Hard GCSE Congruence Questions

Challenging, exam-style GCSE Congruence questions with worked solutions. Stretch yourself on the hardest congruence conditions, SSS, identifying congruent triangles, SAS problems.

congruence conditionsSSSidentifying congruent trianglesSASincluded angleAAS
GCSE Foundation34 questionsStep-by-step solutions
Question 1
6 markschallenging
Triangle ABCABC and triangle UVWUVW have AA corresponding to UU, BB corresponding to VV and CC corresponding to WW. Which of these sets of information is enough to prove that the two triangles are congruent?
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Worked solution

  1. Recall the five conditions that prove congruence

    SSS, SAS, ASA, AAS, RHS\text{SSS}, \ \text{SAS}, \ \text{ASA}, \ \text{AAS}, \ \text{RHS}

    A set of information proves congruence only if it matches one of SSS, SAS, ASA, AAS or RHS.

  2. Say what all five have in common

    data⇒all three sides fixed\text{data} \Rightarrow \text{all three sides fixed}

    Each of the five fixes all three side lengths — by the cosine rule, the sine rule or Pythagoras — and three sides fix a triangle.

  3. Read off the pattern of the set that works

    S, S, S (three sides)\text{S, S, S (three sides)}

    Written in order round the triangle, one of the sets reads S, S, S (three sides) — that is the pattern SSS.

  4. Mark the correct set on a diagram

    AB=UV,BC=VW,CA=WUAB = UV, \quad BC = VW, \quad CA = WU

    Marking the equal parts on both triangles is the quickest way to see which condition the information matches.

  5. Check the first wrong set

    AB=UV, CA=WU, ∠ABC=∠UVW⇒SSAAB = UV, \ CA = WU, \ \angle ABC = \angle UVW \Rightarrow \text{SSA}

    The set AB=UVAB = UV, CA=WUCA = WU and ∠ABC=∠UVW\angle ABC = \angle UVW is SSA — two sides and an angle that is not between them leaves two possible triangles, so it cannot prove the two triangles are congruent.

  6. Check the second wrong set

    ∠BAC=∠VUW, ∠ABC=∠UVW, ∠ACB=∠UWV⇒AAA\angle BAC = \angle VUW, \ \angle ABC = \angle UVW, \ \angle ACB = \angle UWV \Rightarrow \text{AAA}

    The set ∠BAC=∠VUW\angle BAC = \angle VUW, ∠ABC=∠UVW\angle ABC = \angle UVW and ∠ACB=∠UWV\angle ACB = \angle UWV is AAA — angles alone fix the shape but not the size, so it cannot prove the two triangles are congruent.

  7. Check the third wrong set

    BC=VW, ∠BAC=∠VUW⇒INSUFBC = VW, \ \angle BAC = \angle VUW \Rightarrow \text{INSUF}

    The set BC=VWBC = VW and ∠BAC=∠VUW\angle BAC = \angle VUW is not enough — there are simply not enough parts to fix a triangle, so it cannot prove the two triangles are congruent.

  8. Check the last wrong set

    AB=UV, BC=VW⇒INSUFAB = UV, \ BC = VW \Rightarrow \text{INSUF}

    The set AB=UVAB = UV and BC=VWBC = VW is not enough — there are simply not enough parts to fix a triangle, so it cannot prove the two triangles are congruent.

  9. Say why the failing sets fail

    SSA⇒2 triangles,AAA⇒any size\text{SSA} \Rightarrow 2 \text{ triangles}, \quad \text{AAA} \Rightarrow \text{any size}

    SSA leaves two possible triangles and AAA leaves a whole family of similar triangles, so neither can prove congruence.

  10. Note the exception

    right angle⇒RHS\text{right angle} \Rightarrow \text{RHS}

    The one time two sides and a non-included angle do work is when that angle is a right angle. That case has its own name: RHS.

  11. Note that the order of the letters matters

    SAS≠SSA\text{SAS} \ne \text{SSA}

    SAS and SSA use the same three parts; only the position of the angle differs, and that is the difference between a proof and no proof.

  12. Write the congruence statement for the correct set

    △ABC≅△UVW\triangle ABC \cong \triangle UVW

    With that information triangle ABCABC and triangle UVWUVW are congruent, so every remaining pair of parts is equal as well.

