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Congruent

Congruent

Congruent shapes

Rules Core + Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. What congruent means
  4. The four conditions for triangles
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 4.1: congruent shapes
  • 4.1Know that congruent shapes are the same shape AND the same size (a reflection is allowed).
  • 4.1Recognise congruent triangles from SSS, SAS, ASA and RHS.
  • 4.1Use congruence to find missing sides and angles, matching the vertices in order.

Next: Similarity - same shape, different size.

Formula bank

SSSLearn itthree pairs of equal sidesUse it when all three sides given.
SASLearn ittwo sides and the angle BETWEEN themthe angle must be the included one.Use it when two sides and an angle.
ASA (AAS)Learn ittwo angles and a matching sideUse it when two angles and a side.
RHSLearn itright angle, hypotenuse and one other sideUse it when right-angled triangles.

What congruent means

Two congruent trianglesABCPQR
Matching ticks = equal sides. P matches A, Q matches B, R matches C.

Congruent shapes have every side and every angle equal; one may be turned over or moved. 'Triangle ABC is congruent to triangle PQR' matches the letters in order: A ↔ P, B ↔ Q, C ↔ R.

The four conditions for triangles

three sides

SSS
Side, side, side.

All three sides match.

two sides + the angle between

SAS
Side, angle, side.

Two sides and the INCLUDED angle match.

two angles + a side

ASA
Angle, side, angle.

Two angles and a corresponding side match.

right angle, hypotenuse, side

RHS
Right angle, hypotenuse, side.

Both right-angled, equal hypotenuses, one other side equal.

AAA is NOT a condition: equal angles only make triangles similar (they can be different sizes). SSA (angle not between the sides) is not enough either.

General methodHow to answer a congruence question

  1. List what is equal: sides (ticks) and angles (arcs or values).
  2. Match it to SSS, SAS (angle between the sides), ASA/AAS or RHS.
  3. Write the triangles with the vertices in matching order.
  4. Read a missing side or angle from its partner in the other triangle.

Example Triangle ABC is congruent to triangle DEF. AB = 8 cm, BC = 11 cm, angle ABC = 64°. Find EF and angle DEF.

Same order: A ↔ D, B ↔ E, C ↔ F.

  1. EF ↔ BC
  2. angle DEF ↔ angle ABC

Answer EF = 11 cm, angle DEF = 64°

Try it yourself

1
Triangle ABC ≅ triangle PQR; angle A = 48°, angle B = 73°. Find angle R.
Show answer

Answer 59

2
Two triangles each have sides 6 cm and 9 cm with 40° between them. Which condition? Type SSS, SAS, ASA or RHS.
Show answer

Answer SAS

Ways they test it

Type 1 Pick the congruent pair

"Which two of the triangles A, B and C are congruent?" (1 mark)

Spot it several triangles with measurements.

  1. Check where each angle sits relative to the sides.
  2. Match one condition exactly.

Type 2 Name the condition

"Give the congruence condition." (1 mark)

Spot it two triangles with marked equal parts.

  1. SSS, SAS, ASA or RHS.

For example right angle, hypotenuse 10, side 6 in both

    Answer RHS

    Type 3 Complete the statement

    "Triangle ABC is congruent to triangle ..." (1 mark)

    Spot it a statement with a gap.

    1. Write the vertices in matching order.

    Mistakes to avoid

    • ✗ AAA proves congruence✓ AAA = similar onlyThe size isn't fixed.
    • ✗ SAS with the angle not between the sides✓ the angle must be between themSSA can give two different triangles.
    • ✗ Reading the vertices out of order✓ match letter by letterABC ↔ DEF.

    Summary

    • Congruent = same shape and same size (reflections allowed).
    • Conditions: SSS, SAS (included angle), ASA/AAS, RHS.
    • AAA gives similar, not congruent.
    • Match the vertices in order before reading off parts.

    If you see …

    • two sides and an angleSAS only if the angle is between the sides.
    • right-angled trianglesthink RHS.

    Practice

    Practice: easy, medium, hard

    Easy

    1
    Which is NOT a congruence condition: SSS, SAS, AAA or RHS?
    Show answer

    Answer AAA

    2
    Triangle ABC ≅ triangle XYZ. Which side equals YZ? Type two letters.
    Show answer

    Answer BC

    3
    Right-angled triangles with hypotenuse 10 cm and another side 6 cm. Condition?
    Show answer

    Answer RHS

    Medium

    1
    ABC ≅ DEF with AB = DE = 7 cm, angle A = angle D = 52°, angle B = angle E = 61°. Condition?
    Guide me

    What is it asking? a condition

    What do we know? two angles and the side between

    Plan ASA

    Show working
    1. ASA

    Answer ASA

    2
    In the same triangles, find angle F.
    Guide me

    What is it asking? an angle

    What do we know? F ↔ C

    Plan 180 − 52 − 61

    Show working
    1. 67°

    Answer 67

    3
    Triangle PQR is isosceles with PQ = PR; M is the midpoint of QR. Which condition shows PQM ≅ PRM?
    Guide me

    What is it asking? a condition

    What do we know? PQ = PR, QM = RM

    Plan PM is shared

    Show working
    1. three pairs of equal sides: SSS

    Answer SSS

    Hard

    1
    In that isosceles triangle, find angle PMQ.
    Guide me

    What is it asking? an angle

    What do we know? angle PMQ = angle PMR

    Plan they add to 180°

    Show working
    1. 90°

    Answer 90

    PartAnswer / workingMarksGuidance
    90 with reasonB1
    Total: 1 mark
    2
    Triangle X has sides 5 cm and 7 cm with 50° between them; triangle Z has sides 7 cm and 5 cm with 50° between them. Congruent? Type yes or no.
    Guide me

    What is it asking? yes or no

    What do we know? two sides + included angle

    Plan SAS

    Show working
    1. yes (SAS)

    Answer yes

    PartAnswer / workingMarksGuidance
    yes, SASB1
    Total: 1 mark
    3
    Triangle Y has sides 5 cm and 7 cm with 50° opposite the 7 cm side. Must it be congruent to X? Type yes or no.
    Guide me

    What is it asking? yes or no

    What do we know? the angle is not between the sides

    Plan SSA is not a condition

    Show working
    1. no

    Answer no

    PartAnswer / workingMarksGuidance
    noB1
    Total: 1 mark

    Can I do this?

    Anything unticked: revise that section first. All ticked: take the quiz.

    Quiz

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