Practice Maths

Congruent Figures

Key Ideas

Key Terms

Congruent
Two figures are congruent (≅) if they are exactly the same shape and size — one can be flipped, rotated or slid to match the other perfectly.
Corresponding order
When writing a congruence statement, vertices must be listed in matching order so that equal sides and angles are correctly identified.
SSS
Side-Side-Side — three pairs of equal corresponding sides prove two triangles are congruent.
SAS
Side-Angle-Side — two pairs of equal sides and the included angle (between those two sides) prove congruence.
AAS
Angle-Angle-Side — two pairs of equal angles and one pair of equal corresponding sides prove congruence.
RHS
Right angle-Hypotenuse-Side — for right-angled triangles, equal hypotenuse and one other side prove congruence.
Corresponding sides
Sides in the same position in congruent (or similar) figures — equal in length for congruent figures.
Corresponding angles
Angles in the same position in congruent (or similar) figures — always equal in both congruent and similar figures.

Naming Congruent Triangles

If ▵ABC ≅ ▵DEF, then A corresponds to D, B to E, C to F.
This means AB = DE, BC = EF, AC = DF, and ∠A = ∠D, ∠B = ∠E, ∠C = ∠F.

Hot Tip Always check which congruence test applies before writing your proof. Write the test (SSS, SAS, AAS or RHS) as part of your reasoning. Order matters when writing the congruence statement.

Worked Example

Question: ▵PQR has PQ = 5 cm, QR = 7 cm, PR = 9 cm. ▵XYZ has XY = 5 cm, YZ = 7 cm, XZ = 9 cm. Are the triangles congruent? If so, state the test and write the congruence statement.

Step 1 — Compare the sides.
PQ = XY = 5 cm, QR = YZ = 7 cm, PR = XZ = 9 cm. All three pairs of sides are equal.

Step 2 — State the test.
Three sides equal → SSS congruence test.

Step 3 — Write the congruence statement in matching order.
▵PQR ≅ ▵XYZ (SSS)

What Does Congruent Mean?

Two figures are congruent if they are exactly the same shape AND the same size. One can be placed on top of the other to match perfectly — possibly after being flipped, rotated, or slid into position. The symbol for congruence is ≅.

Congruence means all corresponding sides are equal in length and all corresponding angles are equal in measure. A shape that has been reflected (flipped), rotated (turned), or translated (slid) is still congruent to the original — only its position or orientation has changed, not its actual size or shape. A shape that has been enlarged or shrunk is NOT congruent (it is similar).

Corresponding Parts

When two figures are congruent, every side and every angle in one figure corresponds to a specific side or angle in the other. These are called corresponding parts. In a written congruence statement, the order of the vertices tells you which parts correspond.

Example: If triangle ABC ≅ triangle DEF, then: A corresponds to D, B corresponds to E, C corresponds to F. This means: AB = DE, BC = EF, AC = DF, and ∠A = ∠D, ∠B = ∠E, ∠C = ∠F.

Always match up the corresponding vertices carefully before making any statements about sides or angles.

Congruence Tests for Triangles

For triangles, you don't need to check all six measurements (three sides and three angles) to prove congruence. There are four shortcut tests:

  • SSS (Side-Side-Side): All three pairs of corresponding sides are equal.
  • SAS (Side-Angle-Side): Two pairs of corresponding sides are equal, and the included angle (between those two sides) is equal.
  • AAS (Angle-Angle-Side): Two pairs of corresponding angles are equal and one pair of corresponding sides is equal (must be the non-included side, or use ASA if the included side is between the angles).
  • RHS (Right angle-Hypotenuse-Side): Both triangles have a right angle, and the hypotenuse and one other side are equal.

A warning: AAA (three equal angles) does NOT prove congruence — it only proves similarity (same shape, potentially different sizes).

Why Are the Congruence Tests Important?

In geometry, we often need to prove that two triangles are congruent in order to conclude that certain sides or angles are equal. The congruence tests give us a logical way to justify these claims with the minimum amount of information. This is the foundation of geometric proof.

For example, to prove that two bridge supports are exactly the same, an engineer would check whether the relevant triangles satisfy SSS or SAS — checking three measurements is much more practical than checking all six.

