Write a Two-Column Proof About Angles | Geometry | LevelKite

You have already written two-column proofs about segments. A proof about angles has the same shape: you list statements in order and name the reason for each one. The finished proof holds for every figure that fits the Given, whatever the actual angle measures are. That is why you work with names and rules, not with numbers from a picture.

Angle proofs use two kinds of statement. $\angle ABC \cong \angle DEF$ says the angles are congruent, which means they have equal measures. $m\angle ABC = m\angle DEF$ says their measures, the numbers of degrees in them, are equal. You can add and subtract measures because they are numbers, but you can't add the angles themselves. So a proof often moves from angles to measures, does the arithmetic, and then moves back.

Key concept

A proof is a chain of named reasons

A two-column proof starts with the Given and ends with the Prove. Every row in between needs one reason, and it may use only what the rows above it have shown. A reason is the exact name of a definition, a postulate, a property of equality or congruence, or a theorem already proved, or else the word Given. A step can never rest on its own conclusion, on a fact nobody stated, or on how the figure looks.

Here are the main reasons this lesson uses, each with the condition it needs.

Definition of congruent angles: two angles are congruent if and only if their measures are equal. You can use it in either direction.
Angle Addition Postulate: if $D$ is in the interior of $\angle ABC$ , which means between its two rays, then $m\angle ABD + m\angle DBC = m\angle ABC$ .
Vertical Angles Theorem: if two angles are vertical angles, the pair of opposite angles formed when two lines intersect, then they are congruent.
Supplement Theorem: if two angles form a linear pair, meaning they share a vertex and a side and their other sides are opposite rays, then they are supplementary, meaning their measures add to $180^\circ$ .
Congruent Supplements Theorem: if two angles are supplementary to the same angle or to congruent angles, then they are congruent.

Each theorem runs in the direction written. Its converse is a different claim, so you can't assume it.

Before you fill in a proof, make a plan: compare the Given with the Prove and look for the missing link. Then write the proof row by row.

Your first proof uses the figure below. It shows lines $AC$ and $BD$ crossing at $E$ , plus a separate angle, $\angle FGH$ , formed by rays $GF$ and $GH$ . In any figure, only tick, arrow and right-angle marks and the facts stated in the text count as givens. This figure has no such marks and is not drawn to scale, so every fact about its angles, including which ones are equal, comes from the text.

Lines

AC

and

BD

cross at

E

, and rays

GF

and

GH

form

\angle FGH

. The angle facts for each proof are stated in the text.

Worked example

Example 1: Chain two congruences

Given: $\angle AED$ and $\angle BEC$ are vertical angles, and $\angle AED \cong \angle FGH$ . Prove: $\angle BEC \cong \angle FGH$ .

Reason: Given. Start with the stated facts. This one tells you the Vertical Angles Theorem is available.
$\angle AED \text{ and } \angle BEC \text{ are vertical angles}$
Reason: Given. This links angle AED to angle FGH, but the Prove needs angle BEC. So your plan is to link angle BEC to angle AED, then join the two links.
$\angle AED \cong \angle FGH$
Reason: Vertical Angles Theorem. Step 1 meets the theorem's condition, so these angles are congruent. The theorem goes only from vertical to congruent: angle AED and angle FGH are congruent, but they have different vertices and are not vertical angles.
$\angle BEC \cong \angle AED$
Reason: Transitive Property of Congruence. Step 3 says angle BEC is congruent to angle AED, and step 2 says angle AED is congruent to angle FGH. Chain them to reach the Prove.
$\angle BEC \cong \angle FGH$

Answer

Since $\angle BEC \cong \angle AED$ and $\angle AED \cong \angle FGH$ , you have proved that $\angle BEC \cong \angle FGH$ .

The finished proof for Example 1

Statement	Reason
$\angle AED$ and $\angle BEC$ are vertical angles	Given
$\angle AED \cong \angle FGH$	Given
$\angle BEC \cong \angle AED$	Vertical Angles Theorem
$\angle BEC \cong \angle FGH$	Transitive Property of Congruence

Worked example

Example 2: Use a linear pair

Use the figure of lines $AC$ and $BD$ again, with new facts. Given: $\angle AEB$ and $\angle BEC$ form a linear pair, and $\angle AEB$ and $\angle FGH$ are supplementary. Prove: $\angle BEC \cong \angle FGH$ .