  13. Say what congruence then gives you

    all 6 pairs of parts equal\text{all 6 pairs of parts equal}

    Once two triangles are proved congruent, all three pairs of sides and all three pairs of angles are equal — that is what makes congruence worth proving.

  14. Summarise the method

    list parts→match a condition→conclude\text{list parts} \rightarrow \text{match a condition} \rightarrow \text{conclude}

    List the equal parts, see which of the five conditions they match, and only then write the conclusion.

  15. State the answer

    SSS\text{SSS}

    So the set that is enough is the SSS set: AB=UVAB = UV, BC=VWBC = VW and CA=WUCA = WU

Answer
SSS\text{SSS}
Question 2
5 markschallenging
Triangle ABCABC has sides of length 66 cm, 77 cm and 99 cm. Triangle PQRPQR has exactly the same three angles as triangle ABCABC, but triangle PQRPQR is not congruent to triangle ABCABC. Which of these could be the side lengths of triangle PQRPQR?
Show worked solution

Worked solution

  1. Say what equal angles do and do not give you

    AAA⇒similar\text{AAA} \Rightarrow \text{similar}

    Two triangles with the same three angles are similar: same shape, but not necessarily the same size. That is why AAA is not a congruence condition.

  2. Write down what similar means for the sides

    PQAB=QRBC=RPCA=k\frac{PQ}{AB} = \frac{QR}{BC} = \frac{RP}{CA} = k

    In similar triangles the sides are all in the same ratio kk, the scale factor of the enlargement.

  3. Check the angles really are the same

    cos⁡A=b2+c2−a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc}

    The cosine rule gives each angle from the ratios of the sides, and an enlargement leaves every ratio unchanged, so every angle is unchanged.

  4. Rule out a scale factor of one

    k≠1k \ne 1

    A scale factor of 11 would make the triangles congruent, and the question says they are not, so kk must be some other positive number.

  5. Test the option with a constant scale factor

    126=147=189=2\frac{12}{6} = \frac{14}{7} = \frac{18}{9} = 2

    Every side has been multiplied by the same number, 22, so this triangle is similar to the first one but a different size.

  6. Check the triangles are not congruent

    9≠189 \ne 18

    Congruent triangles have equal sides. These do not, so they are similar but not congruent — exactly the situation the question describes.

  7. Check a wrong option in detail

    76≠109\frac{7}{6} \ne \frac{10}{9}

    The first and last sides of that option are not scaled by the same factor, so it cannot be similar to the original triangle.

  8. Check another wrong option in detail

    66≠109\frac{6}{6} \ne \frac{10}{9}

    Again the ratios disagree, so the angles of that triangle are not the same as the angles of the original.

  9. Test the other options

    ratios not equal\text{ratios not equal}

    In each of the other options the three ratios are not all the same, so those triangles are not even similar to the first one — their angles are different.

  10. Say what extra information would force congruence

    AAA+one equal side⇒congruent\text{AAA} + \text{one equal side} \Rightarrow \text{congruent}

    Add any one pair of equal corresponding sides to equal angles and the scale factor is forced to be 11: that is the condition AAS (or ASA).

  11. Summarise

    k≠1⇒similar, not congruentk \ne 1 \Rightarrow \text{similar, not congruent}

    Any scale factor other than 11 gives a triangle with the same angles and different sides — an endless supply of counter-examples to AAA.

  12. Note the common mistake

    same angles≠same triangle\text{same angles} \ne \text{same triangle}

    Equal angles feel like enough, but they only fix the shape. Congruence always needs at least one length.

  13. Say why this proves AAA is not a condition

    AAA is not a congruence condition\text{AAA is not a congruence condition}

    One counter-example is enough: two triangles with identical angles and different sides. So AAA can never prove congruence.

  14. Restate the answer

    12, 14, 1812 , \ 14 , \ 18

    The side lengths are 1212 cm, 1414 cm and 1818 cm

  15. State the answer

    12, 14, 18 cm12 , \ 14 , \ 18 \text{ cm}

    So triangle PQRPQR could have sides 1212 cm, 1414 cm and 1818 cm — the same angles as the 6,, 7,, 9 triangle, but every length multiplied by 22.

