GCSE Maths: Quadrilaterals (Higher)
Part of the GCSE Maths angles, polygons and symmetry chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Quadrilaterals
A quadrilateral is a polygon with four straight sides. Draw any quadrilateral and cut it in two with a diagonal: you get two triangles. Each triangle has angles adding up to 180°, so the four angles of any quadrilateral add up to 2 × 180° = 360°.
Some quadrilaterals have special names and special properties. You must know them all, and use them to find angles and sides. In these diagrams, matching ticks show equal sides, arrows show parallel sides and a small square shows a right angle.
The table summarises the properties. The last two columns (lines of symmetry and order of rotational symmetry) are worked out from the shapes themselves; see Section 17.7.
| Quadrilateral | Sides | Angles | Diagonals | Lines of symmetry | Order of rotation |
|---|---|---|---|---|---|
| Square | 4 equal sides; opposite sides parallel | all 90° | equal, bisect each other at 90° | 4 | 4 |
| Rectangle | opposite sides equal and parallel | all 90° | equal, bisect each other | 2 | 2 |
| Parallelogram | opposite sides equal and parallel | opposite angles equal | bisect each other | 0 | 2 |
| Rhombus | 4 equal sides; opposite sides parallel | opposite angles equal | bisect each other at 90°; bisect the angles | 2 | 2 |
| Kite | two pairs of equal adjacent sides | one pair of equal angles (opposite each other) | cross at 90° | 1 | 1 |
| Isosceles trapezium | one pair of parallel sides; the other two sides equal | two pairs of equal angles (base angles) | equal | 1 | 1 |
| Trapezium | exactly one pair of parallel sides | angles between the parallel sides add to 180° | no special property | 0 | 1 |
Notice the family tree: a square is a special rectangle (all sides equal), a special rhombus (all angles 90°) and a special parallelogram. A rectangle and a rhombus are both special parallelograms. A parallelogram has two pairs of parallel sides; a trapezium has just one pair.
Using the properties for angles. In a parallelogram, opposite angles are equal and neighbouring angles add up to 180° (they are co-interior angles between the parallel sides). In a kite, one pair of opposite angles (the ones between the unequal sides) are equal. In a trapezium, the two angles at the ends of a non-parallel side add up to 180°. In an isosceles trapezium, the base angles are equal.
Key ideas
- Angles in a quadrilateral add up to 360°.
- Parallelogram: opposite sides parallel and equal; opposite angles equal; neighbouring angles add to 180°.
- Rhombus: a parallelogram with 4 equal sides; diagonals cross at 90° and bisect the angles.
- Rectangle: four right angles; diagonals equal. Square: all sides equal and all angles 90°.
- Kite: two pairs of equal adjacent sides; diagonals cross at 90°.
- Trapezium: one pair of parallel sides (isosceles: the other two sides equal).
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the quadrilateral must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.85 + 100 + 75 = 260°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 260 = 100°
x = 100°
Check. Add all the angles: 85 + 100 + 75 + 100 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Use the properties of a parallelogram: opposite angles are equal and neighbouring angles add to 180°.
- Find x. Angles A and B are neighbours, between the parallel sides AD and BC (co-interior angles).x = 180 − 68 = 112° (co-interior angles add up to 180°)
- Find y. Angle C is opposite angle A.y = 68° (opposite angles of a parallelogram are equal)
- Find z. Angle D is opposite angle B.z = 112° (opposite angles of a parallelogram are equal)
Check. Add all four angles: 68 + 112 + 68 + 112 = 360°. ✓
Watch out. Opposite angles are equal; neighbouring angles add to 180°.
Key words. A kite has two pairs of equal sides next to each other. It has one line of symmetry.
Method. A kite has one pair of equal angles, at B and D. Use the angle sum 360°, then share what is left.
