GCSE Notation and drawing Practice Questions

Free GCSE Notation and drawing practice questions with full step-by-step worked solutions. Covers three-letter angle notation, vertex, polygon names, counting sides. Practise exam-style problems and check your method.

three-letter angle notationvertexpolygon namescounting sidestriangle typesequilateral
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
The angle ∠ABC\angle ABC is marked in a diagram. Which letter names the vertex of this angle?
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Worked solution

  1. Read the three letters in order

    ∠ABC\angle ABC

    The three letters are A, then B, then C.

  2. Recall the rule for three-letter angle notation

    ∠ABC  →  vertex=B\angle ABC \;\rightarrow\; \text{vertex} = B

    In three-letter angle notation the MIDDLE letter is always the vertex, the corner where the two arms meet.

  3. Name the vertex

    B

    The middle letter is B, so the angle is at B, between the arms BA and BC.

Answer
B\text{B}
Question 2
1 markeasy
How many sides does a decagon have?
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Worked solution

  1. Recall the polygon names

    deca-=10\text{deca-} = 10

    The prefix "deca" means ten.

  2. Apply it to the polygon

    decagon→10 sides\text{decagon} \rightarrow 10 \text{ sides}

    A decagon is a polygon with ten straight sides.

  3. State the answer

    1010

    A decagon has 10 sides.

Answer
1010
Question 3
2 marksintermediate
P is the point (2, 1) and Q is the point (2, 6). M is the midpoint of the line segment PQ. Write down the coordinates of M.
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Worked solution

  1. Note what a midpoint is

    PM=MQPM = MQ

    The midpoint is the point exactly halfway along the segment; it bisects PQ.

  2. Compare the x-coordinates

    2 and 22 \text{ and } 2

    Both points have x=2x = 2, so PQ is a vertical segment and M also has x=2x = 2.

  3. Average the y-coordinates

    1+62=72\frac{1 + 6}{2} = \frac{7}{2}

    The midpoint of the y-values is 7 over 2.

  4. Write it as a decimal

    72=3.5\frac{7}{2} = 3.5

    Seven halves is 3.5.

  5. Write the coordinates

    M=(2,  3.5)M = (2,\; 3.5)

    So M is the point (2, 3.5).

  6. Check the two halves

    3.5−1=2.5,6−3.5=2.53.5 - 1 = 2.5, \quad 6 - 3.5 = 2.5

    Both halves are 2.5 units long, so M really is the midpoint.

Answer
M=(2,  3.5)M = (2,\; 3.5)
Question 4
3 markshard
In quadrilateral ABCD, AB∥DCAB \parallel DC and AB=DCAB = DC. What is the strongest conclusion you can draw about ABCD?
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Worked solution

  1. Write the two facts down

    AB∥DC,AB=DCAB \parallel DC, \quad AB = DC

    One pair of opposite sides is both equal in length and parallel.

  2. Turn the facts into a movement

    A→B is the same move as D→CA \rightarrow B \text{ is the same move as } D \rightarrow C

    Equal and parallel means the journey from A to B is exactly the same move as the journey from D to C.

  3. Compare the other two sides

    A→D and B→CA \rightarrow D \text{ and } B \rightarrow C

    The move from A to D must then be the same as the move from B to C, because both are found by subtracting the same move.

  4. Conclude the other pair is parallel and equal

    AD∥BC,AD=BCAD \parallel BC, \quad AD = BC

    So the second pair of opposite sides is equal and parallel as well.

  5. Name the shape

    parallelogram\text{parallelogram}

    Two pairs of parallel sides is exactly the definition of a parallelogram.

  6. Test with an example

    A(1,1),B(4,2),C(6,6),D(3,5)A(1,1), B(4,2), C(6,6), D(3,5)

    From A to B is (3, 1); from D to C is also (3, 1). From A to D is (2, 4), and from B to C is also (2, 4). It is a parallelogram.

