AQA GCSE Maths · 3.4.1

Properties and constructions

50 maths questions with worked answers with answers for 3.4.1 Properties and constructions in AQA GCSE Maths (spec 8300). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official AQA material.

All 50 questions and answers

Question 1 · Angles · G3
Two angles on a straight line are x and 68°. Find x.
Answer:
Angles on a straight line total 180°.
x = 180° − 68° = 112°.
Question 2 · Angles · G3
Angles around a point are 95°, 120° and x. Find x.
Answer:
Angles around a point total 360°.
x = 360° − 95° − 120° = 145°.
Question 3 · Angles · G3
Two vertically opposite angles are 3x and 75°. Find x.
Answer:
Vertically opposite angles are equal: 3x = 75°.
Divide by 3: x = 25°.
Question 4 · Angles · G3
Two parallel lines are crossed by a transversal. One angle is 63°. Find its corresponding angle.
Answer:
Corresponding angles formed by parallel lines are equal.
The corresponding angle is 63°.
Question 5 · Angles · G3
Two parallel lines are crossed by a transversal. One angle is 114°. Find the alternate angle.
Answer:
Alternate angles formed by parallel lines are equal.
The alternate angle is 114°.
Question 6 · Angles · G3
Co-interior angles between parallel lines are x and 72°. Find x.
Answer:
Co-interior angles add to 180°.
x = 180° − 72° = 108°.
Question 7 · Triangles · G3
A triangle has angles 42°, 67° and x. Find x.
Answer:
The angles in a triangle add to 180°.
x = 180° − 42° − 67° = 71°.
Question 8 · Triangles · G4
An isosceles triangle has a vertex angle of 40°. Find each equal base angle.
Answer:
The two equal angles total 180° − 40° = 140°.
Each equal angle = 140° ÷ 2 = 70°.
Question 9 · Triangles · G4
An equilateral triangle has one angle marked x. Find x.
Answer:
All three angles of an equilateral triangle are equal.
180° ÷ 3 = 60°, so x = 60°.
Question 10 · Triangles · G3
An exterior angle of a triangle is 125°. One non-adjacent interior angle is 48°. Find the other.
Answer:
The exterior angle equals the sum of the two non-adjacent interior angles.
Other angle = 125° − 48° = 77°.
Question 11 · Polygons · G3
Find the sum of the interior angles of a pentagon.
Answer:
A pentagon has 5 sides; the sum is (5 − 2) × 180°.
3 × 180° = 540°.
Question 12 · Polygons · G3
Find each interior angle of a regular hexagon.
Answer:
Interior angle sum = (6 − 2) × 180° = 720°.
Each angle = 720° ÷ 6 = 120°.
Question 13 · Polygons · G3
Find each exterior angle of a regular octagon.
Answer:
The exterior angles total 360°.
For a regular octagon, each = 360° ÷ 8 = 45°.
Question 14 · Polygons · G3
A regular polygon has exterior angle 24°. How many sides does it have?
Answer:
The exterior angles total 360° and all are 24°.
Number of sides = 360 ÷ 24 = 15.
Question 15 · Polygons · G3
Find the sum of the interior angles of a decagon.
Answer:
A decagon has 10 sides.
Interior sum = (10 − 2) × 180° = 1,440°.
Question 16 · Polygons · G3
A quadrilateral has angles 80°, 110°, 95° and x. Find x.
Answer:
A quadrilateral's interior angles total 360°.
x = 360° − 80° − 110° − 95° = 75°.
Question 17 · Quadrilaterals · G4
Which quadrilateral has exactly one pair of parallel sides in UK terminology?
Answer:
Identify the quadrilateral with one pair of parallel sides.
In UK terminology it is a trapezium.
Question 18 · Quadrilaterals · G4
What is the sum of adjacent interior angles of a parallelogram?
Answer:
Adjacent sides meet across parallel lines, making co-interior angles.
Co-interior angles add to 180°.
Question 19 · Quadrilaterals · G4
A rectangle has one angle of 90°. What are the other three angles?
Answer:
A rectangle has four right angles.
Therefore each of the other three angles is 90°.
Question 20 · Quadrilaterals · G4
State one property common to all rhombuses.
Answer:
By definition, a rhombus has four equal sides.
Thus all four side lengths are equal.
Question 21 · Symmetry · G1
How many lines of symmetry does a non-square rectangle have?
Answer:
A non-square rectangle is symmetric about the horizontal and vertical centre lines.
It has 2 lines of symmetry.
Question 22 · Symmetry · G1
What is the order of rotational symmetry of a square?
Answer:
A square matches itself after turns of 90°, 180°, 270° and 360°.
Its rotational symmetry has order 4.
Question 23 · Symmetry · G1
How many lines of symmetry does an equilateral triangle have?
Answer:
Each vertex and the midpoint of its opposite side define one symmetry line.
There are 3 such lines.
Question 24 · Symmetry · G1
What is the order of rotational symmetry of a non-square rectangle?
Answer:
A non-square rectangle matches itself after turns of 180° and 360°.
Its rotational symmetry has order 2.
Question 25 · Transformations · G7
Translate (2, −1) by the vector 34. Give the image.
Answer:
Add 3 to the x-coordinate: 2 + 3 = 5.
Add 4 to the y-coordinate: −1 + 4 = 3.
The image is (5, 3).
Question 26 · Transformations · G7
Reflect the point (4, −2) in the x-axis.
Answer:
Reflection in the x-axis keeps x and changes the sign of y.
(4, −2) becomes (4, 2).
Question 27 · Transformations · G7
Reflect the point (3, 5) in the y-axis.
Answer:
Reflection in the y-axis changes the sign of x and keeps y.
(3, 5) becomes (−3, 5).
Question 28 · Transformations · G7
Reflect (2, −3) in the line y = x.
Answer:
Reflection in y = x swaps the coordinates.
(2, −3) becomes (−3, 2).
Question 29 · Transformations · G7
Rotate (2, 1) by 90° anticlockwise about the origin.
Answer:
A 90° anticlockwise rotation about the origin maps (x, y) to (−y, x).
(2, 1) becomes (−1, 2).
Question 30 · Transformations · G7
Rotate (−3, 4) by 180° about the origin.
Answer:
A 180° rotation about the origin changes both coordinate signs.
(−3, 4) becomes (3, −4).
Question 31 · Transformations · G7
Enlarge (2, −1) by scale factor 3 about the origin.
Answer:
Multiply both coordinates by 3.
(2 × 3, −1 × 3) = (6, −3).
Question 32 · Transformations · G7
Enlarge (4, 6) by scale factor 12 about the origin.
Answer:
Multiply both coordinates by 12.
(4 × 12, 6 × 12) = (2, 3).
Question 33 · Transformations · G7 · Higher only
Enlarge (2, 3) by scale factor −2 about the origin.
Answer:
Multiply both coordinates by −2.
(2 × −2, 3 × −2) = (−4, −6).
Question 34 · Transformations · G7
Are two shapes necessarily congruent after a rotation?
Answer:
A rotation preserves every length and angle.
The image has the same shape and size, so it is congruent.
Question 35 · Transformations · G7
Are two shapes similar after enlargement by scale factor 2?
Answer:
An enlargement by 2 doubles every length and preserves every angle.
Corresponding sides are in a constant ratio, so the shapes are similar.
Question 36 · Congruence · G7
What does congruent mean?
Answer:
Compare corresponding sides and angles.
Congruent shapes match exactly in shape and size; orientation may differ.
Question 37 · Congruence · G5
Two triangles have corresponding side lengths 3 cm, 4 cm and 5 cm. Are they congruent?
Answer:
The three pairs of corresponding sides have equal lengths.
Therefore the triangles are congruent by SSS.
Question 38 · Congruence · G6
Do three equal corresponding angles alone prove two triangles congruent?
Answer:
Equal angles establish the same shape but do not fix the side lengths.
The triangles can have different sizes, so this only proves similarity.
Question 39 · Similarity · G7
Similar triangles have corresponding side lengths 3 cm and 9 cm. Find the scale factor from the smaller to the larger.
Answer:
Scale factor = corresponding larger length ÷ smaller length.
9 ÷ 3 = 3.
Question 40 · Similarity · G7
Similar figures have side lengths in ratio 2:5. A small side is 6 cm. Find its corresponding large side.
Answer:
Scale factor from small to large = 52.
6 × 52 = 15 cm.
Question 41 · Circle theorems · G10 · Higher only
What is the angle in a semicircle subtended by a diameter?
Answer:
An angle subtended by a diameter at the circumference is a right angle.
Therefore the angle is 90°.
Question 42 · Circle theorems · G10 · Higher only
An angle at the centre and an angle at the circumference stand on the same arc. The centre angle is 110°. Find the circumference angle.
Answer:
The angle at the centre is twice the angle at the circumference on the same arc.
110° ÷ 2 = 55°.
Question 43 · Circle theorems · G10 · Higher only
Opposite angles of a cyclic quadrilateral are 72° and x. Find x.
Answer:
Opposite angles of a cyclic quadrilateral add to 180°.
x = 180° − 72° = 108°.
Question 44 · Circle theorems · G10 · Higher only
A tangent touches a circle at T. What angle does the tangent make with radius OT?
Answer:
A tangent is perpendicular to the radius at the point of contact.
The angle is 90°.
Question 45 · Circle theorems · G10 · Higher only
Two tangents from the same external point touch a circle. How do their lengths compare?
Answer:
Tangents drawn from the same external point have equal lengths.
Therefore the two tangent segments are equal.
Question 46 · Circle theorems · G10 · Higher only
Two angles in the same segment of a circle stand on the same chord. One is 43°. Find the other.
Answer:
Angles in the same segment standing on the same chord are equal.
The other angle is 43°.
Question 47 · Solids · G12
How many faces does a cube have?
Answer:
A cube has a top, a bottom and four side faces.
2 + 4 = 6 faces.
Question 48 · Solids · G12
How many edges does a triangular prism have?
Answer:
Each triangular end has 3 edges, and 3 edges join the ends.
3 + 3 + 3 = 9 edges.
Question 49 · Solids · G12
How many vertices does a square-based pyramid have?
Answer:
The square base has 4 vertices; the apex adds 1.
4 + 1 = 5 vertices.
Question 50 · Loci · G2
A point is equidistant from two distinct fixed points A and B. Describe its locus.
Answer:
Points the same distance from A and B lie on the line through the midpoint of AB at 90° to AB.
This locus is the perpendicular bisector of AB.

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