01.07 Line and Angle Proofs

The case file

Canvas starts this lesson with courtrooms and paperwork. Same idea here: nobody believes “those angles match” just because it looks that way.

Your job in 01.07 is to prove two kinds of things:

Twins across an X (vertical angles) are congruent.
When two roads stay parallel and a bike path cuts both, certain corners match — or add to 180°.

The forbidden move: you cannot prove a theorem by quoting the theorem. If the question is “prove vertical angles are congruent,” “because they’re vertical” is not allowed.

Quiz-style trap: two answers can have the same number. The right one is the reason (Corresponding Angles Theorem vs a Property of Equality that just happens to use that number).

Twins across the X

Two lines cross. Four angles. The ones that sit across from each other are vertical. They share a vertex, they do not share a side (they are nonadjacent), and they are congruent.

Tap a numbered corner on the picture. Its twin lights up with it.

Tap 1, 2, 3, or 4 on the drawing.

Lookalike words: vertical = across, nonadjacent, congruent. Linear pair = next to each other on a straight line, add to 180°. If a question says “best describes a vertical angle,” “nonadjacent” can be the keyed word.

Prove the twins without saying “twin”

Canvas walks this as a two-column proof. Here it is as peeling off a shared neighbor.

∠1 and ∠2 make a straight line → they add to 180°.
∠3 and ∠2 make a straight line → they also add to 180°.

So both pairs equal 180°. That’s the Transitive Property: two things equal to the same thing are equal to each other.

Now subtract the shared neighbor ∠2 from both sides. What’s left is m∠1 = m∠3. Definition of congruent angles: ∠1 ≅ ∠3.

Eight corners, three letters

Two parallel streets. A diagonal bike path. Eight corners. Canvas names four theorems. The cheat codes:

Tap F, Z, or C — the matching corners light up on the picture. You can also tap a numbered corner.

F — corresponding. Same side of the bike path, same “slot.” If the streets are parallel, they match.
Z — alternate interior. Inside the streets, opposite sides of the path. The Z. They match.
C — same-side interior. Inside the streets, same side of the path. They add to 180° (supplementary), they are not congruent (unless each is 90°).

Alternate exterior is the Z on the outside: opposite sides, outside the streets. They match too.

Lookalike pair: corresponding vs alternate interior both say “congruent if parallel.” The difference is where they sit — same corner of the F, or the inside corners of the Z.

The four theorems (if the streets stay parallel)

Corresponding Angles Theorem. Transversal + two parallels → corresponding angles are congruent.

Alternate Interior Angles Theorem. Those inside, opposite-side angles are congruent.

Alternate Exterior Angles Theorem. Outside, opposite-side angles are congruent.

Same-Side Interior Angles Theorem. Inside, same-side angles are supplementary (add to 180°).

If you only know one angle, these four let you fill in the other seven.

Build the corresponding-angles case

Canvas proves corresponding angles using a straight line, angle addition, and same-side interior. Story version:

Given: AB ∥ CD. Prove: ∠AGE ≅ ∠CHE.

Both of those angles sit next to the same leftover angle ∠AGF, and both pairs make 180°. Subtract the leftover. The two you wanted are equal, so they are congruent.

m∠AGE + m∠AGF = 180° (straight line / angle addition)
m∠CHE + m∠AGF = 180° (same-side interior)
Subtract m∠AGF from both → m∠AGE = m∠CHE → congruent.

Same number, two reasons: both lines end at 180°. One reason is “straight angle + Angle Addition.” The other is “Same-Side Interior Angles Theorem.” If a quiz shows 180° twice, pick the reason that matches that pair, not whichever 180° sentence you saw first.

The Z is a shortcut

Once corresponding angles are proven, alternate interior is a two-step hop:

∠AGF ≅ ∠EGB because they are vertical (the twins you already proved).
∠EGB ≅ ∠EHD because they are corresponding.
So ∠AGF ≅ ∠EHD by the Transitive Property (A matches B, B matches C, so A matches C).

That’s the whole Alternate Interior Angles Theorem. Vertical + corresponding + transitive.

Your turn — pick the reason

Same practice story as the lesson: AC ∥ GD, m∠CBE = 60°, m∠BFG = 120°. Prove ∠BED ≅ ∠BFG.

Watch for lookalike theorems and for “same 60°, different why.”