The undefined terms and the precise definitions built on them — midpoint, bisector, perpendicular, linear pair, vertical angles. Parallel lines and the four angle pairs a transversal makes, with the converses that prove lines parallel. The triangle angle-sum and exterior-angle theorems. Congruence as a rigid motion, and the shortcuts SSS, SAS, ASA, AAS and HL — with the numbers that show why SSA and AAA fail. Two-column proofs with a reason on every line, ending in CPCTC. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
A 30° angle, a side of 10 along the bottom, and a side of 6 swinging down from the top.
That is why SSA is not a congruence criterion — two sides and an angle NOT between them can build two different triangles. SSS, SAS, ASA and AAS lock a triangle down. SSA only works when the angle is a right angle, and then it has its own name: HL.
What people get wrong
⚠️People often think…
SSA proves two triangles congruent.
Sides of 6 and 10 with a 30° angle opposite the 6 build two different triangles — the short side can swing to either of two landing spots. Only a right angle rescues it, and that case gets its own name: HL.
The angle has to be BETWEEN the sides, or be a right angle.
⚠️People often think…
Same-side interior angles are congruent.
With parallel lines they are SUPPLEMENTARY — they add to 180. A 74° angle pairs with 106°, not another 74°. Alternate interior and corresponding angles are the congruent ones.
Same side of the transversal: they add up. Opposite sides: equal.
⚠️People often think…
The transversal angle rules work on any two lines.
The angle PAIRS have names no matter what — alternate interior, corresponding, same-side. But the congruence only holds when the lines are known to be parallel. A picture that looks parallel is not the same as a picture marked parallel.
Look for the arrowheads before you set anything equal.
⚠️People often think…
The letter order in △ABC ≅ △DFE does not really matter.
It is the whole content of the statement. △ABC ≅ △DEF says A matches D, B matches E, C matches F. Write the letters out of order and every CPCTC step after it points at the wrong parts.
Stack the two names and read straight down.
Name the criterion, then use it
1Watch one
Two triangles share two sides and an angle that is not between them. Congruent?
Name the pattern: side, side, angle.
The short side can swing to two landing spots.
Two different triangles fit the same data.
Not congruent. SSA proves nothing.
2Do one with me
Two parallel lines are cut by a transversal. One same-side interior angle measures 74°.
Same-side interior angles are congruent or supplementary?
So they add to
The other angle measures
💬One sentence, then you move on
Why does SAS lock a triangle down when SSA does not?
3Try one
△ABC ≅ △DEF. Which angle in the second triangle corresponds to angle B? Write one letter.
I want a hint first
Stack them: A B C on top of D E F. Now read straight down each column.
💬Last one — then you're done here
Why can't you use CPCTC before the triangles are proved congruent?
Where this goes
Where this lives
Trusses, brackets and bike frames are made of triangles because a triangle is the only shape that cannot be pushed out of shape without breaking a side.
What this feeds
Next unit is similarity and right-triangle trigonometry — same shapes, different sizes.
Find one triangle in the room and say what is holding it rigid.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
📝 Writing a two-column proof — the routine
Copy the GIVEN as your first statements, reason “Given.” Mark the diagram with every fact you know. Turn each given into what it MEANS (a bisector → two congruent angles; a midpoint → two congruent segments; perpendicular → a right angle). Look for a shared side or angle → Reflexive Property. Count what you have: three sides → SSS; two sides and the angle between → SAS; two angles and a side → ASA or AAS. State the triangles congruent with the vertices in matching order. If the prove asks for one more part, finish with CPCTC. Your last statement must be exactly the prove.
Given, mark, unpack, share, count, match, CPCTC.
🔑 The whole unit in one idea
Congruent means a rigid motion — a slide, turn or flip — carries one figure exactly onto the other, so every length and angle matches. You never need to check all six parts of a triangle: three well-chosen parts (SSS, SAS, ASA, AAS, HL) already force the rigid motion to exist. SSA and AAA leave the triangle free to change, so they prove nothing. A proof is just a chain of definitions, postulates and theorems that turns what you were given into that conclusion, one justified line at a time.
Three right parts lock the triangle.
⚠️ Traps the test loves
Calling SSA a criterion. Sides 6 and 10 with a 30° angle opposite the 6 make TWO different triangles. Only a right angle rescues it (HL).
Treating same-side interior angles as congruent. With parallel lines they are SUPPLEMENTARY: 74° pairs with 106°, not 74°.
Using the transversal theorems when the lines are NOT known to be parallel. The angle pairs have names either way; the congruence needs ∥.
Writing △ABC ≅ △DFE when B matches E. The letters must be in corresponding order or every CPCTC step after it is wrong.
Using CPCTC BEFORE the triangles are proved congruent, or using the theorem you are proving as one of its own reasons.
Forgetting the shared side. Two triangles that share BD already have one pair of congruent sides — BD ≅ BD, Reflexive Property.
∥ Parallel lines and a transversal — the chart
Angles 1–4 at the first crossing, 5–8 at the second, numbered the same way at each: 1 upper left, 2 upper right, 3 lower left, 4 lower right.
Pair
Where
Examples
If lines are parallel
Corresponding
same position at each crossing
1 & 5, 2 & 6, 3 & 7, 4 & 8
congruent (postulate)
Alternate interior
between the lines, opposite sides of the transversal
3 & 6, 4 & 5
congruent
Alternate exterior
outside the lines, opposite sides
1 & 8, 2 & 7
congruent
Same-side interior
between the lines, same side
3 & 5, 4 & 6
supplementary
Vertical
across one crossing
1 & 4, 2 & 3, 5 & 8, 6 & 7
always congruent — no parallel needed
Linear pair
adjacent along one line
1 & 2, 1 & 3, 5 & 6, …
always supplementary — no parallel needed
🎯 How the test will ask
A number puzzle: parallel lines, two angles as expressions in x. Decide congruent or supplementary FIRST, then write the equation.
Two marked triangles — “which criterion?” Read the marks in order around the triangle; ask whether the angle is between the two sides.
A proof with blanks — fill the missing statement or reason. The reason for a definition-type step is “definition of…”; for an algebra step, a property of equality.
“Are the triangles congruent? Explain.” — when the answer is NO, name SSA or AAA and say why the triangle is not locked.
A triangle with one angle missing, or an exterior angle — angle sum 180°, or exterior = sum of the two remote interior angles.
“Which transformation is rigid?” — translation, rotation, reflection preserve length; a dilation does not.
✅ Before the test, can you…
State the definitions of midpoint, angle bisector, perpendicular, complementary, supplementary and linear pair, and use each as a reason?
Prove the Vertical Angles Theorem in six lines without looking?
Name all four transversal angle pairs from a numbered diagram and say which are congruent and which supplementary?
Tell SSS, SAS, ASA, AAS and HL apart from the markings, and reject SSA and AAA with a reason?
Write a congruence statement with the letters in corresponding order and read any side or angle off it?
Find a missing angle with the angle-sum, exterior-angle or isosceles theorem, and check that the three angles total 180°?
Write a complete two-column proof that ends in SAS and then uses CPCTC?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a geometer: put a two-column proof in order, decide from the markings which congruence criterion applies, name the angle pairs a transversal makes, then argue why SSA proves nothing. Every activity checks itself, and hints are free.
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