057i - Conditions for Congruent Triangles

Lesson 57 — Explicit Instruction: Conditions for Congruent Triangles

Strand: Space | Descriptor: AC9M8SP01 | Duration: 45 minutes

Learning Intentions

• To understand that congruent triangles are identical in both shape and size.
• To identify and justify the four minimal conditions that guarantee triangle congruence: SSS, SAS, AAS, RHS.

Success Criteria

I can:

1. State the four congruence conditions for triangles: SSS, SAS, AAS, RHS.
2. Determine which condition, if any, proves a given pair of triangles congruent.
3. Explain, using a counterexample, why SSA and AAA do not prove congruence.
4. Correctly match corresponding vertices when writing a congruence statement, e.g. .

Warmup

(5 minutes — “same or different?”, mini whiteboards)

For each pair of triangles described below, judge by eye whether they look like they must be the same shape and size, or whether you’re not sure.

1. Triangle 1 has sides , , cm. Triangle 2 has sides , , cm.
2. Triangle 3 has sides , cm with a angle between them. Triangle 4 has sides , cm with a angle between them.
3. Triangle 5 has sides , cm and a angle not between them. Triangle 6 has sides , cm and a angle not between them, in the same positions.
4. Triangle 7 has angles , , . Triangle 8 has angles , , , but is visibly larger when drawn to scale.

Teacher note: Don’t resolve Q3 or Q4 yet — flag them as “the two traps we’ll test properly today,” and return to them explicitly in Activity 2.

Activities

Activity 1 — Explicit Instruction: SSS and SAS (12 min)

I do: Define congruent: two shapes are congruent if they are identical in shape and size — one can be placed exactly on top of the other by some combination of translation, reflection and rotation (we’ll explore this fully in Lesson 60). Introduce standard markings: tick marks for equal sides, arc marks for equal angles.

Condition 1 — SSS (Side-Side-Side): if all three pairs of corresponding sides are equal, the triangles are congruent. Model: with , , and with , , . There is only one possible triangle shape with three given side lengths — the sides “lock” the angles in place, so .

Condition 2 — SAS (Side-Angle-Side): if two pairs of corresponding sides are equal, and the angle between them (the included angle) is equal, the triangles are congruent. Model: with , , , matched by with the same measurements in the same arrangement.

We do: Together, check: has , , ; has , , . Which condition applies? (SSS.) Then: has , , ; has , , . Which condition applies? (SAS.)

You do: For each pair, state SSS, SAS, or “not enough information” (we’ll return to why some fail):

1. : , , ; : , , .
2. : , , ; : , , .
3. : , , (angle not between the two given sides); : , , (same arrangement).

Activity 2 — Explicit Instruction: AAS, RHS, and why SSA and AAA Fail (12 min)

I do — Condition 3, AAS (Angle-Angle-Side): if two pairs of corresponding angles are equal, and one pair of corresponding sides (not necessarily between them) is equal, the triangles are congruent. Why does this work even though the side isn’t included? Because if two angles are fixed, the third angle is automatically fixed too (angle sum ) — so AAS is really “ASA in disguise,” and the side can be relabelled as the included side between the two now-known angles.

Condition 4 — RHS (Right angle-Hypotenuse-Side): a special case for right-angled triangles only — if the hypotenuse and one other side are equal, the triangles are congruent, even though this looks like SSA. Why is RHS an exception? Because once the right angle and hypotenuse are fixed, Pythagoras’ theorem forces the third side to be one specific length — there is no freedom left for a second, different triangle.

Non-example 1 — SSA fails in general: Draw with , , and (angle not between the two given sides). Show that two different triangles can be drawn from this same data — swing from and it can meet at two different points, producing two different triangle shapes. This is why SSA (unlike RHS) does not guarantee congruence.

Non-example 2 — AAA fails: Draw two triangles with identical angles but different sizes (like a photocopy enlarged to ). Same shape, different size — this proves the triangles are similar (Lesson 58), not congruent.

We do: Together, classify: : , , ; : , , . (AAS.) Then: two right triangles share hypotenuse and one leg . (RHS.)

You do: For each, state SSS, SAS, AAS, RHS, or “not enough information”:

1. Two triangles share two angles and the side between them.
2. Two right-angled triangles share the hypotenuse only, with no other equal side or angle given.
3. Two triangles share two sides and a non-included angle.
4. Two triangles share all three angles, with no side information.

Activity 3 — Inquiry Task: Investigating the Ambiguous Case (11 min)

Pairs, then whole-class share.

Draw . From one arm, mark cm. From , you need to draw a side of length cm that meets the other arm of the angle at some point . Investigate: how many different positions can take, and does this depend on the length you’re swinging?

