | Equal corresponding angles. |
| Two angles that total 180° form a linear pair. | False | 80° and 100° drawn apart. |
| On parallel lines, co-interior angles are equal. | False | They total 180°. They are equal only at 90° each. |
| Lines that look parallel are parallel. | False | Only equal angles show it. |
This chapter starts from two facts. First, a straight angle is 180°. Second, a transversal cuts two lines. The lines are parallel exactly when corresponding angles are equal.