Congruence in Everyday Life

Congruent shapes appear everywhere in real life. Mass-produced objects (tiles, bricks, circuit boards, coins) are all congruent to each other by design. Architects and engineers rely on congruence to ensure that identical components fit together perfectly. In art and design, congruent shapes create symmetric, balanced patterns.

Key tip: When writing a congruence statement, the order of vertices matters! Always list the vertices in corresponding order. If you write "triangle ABC ≅ triangle PQR," you are claiming A corresponds to P, B to Q, and C to R. Getting the order wrong will lead to incorrect statements about which sides and angles are equal. Take the time to match the shapes up carefully before writing the statement.

Mastery Practice

  1. State whether each pair of figures is congruent. Give a reason for each answer. Fluency

    1. Two squares, both with side length 6 cm.
    2. A rectangle 4 cm × 7 cm and a rectangle 7 cm × 4 cm.
    3. A triangle with sides 3, 4, 5 cm and a triangle with sides 3, 4, 6 cm.
    4. Two circles, both with radius 5 cm.
    5. A triangle with angles 40°, 60°, 80° and sides 5, 7, 8 cm, and another triangle with angles 40°, 60°, 80° and sides 10, 14, 16 cm.
    6. Two equilateral triangles, both with side length 9 cm.

  2. Given that ▵ABC ≅ ▵PQR, write the value of each measurement. Fluency

    AB = 8 cm, BC = 11 cm, AC = 6 cm, ∠A = 55°, ∠B = 70°, ∠C = 55°

    1. PQ = ?
    2. QR = ?
    3. PR = ?
    4. ∠P = ?
    5. ∠Q = ?
    6. ∠R = ?

  3. For each pair of triangles, state which congruence test applies (SSS, SAS, AAS or RHS) or explain why no test applies. Fluency

    1. Two triangles each with sides 5 cm, 8 cm, 10 cm.
    2. Two triangles each with two sides 6 cm and 9 cm, and the included angle 48°.
    3. Two triangles each with angles 35° and 75°, and the side between these angles 7 cm.
    4. Two right-angled triangles each with hypotenuse 13 cm and one leg 5 cm.
    5. Two triangles each with angles 50°, 60°, 70° but one has sides 4, 5, 6 cm and the other has sides 8, 10, 12 cm.
    6. Two triangles with two equal angles and a non-corresponding equal side.
    7. Two right-angled triangles with equal legs of 4 cm and 3 cm.
    8. Two triangles with two equal sides and a non-included angle equal.

  4. Use congruence to find the unknown values. Understanding

    1. ▵KLM ≅ ▵STU. KL = 2x + 1, ST = 13. Find x.
    2. ▵DEF ≅ ▵GHI. ∠D = 3y − 5, ∠G = 70°. Find y.
    3. ▵ABC ≅ ▵XYZ. AC = 4a, XZ = 20. AB = b + 3, XY = 11. Find a and b.
    4. In quadrilateral ABCD, diagonal BD divides it into ▵ABD and ▵CBD. If ▵ABD ≅ ▵CBD, and AB = 7 cm, find CB.
    5. ▵PQR ≅ ▵LMN. ∠P = 42°, ∠Q = 95°. Find ∠N.

  5. For each pair of triangles, write a full congruence proof. State the test used and write the congruence statement. Understanding

    1. In ▵ABC and ▵DEF: AB = DE = 9 cm, BC = EF = 12 cm, AC = DF = 15 cm.
    2. In ▵PQR and ▵STU: PQ = ST = 7 cm, ∠PQR = ∠STU = 63°, QR = TU = 10 cm.
    3. In ▵XYZ and ▵MNO: ∠X = ∠M = 40°, ∠Y = ∠N = 85°, XY = MN = 6 cm.
    4. In right-angled ▵ABC (right angle at B) and right-angled ▵DEF (right angle at E): AC = DF = 17 cm, AB = DE = 8 cm.

