Proving Theorems about Lines and Angles

In Parallel Lines Cut by a Transversal you saw that parallel lines cut by a transversal make pairs of equal angles. Now you will prove those facts with the tools from Writing a Proof, and also prove their converses, which run the other way.

Vertical angles

Vertical Angles Theorem: vertical angles are congruent. Proof: When two lines cross, $\angle 1$ and $\angle 2$ form a linear pair, so $m\angle 1 + m\angle 2 = 180^\circ$. Also $\angle 2$ and $\angle 3$ form a linear pair, so $m\angle 2 + m\angle 3 = 180^\circ$. By substitution $m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3$, and subtracting $m\angle 2$ gives $m\angle 1 = m\angle 3$.

Parallel lines

Por qué funciona

We accept one fact about parallel lines as a postulate: if two parallel lines are cut by a transversal, corresponding angles are congruent, along with its converse. Everything else follows. For example, an alternate interior angle is vertical to a corresponding angle, so it is congruent too: the Alternate Interior Angles Theorem is two short steps from the postulate.

Drag the transversal and switch between the pairs. With parallel lines, each pair is either congruent or supplementary.

Angle pairs formed by two parallel lines and a transversal.
Pair	Where the angles sit	With parallel lines
Corresponding	same position at each crossing	congruent
Alternate interior	between the lines, opposite sides of the transversal	congruent
Alternate exterior	outside the lines, opposite sides	congruent
Same-side interior	between the lines, same side	supplementary

Ejemplo resuelto: Proving the Alternate Interior Angles Theorem

Given: $\ell \parallel m$, cut by transversal $t$. Prove: $\angle 3 \cong \angle 6$ (alternate interior angles).

1. $\ell \parallel m$ — Given.
2. $\angle 2 \cong \angle 6$ — Corresponding Angles Postulate (both are above their line, right of $t$).
3. $\angle 2 \cong \angle 3$ — Vertical Angles Theorem.
4. $\angle 3 \cong \angle 6$ — Transitive Property of Congruence.

Respuesta: Alternate interior angles are congruent.

Ejemplo resuelto: Using a converse

Now suppose we do not know whether $\ell \parallel m$. In the same numbering, $m\angle 3 = (5x - 12)^\circ$ and $m\angle 6 = (3x + 20)^\circ$. For what $x$ is $\ell \parallel m$?

If alternate interior angles are congruent, the lines are parallel (the converse). So set them equal: $5x - 12 = 3x + 20$.
$2x = 32$, so $x = 16$. Then both angles measure $68^\circ$.

Respuesta: $x = 16$ makes $\ell \parallel m$.

Ejemplo resuelto: An angle chase

$\ell \parallel m$ and $m\angle 1 = 118^\circ$. Find $m\angle 6$.

$\angle 1$ and $\angle 5$ are corresponding, so $m\angle 5 = 118^\circ$.
$\angle 5$ and $\angle 6$ form a linear pair: $m\angle 6 = 180 - 118 = 62^\circ$.

Respuesta: $62^\circ$. The angles $\angle 1$ and $\angle 6$ are not one of the named pairs, so the chase took two steps.

Idea clave

Parallel lines give congruent corresponding, alternate interior and alternate exterior angles, and supplementary same-side interior angles. The converses say the reverse: if one of those angle facts holds, the lines are parallel.

Error común

Running the logic backward. To prove two lines are parallel, you cannot use "parallel lines make congruent alternate interior angles": that assumes what you want. Use the converse: "if alternate interior angles are congruent, then the lines are parallel." Check by asking which part you know (angles or lines) and which part you want.

Vertical angles

Parallel lines

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