Prove Angle Relationships Formed by Parallel Lines | Geometry | LevelKite

You have already used parallel-line angle relationships to find angle measures. Now you will prove that those relationships are true. A proof is a chain of statements that leads from what you are told to what you want to show, with a reason for every statement.

Your chain starts with a postulate, a statement accepted without proof. The Corresponding Angles Postulate says: if two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent. A transversal is a line that crosses two other lines at two different points. Corresponding angles sit in the same position at each crossing.

You also have three theorems, which are statements already proved: the Vertical Angles Theorem, the Supplement Theorem (a linear pair is supplementary, so its measures add to $180^\circ$ ) and the Congruent Supplements Theorem (angles supplementary to the same angle are congruent).

Key concept

Named reasons, in the right direction

A proof moves from the Given (what you are told) to the Prove (what you must show). In a two-column proof, each row pairs a statement with its reason: the word Given or the exact name of a definition, postulate, property of equality or congruence, or proved theorem. How a figure looks is never a reason, and neither is the conclusion.

Check the direction first. Theorem direction: the two lines are given parallel, so you conclude an angle relationship. Converse direction: an angle condition is given (congruent corresponding, alternate interior or alternate exterior angles, or supplementary consecutive interior angles), so you conclude the lines are parallel. A converse swaps the if and then parts of a statement, so the directions are different claims.

Your first proof uses the figure below, which is not drawn to scale. Line $AB$ , written $\overleftrightarrow{AB}$ , and line $CD$ are cut by transversal $\overleftrightarrow{EH}$ . Point $F$ lies between $A$ and $B$ , point $G$ lies between $C$ and $D$ , and $E$ , $F$ , $G$ , $H$ lie in that order on the transversal. Points $A$ and $C$ are on one side of the transversal, and points $B$ and $D$ are on the other.

The arrows on the two lines state the given fact: $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ . The figure also marks $\angle AFG$ , $\angle EFB$ , $\angle BFG$ and $\angle FGD$ , the angles you need for the next two proofs. A marked angle only shows you where to look. It does not mean that angle equals any other angle.

The arrows give

\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}

. The marked angles show where to look, not which angles are equal.

Worked example

Alternate interior angles are congruent

The Alternate Interior Angles Theorem says: if two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent. Alternate interior angles lie between the two lines, on opposite sides of the transversal. Given: $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ , cut by transversal $\overleftrightarrow{EH}$ at $F$ and $G$ . Prove: $\angle AFG \cong \angle FGD$ .

Start with what you are told. The lines are given parallel, so you are working in the theorem direction. Reason: Given.
$\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}, \text{ cut by } \overleftrightarrow{EH}$
The postulate does not link angle AFG and angle FGD directly, so plan to use angle EFB as a bridge, an angle connected to both. Rays FA and FB are opposite, and rays FG and FE are opposite, so angle AFG and angle EFB are vertical angles. Reason: Vertical Angles Theorem.
$\angle AFG \cong \angle EFB$
Points B and D are on the same side of the transversal, and rays FE and GF point the same way along it. So angle EFB and angle FGD sit in the same position at F and at G, which makes them corresponding angles. The lines are given parallel, so the postulate applies. Reason: Corresponding Angles Postulate.
$\angle EFB \cong \angle FGD$
Both target angles are now congruent to the bridge angle, so join the last two statements. Reason: Transitive Property of Congruence.
$\angle AFG \cong \angle FGD$

Answer

Because the lines are given parallel, $\angle AFG \cong \angle FGD$ . Matching steps with $\angle EFA$ as the bridge show $\angle BFG \cong \angle FGC$ , so you have proved the Alternate Interior Angles Theorem.

Two-column proof of the Alternate Interior Angles Theorem for this pair

Statement	Reason
$\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ , cut by $\overleftrightarrow{EH}$	Given
$\angle AFG \cong \angle EFB$	Vertical Angles Theorem
$\angle EFB \cong \angle FGD$	Corresponding Angles Postulate
$\angle AFG \cong \angle FGD$	Transitive Property of Congruence

Worked example

Consecutive interior angles are supplementary

Use the first figure again. The Consecutive Interior Angles Theorem says: if two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary. Consecutive interior angles lie between the two lines, on the same side of the transversal. Given: $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ , cut by transversal $\overleftrightarrow{EH}$ at $F$ and $G$ . Prove: $\angle BFG$ and $\angle FGD$ are supplementary.