Answer
12,14,18 cm12, 14, 18 \text{ cm}
Question 3
5 markschallenging
Triangle ABCABC and triangle PQRPQR are drawn so that ∠ABC=∠PQR=90∘\angle ABC = \angle PQR = 90^\circ, CA=RP=41CA = RP = 41 cm and BC=QR=40BC = QR = 40 cm. Which statement about triangle ABCABC and triangle PQRPQR is correct?
Show worked solution

Worked solution

  1. Write down the pairs of equal parts that the question gives

    ∠ABC=∠PQR=90∘,CA=RP=41 cm,BC=QR=40 cm\angle ABC = \angle PQR = 90^\circ, \quad CA = RP = 41 \text{ cm}, \quad BC = QR = 40 \text{ cm}

    The question gives ∠ABC\angle ABC, CACA, BCBC as equal pairs. Everything else has to be deduced.

  2. Write down how the vertices correspond

    A↔P, B↔Q, C↔RA \leftrightarrow P, \ B \leftrightarrow Q, \ C \leftrightarrow R

    The equal parts pair AA with PP, BB with QQ and CC with RR. Getting the correspondence right is what makes the rest of the argument safe.

  3. Count the pairs of equal sides

    equal sides=2\text{equal sides} = 2

    There are 22 pairs of equal sides.

  4. Count the pairs of equal angles

    equal angles=1\text{equal angles} = 1

    There is 11 pair of equal angles. Sides and angles together are what a congruence condition is built from.

  5. Describe the pattern of the given information

    R, H, S (right angle, hypotenuse, one other side)\text{R, H, S (right angle, hypotenuse, one other side)}

    Written in order round the triangle the information is R, H, S (right angle, hypotenuse, one other side).

  6. Test the information against SSS

    SSS: needs 3 sides, 2 given⇒no\text{SSS: needs 3 sides, 2 given} \Rightarrow \text{no}

    SSS needs all three pairs of sides to be equal.

  7. Test the information against SAS

    SAS: 2 sides + the angle between them, not that pattern⇒no\text{SAS: 2 sides + the angle between them, not that pattern} \Rightarrow \text{no}

    SAS needs the equal angle to sit between the two equal sides.

  8. Test the information against ASA

    ASA: 2 angles + the side between them, not that pattern⇒no\text{ASA: 2 angles + the side between them, not that pattern} \Rightarrow \text{no}

    ASA needs the equal side to sit between the two equal angles.

  9. Test the information against AAS

    AAS: 2 angles + a side not between them, not that pattern⇒no\text{AAS: 2 angles + a side not between them, not that pattern} \Rightarrow \text{no}

    AAS needs two angles and a side that is not between them.

  10. Test the information against RHS

    RHS: right angle + hypotenuse + one other side, that is the pattern⇒yes\text{RHS: right angle + hypotenuse + one other side, that is the pattern} \Rightarrow \text{yes}

    A right angle, equal hypotenuses and one other equal side: RHS applies.

  11. Check the information is not one of the patterns that fails

    SSA and AAA are not conditions\text{SSA and AAA are not conditions}

    The two patterns that never prove congruence are SSA (two sides and an angle that is not between them) and AAA (angles only). The information here is not either of them.

  12. Say why angles alone are never enough

    AAA⇒similar, not congruent\text{AAA} \Rightarrow \text{similar, not congruent}

    Two triangles with the same three angles are similar: one is an enlargement of the other. Congruence needs at least one length.

  13. State the condition that matches the given information

    given information=RHS⇒congruent\text{given information} = \text{RHS} \Rightarrow \text{congruent}

    The information matches RHS, so triangle ABCABC and triangle PQRPQR are congruent.

  14. Write down what can be concluded

    △ABC≅△PQR\triangle ABC \cong \triangle PQR

    Triangle ABCABC is congruent to triangle PQRPQR, so every corresponding side and angle is equal.

  15. State the answer

    RHS\text{RHS}

    So the correct statement is: Congruent by RHS: a right angle, an equal hypotenuse and one other equal side are given.