Socratic scaffolding (Polya’s How to Solve It):

Prompt	Purpose
Understand: what is fixed, and what is being tested?	The angle and one side () are fixed; the second side length ( cm) is “swung” like a compass arc from to find where it meets the other arm.
Devise a plan	Physically draw the fixed angle and side, then use a compass set to cm to find all intersection points with the other arm.
Carry it out	Two intersection points are usually found — two different possible positions for , giving two different (non-congruent) triangles from the same SSA data.
Looking back — does this always happen?	No — investigate what happens if the swung side is very long (only one intersection) or very short (no intersection at all).
Now repeat with . What changes?	With a right angle, the “swing” only ever produces one valid triangle — this is exactly why RHS is a safe exception to the general SSA problem.
Why does the right angle rescue the situation?	Pythagoras’ theorem fixes the third side uniquely once the right angle and hypotenuse are known — there’s no room for a second solution.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. State the congruence condition proved by two triangles sharing three equal sides.
2. Two triangles share two equal angles and the side between them equal. Name the condition.
3. Explain, using a sketch description, why SSA is not accepted as a valid congruence condition.
4. Reasoning. Explain why RHS is a valid condition even though it looks like SSA.

Answers: 1. SSS; 2. This is ASA, which is a special case covered by the same reasoning as AAS (two angles fix the third, and combined with any one matching side, the triangle is fixed) — accept ASA or AAS as equivalent reasoning here; 3. Two sides and a non-included angle can be satisfied by two different triangle shapes, since the side being swung can meet the far arm at two different points; 4. Because the right angle and hypotenuse together fix the third side uniquely via Pythagoras’ theorem, removing the ambiguity that SSA normally has.

Common Misconceptions

Misconception	How to pre-empt it
SSA is treated as a valid congruence condition.	Sketch two different triangles satisfying the same SSA data side by side — a concrete counterexample (Activity 3).
AAA proves congruence.	Contrast with a photocopy enlarged to : identical angles, different size.
Any correspondence order is acceptable when naming congruent triangles.	Insist means , , — order encodes the matching.
RHS is just “another name” for SSA and should also fail.	Explicitly connect RHS to Pythagoras’ theorem to show why the right angle removes the ambiguity that SSA has in general.
The included angle in SAS can be any of the triangle’s three angles.	Emphasise “included” means specifically the angle between the two named sides — mark it on the diagram every time.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Two triangles are congruent by SAS. One has sides cm and cm with an included angle of . What is the included angle in the second triangle?

Answer

— congruent triangles have identical corresponding angles as well as identical corresponding sides.

E2 (Kangaroo style). A right-angled triangle has hypotenuse cm and one leg cm. A second right-angled triangle has hypotenuse cm and one leg cm. What condition proves them congruent, and what is the length of the remaining leg in each?

Answer

RHS proves congruence. By Pythagoras: cm in each triangle.

E3 (Challenge). Explain why “AAA” is sometimes jokingly called “the condition that proves the wrong thing.” What does AAA actually prove, and why is this still a useful fact?

Answer

AAA proves the triangles are similar (same shape), not congruent (same shape and size) — the missing side information leaves the overall size unfixed. It is still useful because similarity lets us compare ratios of sides and solve problems like indirect measurement (Lesson 58 and Lesson 61).

Homework

1. State the condition (SSS, SAS, AAS, RHS) that proves each pair of triangles congruent: (a) three matching sides (b) two matching sides with the angle between them equal (c) two matching angles and a non-included matching side (d) equal hypotenuse and one equal leg in right-angled triangles.
2. Sketch (in words) a labelled example of an SSA situation and explain why it does not fix a unique triangle.
3. Two triangles have angles and , with no side lengths given. State whether they must be congruent, and justify your answer.
4. Write the correct congruence statement for two triangles and , given that corresponds to , corresponds to , and corresponds to .
5. Reasoning. Explain, using the idea of “how many different triangles fit the data,” why SSS, SAS, AAS and RHS all guarantee a unique triangle shape, while SSA and AAA do not.
6. Challenge. Two right-angled triangles share a common leg of cm, but one has a hypotenuse of cm and the other has a hypotenuse of cm. Explain, without assuming they are congruent, why these triangles are not congruent, using Pythagoras’ theorem to justify your answer.

Answers: 1. (a) SSS (b) SAS (c) AAS (d) RHS; 2. Any correct description showing two sides and a non-included angle can produce two different triangle shapes, e.g. swinging the second side to meet the base at two different points; 3. Not necessarily congruent — matching angles only (AAA) proves similarity, not congruence, since size is unconstrained; 4. ; 5. SSS, SAS, AAS and RHS each provide enough information to fix every side and angle uniquely (directly or via angle sum / Pythagoras), leaving no freedom for a second shape; SSA and AAA each leave at least one genuine degree of freedom (an ambiguous swing, or an unconstrained overall size); 6. By Pythagoras, the other leg lengths are cm and cm — different remaining legs mean the triangles are not identical in every side length, so they are not congruent, even though they share the cm leg.