  6. Problem solving with congruence. Problem Solving

    1. A ladder leans against a vertical wall. The ladder, wall, and ground form a right-angled triangle. A second identical ladder is placed against the same wall at the same angle. Explain why the two triangles formed are congruent and state the congruence test.
    2. In the diagram, ABCD is a rectangle. Diagonal AC divides it into ▵ABC and ▵CDA.
      1. Write all pairs of equal sides and equal angles.
      2. State the congruence test and write the full congruence statement.
    3. Two triangular garden beds are to be built. The first has measurements: sides 3.5 m, 4 m, 5 m. The builder claims the second bed (sides 5 m, 4 m, 3.5 m) is congruent to the first. Is the builder correct? Explain why, and identify the congruence test.
    4. A student says “If two triangles have two equal angles, they must be congruent.” Is this statement correct? If not, give a counterexample.
  7. Algebraic Congruence. Use congruence statements and algebra together.

    Use congruence to find all unknown values. Show full working. Problem Solving

    1. ▵ABC ≅ ▵PQR. AB = 3x − 2, PQ = x + 6, BC = 2y + 1, QR = 13. Find x and y, then state the lengths AB and BC.
    2. ▵DEF ≅ ▵GHI. ∠D = 4a + 10, ∠G = 90°, ∠E = b + 15, ∠H = 55°. Find a and b, then find ∠F.
    3. In isosceles triangle ABC, AB = AC. M is the midpoint of BC. Show that ▵ABM ≅ ▵ACM, and state the congruence test used.
    4. ▵KLM ≅ ▵NOP. KL = 2(m + 1), NO = 3m − 4, KM = n + 7, NP = 2n − 3. Find m and n.
  8. Congruence Proof. Use known geometric properties to build a congruence argument.

    For each situation, identify the equal sides and angles, state the congruence test, and write the congruence statement. Problem Solving

    1. In parallelogram ABCD, diagonal AC is drawn. Explain why ▵ABC ≅ ▵CDA, naming the equal sides and angles and the congruence test used.
    2. Two right-angled triangles, ▵PQR (right angle at Q) and ▵STU (right angle at T), are given with PQ = ST = 6 cm and PR = SU = 10 cm. Write a congruence proof.
    3. In kite ABCD with AB = AD and CB = CD, diagonal BD is drawn. Prove that ▵ABD ≅ ▵ACD, stating the test used.
    4. Triangles XYZ and MNO have XY = MN, ∠Y = ∠N, YZ = NO. Write a full proof that the triangles are congruent, including the congruence statement and test.
  9. Real-World Congruence. Congruence appears in construction, engineering and design.

    Solve each real-world congruence problem. Problem Solving

    1. A bridge uses a truss design with two triangles. Triangle 1 has sides 4 m, 5 m, 6 m. Triangle 2 has sides 6 m, 4 m, 5 m. Are the triangles congruent? Name the test and explain why the structure is symmetric.
    2. A tiler wants to cut congruent right-angled triangles from a rectangular tile 8 cm × 6 cm by cutting along the diagonal. What are the measurements of each triangle? Are the two triangles congruent? Justify with a congruence test.
    3. A student claims that two triangles are congruent because they have the same area. Give a counterexample to disprove this claim.
    4. A surveyor needs to find the width of a river without crossing it. She marks points A and B on one bank, and point C directly opposite A on the other bank. She measures AB = 30 m, and finds a point D on her side such that ▵ABD ≅ ▵CAB. Explain what measurement she can now make, and why.
  10. Extended Reasoning. Combine congruence with angle properties.

    Answer each question using congruence reasoning and angle laws. Problem Solving

    1. In ▵ABC, ∠B = 90°, AB = 5 cm, BC = 12 cm. A second triangle ▵DEF has ∠E = 90° and DF = 13 cm, DE = 5 cm. Prove ▵ABC ≅ ▵DEF using RHS. (Hint: find AC first.)
    2. ABCD is a square. Diagonal AC divides it into two triangles.
      1. Name the two triangles and write a congruence statement for them.
      2. Which congruence test applies?
      3. What does this tell you about the angles that AC makes with AB and AD?
    3. Two triangles share the same base PQ = 8 cm. Their apex vertices R and S both lie on a line parallel to PQ. Without knowing their exact positions, can you say the triangles are congruent? Explain your reasoning.
    4. Explain the difference between two figures being congruent and two figures being similar. Give one example of each from everyday life.
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