The lines are given parallel, so this is the theorem direction again. Reason: Given.
$\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}, \text{ cut by } \overleftrightarrow{EH}$
You just proved this theorem, so you may now use it as a reason. Its condition holds because the lines are given parallel. Reason: Alternate Interior Angles Theorem.
$\angle AFG \cong \angle FGD$
Congruent angles have equal measures. Reason: Definition of congruent angles.
$m\angle AFG = m\angle FGD$
Rays FA and FB are opposite and the angles share ray FG, so they form a linear pair. Reason: Supplement Theorem.
$\angle AFG \text{ and } \angle BFG \text{ are supplementary}$
Supplementary angles have measures that add to 180 degrees. Reason: Definition of supplementary angles.
$m\angle AFG + m\angle BFG = 180^\circ$
Replace the measure of angle AFG with the equal measure of angle FGD. Reason: Substitution Property of Equality.
$m\angle FGD + m\angle BFG = 180^\circ$
Two angles whose measures add to 180 degrees are supplementary. Reason: Definition of supplementary angles.
$\angle BFG \text{ and } \angle FGD \text{ are supplementary}$

Answer

Because the lines are given parallel, $\angle BFG$ and $\angle FGD$ are supplementary. Matching steps prove the other pair, $\angle AFG$ and $\angle FGC$ , so you have proved the Consecutive Interior Angles Theorem.

Two-column proof of the Consecutive Interior Angles Theorem for this pair

Statement	Reason
$\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ , cut by $\overleftrightarrow{EH}$	Given
$\angle AFG \cong \angle FGD$	Alternate Interior Angles Theorem
$m\angle AFG = m\angle FGD$	Definition of congruent angles
$\angle AFG$ and $\angle BFG$ are supplementary	Supplement Theorem
$m\angle AFG + m\angle BFG = 180^\circ$	Definition of supplementary angles
$m\angle FGD + m\angle BFG = 180^\circ$	Substitution Property of Equality
$\angle BFG$ and $\angle FGD$ are supplementary	Definition of supplementary angles

Now switch to the converse direction. This course also accepts the Converse of the Corresponding Angles Postulate: if two lines are cut by a transversal so that a pair of corresponding angles is congruent, then the lines are parallel.

The figure below uses the same points in the same order, but it has no arrows, because you are not told that the lines are parallel. They may look parallel, but that is not a reason. The only given is an angle condition: $\angle BFG$ and $\angle FGD$ are supplementary. The figure marks those two angles and $\angle EFB$ , which will be your bridge.

No arrows:

\overleftrightarrow{AB}

and

\overleftrightarrow{CD}

are not given parallel, so the proof must show it.

Worked example

Proving lines parallel with a converse

The Consecutive Interior Angles Converse says: if two lines are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the lines are parallel. Transversal $\overleftrightarrow{EH}$ cuts $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ at $F$ and $G$ . Given: $\angle BFG$ and $\angle FGD$ are supplementary. Prove: $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ .

Only an angle condition is given, so this is the converse direction. You cannot use a fact that needs the lines to be parallel. Reason: Given.
$\angle BFG \text{ and } \angle FGD \text{ are supplementary}$
The converse postulate needs congruent corresponding angles, so aim for angle EFB and angle FGD. Rays FE and FG are opposite and the angles share ray FB, so angle EFB and angle BFG form a linear pair. Reason: Supplement Theorem.
$\angle EFB \text{ and } \angle BFG \text{ are supplementary}$
Angle EFB and angle FGD are both supplementary to the same angle, angle BFG. Reason: Congruent Supplements Theorem.
$\angle EFB \cong \angle FGD$
Angle EFB and angle FGD are corresponding angles, and they are now shown congruent. That is the condition the converse needs, so you may conclude the lines are parallel. Reason: Converse of the Corresponding Angles Postulate.
$\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$

Answer

Because $\angle BFG$ and $\angle FGD$ are supplementary, $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$ . Matching steps work for the other pair, so you have proved the Consecutive Interior Angles Converse.

Two-column proof of the Consecutive Interior Angles Converse for this pair

Statement	Reason
$\angle BFG$ and $\angle FGD$ are supplementary	Given
$\angle EFB$ and $\angle BFG$ are supplementary	Supplement Theorem
$\angle EFB \cong \angle FGD$	Congruent Supplements Theorem
$\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$	Converse of the Corresponding Angles Postulate

Takeaway: A proof is a chain of named reasons from the Given to the Prove. When two lines are given parallel, use the Corresponding Angles Postulate or a theorem you have proved to conclude an angle relationship. When only an angle condition is given, build toward a converse to conclude that the lines are parallel. Next, you will use these relationships to solve for a variable and to identify mistakes in parallel-line angle problems.