Answer
RHS\text{RHS}
Question 4
5 markschallenging
Triangle ABCABC and triangle PQRPQR are drawn so that AB=PQ=14AB = PQ = 14 cm, CA=RP=9CA = RP = 9 cm and ∠ABC=∠PQR=30∘\angle ABC = \angle PQR = 30^\circ. Which statement about triangle ABCABC and triangle PQRPQR is correct?
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Worked solution

  1. Write down the pairs of equal parts that the question gives

    AB=PQ=14 cm,CA=RP=9 cm,∠ABC=∠PQR=30∘AB = PQ = 14 \text{ cm}, \quad CA = RP = 9 \text{ cm}, \quad \angle ABC = \angle PQR = 30^\circ

    The question gives ABAB, CACA, ∠ABC\angle ABC as equal pairs. Everything else has to be deduced.

  2. Write down how the vertices correspond

    A↔P, B↔Q, C↔RA \leftrightarrow P, \ B \leftrightarrow Q, \ C \leftrightarrow R

    The equal parts pair AA with PP, BB with QQ and CC with RR. Getting the correspondence right is what makes the rest of the argument safe.

  3. Count the pairs of equal sides

    equal sides=2\text{equal sides} = 2

    There are 22 pairs of equal sides.

  4. Count the pairs of equal angles

    equal angles=1\text{equal angles} = 1

    There is 11 pair of equal angles. Sides and angles together are what a congruence condition is built from.

  5. Describe the pattern of the given information

    S, S, A (the angle is not between the two sides)\text{S, S, A (the angle is not between the two sides)}

    Written in order round the triangle the information is S, S, A (the angle is not between the two sides).

  6. Test the information against SSS

    SSS: needs 3 sides, 2 given⇒no\text{SSS: needs 3 sides, 2 given} \Rightarrow \text{no}

    SSS needs all three pairs of sides to be equal.

  7. Test the information against SAS

    SAS: 2 sides + the angle between them, not that pattern⇒no\text{SAS: 2 sides + the angle between them, not that pattern} \Rightarrow \text{no}

    SAS needs the equal angle to sit between the two equal sides.

  8. Test the information against ASA

    ASA: 2 angles + the side between them, not that pattern⇒no\text{ASA: 2 angles + the side between them, not that pattern} \Rightarrow \text{no}

    ASA needs the equal side to sit between the two equal angles.

  9. Test the information against AAS

    AAS: 2 angles + a side not between them, not that pattern⇒no\text{AAS: 2 angles + a side not between them, not that pattern} \Rightarrow \text{no}

    AAS needs two angles and a side that is not between them.

  10. Test the information against RHS

    RHS: right angle + hypotenuse + one other side, not that pattern⇒no\text{RHS: right angle + hypotenuse + one other side, not that pattern} \Rightarrow \text{no}

    RHS needs a right angle, equal hypotenuses and one other equal side.

  11. Show that two different triangles fit the same information

    BC=−42+73orBC=42+73BC = - 4 \sqrt{2} + 7 \sqrt{3} \quad \text{or} \quad BC = 4 \sqrt{2} + 7 \sqrt{3}

    With ∠ABC=30∘\angle ABC = 30^\circ, AB=14AB = 14 cm and CA=9CA = 9 cm, the third side can take either of two values, −42+73- 4 \sqrt{2} + 7 \sqrt{3} or 42+734 \sqrt{2} + 7 \sqrt{3}. Two different triangles fit the same SSA information, so it cannot prove congruence.

  12. Say why angles alone are never enough

    AAA⇒similar, not congruent\text{AAA} \Rightarrow \text{similar, not congruent}

    Two triangles with the same three angles are similar: one is an enlargement of the other. Congruence needs at least one length.

  13. State the condition that matches the given information

    SSA∉{SSS, SAS, ASA, AAS, RHS}\text{SSA} \notin \{ \text{SSS, SAS, ASA, AAS, RHS} \}

    The information is SSA, which does not fix the triangle, so the two triangles need not be congruent.

  14. Write down what can be concluded

    △ABC≠△PQR necessarily\triangle ABC \ne \triangle PQR \text{ necessarily}

    Triangle ABCABC and triangle PQRPQR may be congruent, but the information given does not prove it.

  15. State the answer

    Not necessarily congruent (SSA)\text{Not necessarily congruent (SSA)}

    So the correct statement is: Not necessarily congruent: the given information is SSA (two sides and an angle that is not between them), which does not fix the triangle.

