Most tested M5.5

Geometric Reasoning and Proof

This topic tests your ability to solve multi-step geometry problems by combining fundamental rules about angles, triangles, and quadrilaterals. Success depends on systematically breaking down complex diagrams and applying the correct properties to find unknown lengths and angles.

Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • Always start by marking all known and deduced information directly onto the diagram, such as equal sides, right angles, and parallel lines.
  • Identify all the simple shapes (e.g., triangles, rectangles) that make up the larger, more complex figure.
  • Use triangle congruence (SSS, SAS, ASA, RHS) to prove that two triangles are identical, which means their corresponding sides and angles are equal.
  • Apply triangle similarity (AAA) to find side lengths in scaled shapes by setting up and solving proportions.
  • Systematically work through angle facts: angles on a straight line sum to 180°, angles around a point sum to 360°, and angles in a quadrilateral sum to 360°.
  • Look for 'hidden' isosceles triangles, which are often created when shapes with equal side lengths are placed next to each other.

Formulae

SAS (Side-Angle-Side) Congruence

To prove two triangles are identical when you know two sides and the angle between them are equal in both triangles.

SSS (Side-Side-Side) Congruence

To prove two triangles are identical when you know all three corresponding sides are equal.

ASA (Angle-Side-Angle) Congruence

To prove two triangles are identical when you know two angles and the side between them are equal in both triangles.

AAA (Angle-Angle-Angle) Similarity

To prove two triangles are similar. If two angles are equal, the third must also be equal, which is sufficient.

Definitions

Congruent Triangles
Triangles that are identical in both shape and size. All corresponding sides and angles are equal.
Similar Triangles
Triangles that have the same shape but may be different sizes. Their corresponding angles are equal, and the ratio of their corresponding side lengths is constant.

Worked example

PQRS is a square. An equilateral triangle PQT is constructed on the side PQ, with the triangle lying inside the square. The line T R is drawn. What is the size of angle STR?

  1. 1

    Draw the diagram.

    Since PQRS is a square, all sides are equal (PQ = QR = RS = SP) and all angles are 90°.

  2. 2

    The triangle PQT is equilateral, so PQ = QT = TP and all its angles are 60°.

  3. 3

    From the first two points, we can deduce that TP = PQ = QR.

    Therefore, triangle TQR is an isosceles triangle.

  4. 4

    Calculate the angle TQR.

    Angle PQR is 90° (corner of the square) and angle PQT is 60° (equilateral triangle).

    So, angle TQR = 90° - 60° = 30°.

  5. 5

    In isosceles triangle TQR, the base angles QTR and QRT are equal.

    Their sum is 180° - 30° = 150°.

    Therefore, angle QTR = 150° / 2 = 75°.

  6. 6

    Similarly, SP = PQ = PT, making triangle SPT an isosceles triangle with angle SPT = 90° - 60° = 30°.

    The base angles PST and PTS are both 75°.

  7. 7

    The angle STR is part of the 360° around point T inside the square.

    We know angle PTQ = 60° and angle QTR = 75° and angle PTS = 75°.

    The angles must sum to 360°, but that's not helpful.

    Instead let's find the angle STR in triangle STR, but that seems too complex.

    Let's look at the angle at T.

    The total angle at T is not 360 since T is not the center.

    Let's re-evaluate.

    The sum of angles in quadrilateral TQRS is 360°.

    We have TQR=30, QRS=90, RST=90-PST=90-75=15

    We need angle QTS.

    This is getting complex.

    Let's restart from step 5.

  8. 8

    Let's check our isosceles triangles again.

    TQR is isosceles, angle TQR=30°, base angles QTR and QRT are both (180-30)/2 = 75°.

    This is correct.

  9. 9

    We want angle STR.

    We know angle QRS is 90°.

    Angle STR = Angle QRS - Angle QRT

    Wait, that's not right.

    We want the angle STR.

  10. 10

    Let's focus on the line segment TR and consider triangle TRS.

    We know RS is a side of the square.

    We need side TR and TS to see if it is isosceles.

    By symmetry, triangle TQR and triangle TPS are congruent (SAS:

    TQ=TP, QR=PS, angle TQR = angle TPS = 30)
    So TR = TS

    This means triangle TRS is isosceles.

  11. 11

    Now we need an angle in triangle TRS.

    Let's find angle RTS.

    Angle PTQ = 60°
    Angle PTS = 75°
    Angle QTR = 75°

    Angle RTS is the angle needed.

    We know the angle PTS is 75° from our earlier calculation.

    The point T lies on the line of symmetry of the square through PQ and SR.

    So the triangle TSR is isosceles with TS=TR.

    Angle TSR = Angle TRS

    The angle at the top is Angle STR.

    No, Angle RTS is the angle at the vertex T.

    Angle RTS = 360° - (Angle PTQ + Angle PTS + Angle QTR) - This is only if T is the center

    Let's go back.

    The angle of the square is 90°.

    Angle PQR = 90
    Angle PQT = 60
    Angle TQR = 30

    Triangle TQR is isosceles (TQ=QR).

    So angle QTR = (180-30)/2 = 75°.

  12. 12

    Now consider angle TRS.

    The full angle at R is QRS = 90°.

    We found angle QRT = 75°.

    So angle TRS = 90° - 75° = 15°.

  13. 13

    By symmetry, triangle SPT is congruent to triangle TQR.

    So angle PST = angle QRT = 75°.

    Angle TSR = angle PSR - angle PST = 90 - 75 = 15°
  14. 14

    Now we have triangle STR.

    We know two angles are 15° (angle TRS and angle TSR).

    So angle STR = 180° - 15° - 15° = 150°.

Answer: 150°

Common mistakes

  • ×Forgetting one of the key properties of a given shape, such as opposite angles being equal in a parallelogram or diagonals bisecting at 90° in a kite or rhombus.
  • ×Making arithmetic mistakes when calculating angles, particularly when subtracting from 180 or 360 and then dividing by 2 for isosceles triangles.
  • ×Assuming two triangles are congruent or similar without rigorously checking that the conditions (e.g., SSS, SAS, AAA) are met.
  • ×Incorrectly applying similarity ratios, for instance by dividing corresponding sides the wrong way around or mixing up which sides correspond.

No-calculator tips

  • ✓Break down angle arithmetic into easier steps. For (180 - 46)/2, calculate 180 - 40 = 140, then 140 - 6 = 134. Half of 134 is half of 100 (50) plus half of 34 (17), which is 67.
  • ✓Visually scan the completed diagram after marking all initial information. Often, new isosceles triangles or other useful shapes become apparent that weren't obvious from the text alone.
  • ✓If a calculation seems very complex, re-read the question. You may have missed a simpler property or a piece of information that provides a shortcut.

Read this topic in the official UAT-UK ESAT guide →

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