  7. Check it need not be a rectangle

    (3,1)⋅(2,4)=10≠0(3,1)\cdot(2,4) = 10 \ne 0

    In that example the sides at A are not perpendicular, so the shape is not a rectangle. So "rectangle" is too strong.

  8. Check it need not be a rhombus

    AB2=10,AD2=20AB^2 = 10, \quad AD^2 = 20

    The adjacent sides are not equal, so it is not a rhombus. "Rhombus" is too strong as well.

  9. Check it cannot be a one-pair trapezium

    AD∥BCAD \parallel BC

    Both pairs of sides are parallel, so it cannot be a trapezium with EXACTLY one pair of parallel sides.

  10. State the strongest conclusion

    ABCD is a parallelogramABCD \text{ is a parallelogram}

    ABCD must be a parallelogram, and nothing stronger can be guaranteed.

Answer
ABCD must be a parallelogramABCD \text{ must be a parallelogram}
Question 5
6 markschallenging
A quadrilateral has two pairs of parallel sides, all four sides equal in length, and diagonals that are equal in length. Give its name.
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Worked solution

  1. Write down the three clues

    2 pairs parallel;  4 equal sides;  equal diagonals2 \text{ pairs parallel};\; 4 \text{ equal sides};\; \text{equal diagonals}

    All three must hold at once.

  2. Use the first clue

    2 pairs parallel⇒parallelogram2 \text{ pairs parallel} \Rightarrow \text{parallelogram}

    Two pairs of parallel sides is the definition of a parallelogram.

  3. Eliminate the kite

    kite: 0 pairs parallel\text{kite: 0 pairs parallel}

    A kite that is not a rhombus has no parallel sides at all, so it fails the first clue.

  4. Eliminate the one-pair trapezium

    1≠21 \ne 2

    A trapezium with exactly one pair of parallel sides fails the first clue.

  5. Eliminate the isosceles trapezium

    1≠21 \ne 2

    An isosceles trapezium also has only one pair of parallel sides, so it fails too.

  6. Use the second clue

    4 equal sides⇒rhombus4 \text{ equal sides} \Rightarrow \text{rhombus}

    A parallelogram with all four sides equal is a rhombus.

  7. Eliminate the rectangle that is not a square

    6≠36 \ne 3

    A rectangle like (0, 0), (6, 0), (6, 3), (0, 3) has sides 6, 3, 6, 3, which are not all equal.

  8. Eliminate the parallelogram with no right angles

    AB2=25,  BC2=13AB^2 = 25, \; BC^2 = 13

    The parallelogram (0, 0), (5, 0), (7, 3), (2, 3) has unequal adjacent sides, so its four sides are not all equal.

  9. Take stock

    it must be a rhombus\text{it must be a rhombus}

    The first two clues force the shape to be a rhombus. The third clue must now separate the square from the other rhombuses.

  10. Test a rhombus that is not a square

    (0,0),(5,0),(8,4),(3,4)(0,0), (5,0), (8,4), (3,4)

    This rhombus has diagonals with squared lengths 80 and 20.

  11. Eliminate it

    80≠2080 \ne 20

    Its diagonals are NOT equal, so it fails the third clue. A rhombus that is not a square is ruled out.

  12. Explain why equal diagonals force right angles

    parallelogram+equal diagonals⇒rectangle\text{parallelogram} + \text{equal diagonals} \Rightarrow \text{rectangle}

    In a parallelogram the diagonals are equal exactly when all four angles are right angles, which makes it a rectangle.

  13. Combine the two names

    rhombus+rectangle=square\text{rhombus} + \text{rectangle} = \text{square}

    The shape is a rhombus (four equal sides) AND a rectangle (four right angles). A shape that is both is a square.

  14. Check with a real square

    (0,0),(4,0),(4,4),(0,4)(0,0), (4,0), (4,4), (0,4)

    Its diagonals are (4, 4) and (-4, 4), both with squared length 32, so they ARE equal. All four sides are 4. Every clue is satisfied.

  15. State the answer

    Square\text{Square}

    The only quadrilateral satisfying all three clues is the square.

Answer
Square\text{Square}

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