Answer
Not necessarily congruent (SSA)\text{Not necessarily congruent (SSA)}
Question 5
5 markschallenging
Triangle ABCABC and triangle PQRPQR are drawn so that ∠BAC=∠QPR=62∘\angle BAC = \angle QPR = 62^\circ, AB=PQ=13AB = PQ = 13 cm and ∠ABC=∠PQR=49∘\angle ABC = \angle PQR = 49^\circ. Which statement about triangle ABCABC and triangle PQRPQR is correct?
Show worked solution

Worked solution

  1. Write down the pairs of equal parts that the question gives

    ∠BAC=∠QPR=62∘,AB=PQ=13 cm,∠ABC=∠PQR=49∘\angle BAC = \angle QPR = 62^\circ, \quad AB = PQ = 13 \text{ cm}, \quad \angle ABC = \angle PQR = 49^\circ

    The question gives ∠BAC\angle BAC, ABAB, ∠ABC\angle ABC as equal pairs. Everything else has to be deduced.

  2. Write down how the vertices correspond

    A↔P, B↔Q, C↔RA \leftrightarrow P, \ B \leftrightarrow Q, \ C \leftrightarrow R

    The equal parts pair AA with PP, BB with QQ and CC with RR. Getting the correspondence right is what makes the rest of the argument safe.

  3. Count the pairs of equal sides

    equal sides=1\text{equal sides} = 1

    There is 11 pair of equal sides.

  4. Count the pairs of equal angles

    equal angles=2\text{equal angles} = 2

    There are 22 pairs of equal angles. Sides and angles together are what a congruence condition is built from.

  5. Describe the pattern of the given information

    A, S, A (the side lies between the two angles)\text{A, S, A (the side lies between the two angles)}

    Written in order round the triangle the information is A, S, A (the side lies between the two angles).

  6. Test the information against SSS

    SSS: needs 3 sides, 1 given⇒no\text{SSS: needs 3 sides, 1 given} \Rightarrow \text{no}

    SSS needs all three pairs of sides to be equal.

  7. Test the information against SAS

    SAS: 2 sides + the angle between them, not that pattern⇒no\text{SAS: 2 sides + the angle between them, not that pattern} \Rightarrow \text{no}

    SAS needs the equal angle to sit between the two equal sides.

  8. Test the information against ASA

    ASA: 2 angles + the side between them, side is between them⇒yes\text{ASA: 2 angles + the side between them, side is between them} \Rightarrow \text{yes}

    The equal side sits between the two equal angles, so ASA applies.

  9. Test the information against AAS

    AAS: 2 angles + a side not between them, not that pattern⇒no\text{AAS: 2 angles + a side not between them, not that pattern} \Rightarrow \text{no}

    AAS needs two angles and a side that is not between them.

  10. Test the information against RHS

    RHS: right angle + hypotenuse + one other side, not that pattern⇒no\text{RHS: right angle + hypotenuse + one other side, not that pattern} \Rightarrow \text{no}

    RHS needs a right angle, equal hypotenuses and one other equal side.

  11. Check the information is not one of the patterns that fails

    SSA and AAA are not conditions\text{SSA and AAA are not conditions}

    The two patterns that never prove congruence are SSA (two sides and an angle that is not between them) and AAA (angles only). The information here is not either of them.

  12. Say why angles alone are never enough

    AAA⇒similar, not congruent\text{AAA} \Rightarrow \text{similar, not congruent}

    Two triangles with the same three angles are similar: one is an enlargement of the other. Congruence needs at least one length.

  13. State the condition that matches the given information

    given information=ASA⇒congruent\text{given information} = \text{ASA} \Rightarrow \text{congruent}

    The information matches ASA, so triangle ABCABC and triangle PQRPQR are congruent.

  14. Write down what can be concluded

    △ABC≅△PQR\triangle ABC \cong \triangle PQR

    Triangle ABCABC is congruent to triangle PQRPQR, so every corresponding side and angle is equal.

  15. State the answer

    ASA\text{ASA}

    So the correct statement is: Congruent by ASA: two pairs of equal angles and the equal side between them are given.

Answer
ASA\text{ASA}

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