Use correspondence and congruence criteria without relying on scale or orientation.
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Try
Two triangles have side lengths 3,4,5. Turn one over and rotate it: corresponding sides still match. Equal size and shape need not mean the same orientation. Write the vertex correspondence before claiming that two particular sides or angles agree.
Predict
Predict whether three matching angles determine size. They do not: a triangle and a doubled copy have the same angles. SSS uses three side equalities; SAS uses two sides and their included angle; ASA uses two angles and their included side. AAS follows because the third angle is determined. For right triangles, equal hypotenuse and one leg give RHS congruence.
Experiment
Trace the two scaled triangles in the diagram and match A with P, B with Q and C with R. List the three side matches. Then imagine doubling only the second triangle. Its angles remain equal, but the equal-length statements fail. This separates congruence from similarity.
Two 3–4–5 triangles drawn at the same scale. Reflection changes their orientation but preserves all three side lengths.
Understand
Worked argument: in quadrilateral ABCD, suppose AB=CD and BC=DA. Draw diagonal AC. Triangles ABC and CDA have AB=CD, BC=DA and AC=CA. By SSS they are congruent with A↔C, B↔D, C↔A. Therefore angle ABC equals CDA. The shared diagonal is a real third piece of evidence, not an omitted detail.
Invalid proof: two sides and any angle always give SAS. The angle must lie between the matched sides. A ray from A can intersect a circle centered at B twice, producing two triangles with the same two lengths and a nonincluded angle. That ambiguity is why SSA is not a general congruence rule.
Apply
Use the practice to distinguish correspondence, criterion and consequence. First list independent given facts, then invoke a sufficient criterion, then derive new corresponding equalities. Do not use the side or angle you are trying to prove as one of the premises.
Continue
Depth: for SSA data, the altitude from the fixed endpoint to the angle ray determines whether the circle has zero, one or two valid intersections. Some special SSA cases are unique, but a general rule must work in every configuration it permits.
Apply · explain · revise
Practice and reasoning
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Triangle ABC is congruent to triangle PQR in that order. BC=13/2 cm. Find QR in cm.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
In triangle ABC, angles A=48° and B=71°. Find C in degrees.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which data establish SAS congruence?
What does equal angle-angle-angle data establish for nondegenerate triangles?
If AB=AC, prove angles ABC and BCA are equal by comparing triangles ABC and ACB.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Explain geometrically why sides AB=c, BC=a and an angle at A need not determine a unique triangle when BC is opposite the given angle.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
