S2 Maths | Congruent Triangles: Five Tests, Matching Order, Proof Layout Tutor Prep Guide

2026-11-14

Tutor lesson-prep guides · Secondary Maths

One student writes two sides and an angle are equal so the triangles are congruent, without checking whether the angle sits between the sides; another writes triangle ABC congruent to triangle EDF and every corresponding side is then wrong. This lesson separates the five usable tests from the two that never work, then locks the matching order with a two-line proof.

Objectives — By the end students can: (1) state the SSS, SAS, ASA, AAS and RHS tests and explain why SSA and AAA are not tests; (2) read corresponding parts from triangle ABC congruent to triangle DEF in letter order, so AB matches DE, BC matches EF and AC matches DF; (3) write a two-line proof that lists the three equal parts first, then the conclusion with the test named in brackets.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Angle sum	In a triangle angle A is 65 degrees and angle B is 48 degrees, what is angle C?	67 degrees
Right angles and hypotenuses	In a right-angled triangle, which side is the hypotenuse?	The side opposite the right angle, the longest one
Matching parts	Triangle ABC is congruent to triangle DEF and A sits opposite BC, so D sits opposite which side?	EF, the same position

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
0 to 5 min	Two counterexamples first	Draw one SSA pair and one AAA pair and ask whether congruence is proved	Offer an opinion and have it overturned
5 to 16 min	The five tests	Model SSS, SAS, ASA, AAS and RHS, each described as three pieces of data	Say which three pieces each test needs
16 to 26 min	Matching order	Write triangle ABC congruent to triangle DEF in colour, then read all six matching parts	Write three matching sides and three matching angles
26 to 38 min	The two-line proof	Show the layout: three given equalities, then the conclusion with the test in brackets	Write one SAS proof in that layout
38 to 43 min	Finding unknown parts	Use congruence for matching angles and sides, then the angle sum for the third angle	Find angle F from 65 and 48 degrees
43 to 45 min	Exit check	Three questions: a test, a matching side, an angle sum	Answer and hand in

03 Teaching script (read it out)

Scene 1: Five tests work, two never do

List SSS, SAS, ASA, AAS and RHS. Each needs three pieces of data and at least one side. Then draw SSA: two sides fixed with the angle outside them, which produces two different triangles, so it proves nothing. AAA is clearer still, since equal angles give the same shape at any size, which is similarity, not congruence. The one sentence students must own is that SSA fails because the angle is not between the two sides.

Ask students to point at the included angle in a diagram and name the three pieces SAS needs

Scene 2: Matching order: position tells the answer

Write triangle ABC congruent to triangle DEF and colour A with D, B with E and C with F in the same pen. Corresponding sides are read by position: AB with DE, BC with EF, AC with DF, and angles follow the same rule. A wrong order is not a typo, it claims a different pair of sides is equal and spoils every later step.

Students write the three matching sides for triangle PQR congruent to triangle XYZ and the tutor checks on the spot

Scene 3: The two-line proof: three equalities, then a conclusion

The layout is two lines. First list the three equal parts, for example AB equals DE, angle B equals angle E, BC equals EF. Then write the conclusion, triangle ABC congruent to triangle DEF, with SAS in brackets, because the test is the whole basis of the reasoning. The included angle is the item students drop most, so ask them to tick three items before they move on.

Students write one SSS proof in two lines and name the test in brackets

04 Question types and worked steps

Type	Example	Steps	Answer
Choose the test	AB equals DE, AC equals DF, angle A equals angle D	Find the two sides and the angle, confirm the angle sits between them, name SAS	SAS
Complete the proof	Angle B equals angle E and BC equals EF are given, one item missing	Read what is already given, note the test needs another angle, add angle C equals angle F	ASA
Find an unknown angle	Triangle ABC congruent to triangle DEF, angle A 65 degrees, angle B 48 degrees	Matching angles are equal, then the angle sum gives the third	Angle F is 67 degrees
Find an unknown side	Triangle PQR congruent to triangle XYZ, PQ 6, QR 8, PR 10	Match sides in letter order, write the lengths straight off, add perimeter if asked	XY 6, YZ 8, XZ 10

Sanity check (teach this): Congruent triangles have equal matching sides, so the perimeters must agree: 6 plus 8 plus 10 is 24, and the other triangle also totals 24. Then 6 squared plus 8 squared is 36 plus 64, which is 100, exactly 10 squared, so this is a right-angled triangle with the right angle between the short sides, meaning RHS also checks the same question. Two routes agreeing means the answer is sound.

05 Independent practice (3 questions)

Question	Answer	What to watch
Triangle ABC congruent to triangle DEF, angle A 65 and angle B 48, find angle F	67 degrees	Whether matching angles were written before using the angle sum
AB equals DE equals 6, BC equals EF equals 8, angle B equals angle E equals 90, which test applies?	SAS, or RHS since a right angle and hypotenuse are present	Whether SSS was forced although an angle is given
Triangle PQR congruent to triangle XYZ, PQ 6, QR 8, PR 10, find the perimeter of XYZ	24	Whether all three matching sides were found

06 Common mistakes and what to say

Mistake	Student writes	Say this
Using SSA as a test	Two sides and a non-included angle are equal, so the triangles are declared congruent	This is SSA, which draws two different triangles from the same data, so it proves nothing; the angle must sit between the sides for SAS.
Wrong matching order	Triangle ABC congruent to triangle EDF	Letters follow correspondence: A with D, B with E, C with F. A wrong order claims the wrong equal sides and every step after it fails.
Using AAA as a test	Three equal angles are taken as congruence	Equal angles only fix the shape, and the size can scale, which is similarity. Congruence needs a side.
Proving with two items only	AB equals DE and AC equals DF, conclusion written	SAS needs three items, two sides and the included angle. Leaving out the angle means the proof does not hold.

07 Homework (paper, section and question numbers)

Unknown sides and angles, questions 1 to 3 — The Pythagoras chapter uses the same 6, 8, 10 right triangle
Transformations and matching positions, questions 4 and 5 — Reading matching points after a slide or turn uses the same ordering idea
Secondary maths subject index, choose two chapters — Choose by school progress and report back next lesson

08 Exit check (last 5 minutes)

Question	Answer	Decision
Angle A is 90 and angle B is 37, find angle C	53 degrees	Angle sum secure
Is SSA a congruence test?	No, the angle must be between the two sides	Test list known
Triangle ABC congruent to triangle DEF, AB is 6, what is DE?	6	Matching sides understood

09 Tutor FAQ

How does a student know whether to write SAS or ASA?

Read the data already given. Two sides with the included angle is SAS; two angles with the included side is ASA. Count the data first, then pick the test rather than recalling one at random.

Why does the matching order matter so much?

Congruence is a claim that six parts match. The order states which parts those are, so a wrong order is a different claim, and examiners award proof marks by the order written.

Is RHS redundant next to SAS?

No. With a right angle and a hypotenuse already known, RHS needs only one more side, while using SAS forces students to justify that the right angle is included. RHS is strictly shorter.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.