Reason: Given. A linear pair points you to the Supplement Theorem.
$\angle AEB \text{ and } \angle BEC \text{ form a linear pair}$
Reason: Given. Angle FGH is supplementary to angle AEB. If angle BEC is too, the Congruent Supplements Theorem can finish the proof.
$\angle AEB \text{ and } \angle FGH \text{ are supplementary}$
Reason: Supplement Theorem. It goes from linear pair to supplementary, and step 1 supplies the linear pair. The converse is false: angle AEB and angle FGH are supplementary, yet they have different vertices, so they do not form a linear pair.
$\angle AEB \text{ and } \angle BEC \text{ are supplementary}$
Reason: Congruent Supplements Theorem. Steps 2 and 3 make angle FGH and angle BEC supplementary to the same angle, angle AEB. That is the condition the theorem needs, and its result is the Prove.
$\angle BEC \cong \angle FGH$

Answer

Both angles are supplementary to $\angle AEB$ , so you have proved that $\angle BEC \cong \angle FGH$ .

The finished proof for Example 2

Statement	Reason
$\angle AEB$ and $\angle BEC$ form a linear pair	Given
$\angle AEB$ and $\angle FGH$ are supplementary	Given
$\angle AEB$ and $\angle BEC$ are supplementary	Supplement Theorem
$\angle BEC \cong \angle FGH$	Congruent Supplements Theorem

Rays

BA

,

BC

,

BD

and

BE

start at

B

. The congruence of

\angle ABC

and

\angle DBE

is stated in the text below, not shown by the drawing.

Worked example

Example 3: Move to measures and back

The figure shows four rays that start at $B$ . Its positions and sizes are not facts until the text states them. Given: $C$ is in the interior of $\angle ABD$ , $D$ is in the interior of $\angle CBE$ , and $\angle ABC \cong \angle DBE$ . Prove: $\angle ABD \cong \angle CBE$ .

Reason: Given. Record the positions first, because the Angle Addition Postulate needs them. Your plan: switch to measures, add the measure of angle CBD to both sides, and rebuild the two larger angles.
$C \text{ is in the interior of } \angle ABD \text{, and } D \text{ is in the interior of } \angle CBE$
Reason: Given. This is the one congruence you start with.
$\angle ABC \cong \angle DBE$
Reason: Definition of congruent angles, in the direction from congruent to equal measures. You need numbers before you can add.
$m\angle ABC = m\angle DBE$
Reason: Addition Property of Equality. Adding the same number to equal numbers keeps them equal. Both larger angles in the Prove contain angle CBD, so add its measure.
$m\angle ABC + m\angle CBD = m\angle DBE + m\angle CBD$
Reason: Angle Addition Postulate. Point C is in the interior of angle ABD, by step 1, so the left side of step 4 equals the measure of angle ABD.
$m\angle ABC + m\angle CBD = m\angle ABD$
Reason: Angle Addition Postulate. Point D is in the interior of angle CBE, by step 1, so the right side of step 4 equals the measure of angle CBE.
$m\angle DBE + m\angle CBD = m\angle CBE$
Reason: Substitution Property of Equality. Replace each side of step 4 with the equal total from steps 5 and 6.
$m\angle ABD = m\angle CBE$
Reason: Definition of congruent angles, now in the direction from equal measures to congruent. This is the Prove.
$\angle ABD \cong \angle CBE$

Answer

Adding $m\angle CBD$ to the equal measures gives equal totals, so you have proved that $\angle ABD \cong \angle CBE$ .

The finished proof for Example 3

Statement	Reason
$C$ is in the interior of $\angle ABD$ , and $D$ is in the interior of $\angle CBE$	Given
$\angle ABC \cong \angle DBE$	Given
$m\angle ABC = m\angle DBE$	Definition of congruent angles
$m\angle ABC + m\angle CBD = m\angle DBE + m\angle CBD$	Addition Property of Equality
$m\angle ABC + m\angle CBD = m\angle ABD$	Angle Addition Postulate
$m\angle DBE + m\angle CBD = m\angle CBE$	Angle Addition Postulate
$m\angle ABD = m\angle CBE$	Substitution Property of Equality
$\angle ABD \cong \angle CBE$	Definition of congruent angles

To write a two-column proof about angles, start from the Given, plan the links you need, and end at the Prove, with one exact reason in every row. Switch to measures when you need to add or subtract, then switch back with the Definition of congruent angles. If you can't name a reason for a step, that step isn't proved yet, however true it looks.