- Say which angles are equal. In a kite the angles between the unequal sides are equal. Here angle ABC = angle ADC, because the two triangles on either side of AC match.angle ABC = angle ADC = x (a kite has one pair of equal angles)
- Use the angle sum of a quadrilateral. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.100 + 60 + x + x = 360
- Collect and subtract. 100 + 60 = 160, and the two equal angles are what is left.2x = 360 − 160 = 200
- Divide by 2. x = 200 ÷ 2 = 100°
Check. Add all four angles: 100 + 100 + 60 + 100 = 360°. ✓
Watch out. The equal angles in a kite are the ones between a short side and a long side, not the ones at the ends of the line of symmetry.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. The angles at the ends of a non-parallel side of a trapezium are co-interior angles, so they add up to 180°.
- Spot the pair of angles. AB and DC are parallel, and AD joins them. So angle DAB and angle ADC are co-interior angles.angle DAB and angle ADC lie between parallel lines on the same side of AD
- State the fact. Co-interior angles add up to 180°.65 + x = 180
- Subtract. x = 180 − 65 = 115°
Check. The two base angles at A and B are both 65° and the two at C and D are both 115°: 65 + 65 + 115 + 115 = 360°. ✓
Watch out. In an isosceles trapezium the angles on each parallel side are equal.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 360°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Write an equation. The angles must add up to 360°, so we add all the expressions and put the answer equal to 360.(2x + 10°) + 3x° + (x + 20°) + (4x − 10°) = 360°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.10x + 20 = 360
- Get the number on the other side. Subtract 20 from both sides.10x = 340
- Divide both sides by 10. 10x means 10 × x, so dividing by 10 leaves x on its own.x = 340 ÷ 10 = 34
- Substitute x = 34 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.2x + 10° = 2 × 34 + 10 = 78°
3x° = 3 × 34 = 102°
x + 20° = 1 × 34 + 20 = 54°
4x − 10° = 4 × 34 − 10 = 126°
Check. Add the angles: 78 + 102 + 54 + 126 = 360°, which is 360°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Watch out
A square is also a rectangle, a rhombus and a parallelogram, and a rectangle is also a parallelogram. When you are asked to “name a quadrilateral” give the most specific name that fits. Also, do not mix up a parallelogram (opposite angles equal) with a kite (only one pair of equal angles).Exercise 17F Quadrilaterals
Non-calculatorKey words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the quadrilateral must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.85 + 100 + 75 = 260°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 260 = 100°
x = 100°
Check. Add all the angles: 85 + 100 + 75 + 100 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the quadrilateral must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.110 + 90 + 70 = 270°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 270 = 90°
x = 90°
Check. Add all the angles: 110 + 90 + 70 + 90 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the quadrilateral must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.90 + 90 + 125 = 305°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 305 = 55°
x = 55°
Check. Add all the angles: 90 + 90 + 125 + 55 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. The angles in the quadrilateral must add up to 360°. Add the angles we know and subtract from 360°.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Add up the angles we already know. We need their total, so we can see how much of the 360° is left.65 + 130 + 82 = 277°
- Subtract that total from 360°. All the angles together must make 360°, so the missing angle is whatever is left over.360 − 277 = 83°
y = 83°
Check. Add all the angles: 65 + 130 + 82 + 83 = 360°, which is 360°. ✓
Watch out. Subtract from the correct total: 180° for a straight line or a triangle, 360° for a full turn or a quadrilateral.
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.square: a quadrilateral with four equal sides and four right angles
- Match the description to the word. All sides are equal and all angles are 90°.The answer is square.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.parallelogram: a quadrilateral with two pairs of parallel sides (opposite sides equal and parallel)
- Match the description to the word. Opposite sides are parallel and equal, and the angles are not 90°.The answer is parallelogram.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.rhombus: a quadrilateral with four equal sides (a squashed square)
- Match the description to the word. All sides are equal but the angles are not 90°.The answer is rhombus.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.trapezium: a quadrilateral with exactly one pair of parallel sides
- Match the description to the word. Only one pair of sides is parallel.The answer is trapezium.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.kite: a quadrilateral with two pairs of equal adjacent sides; one line of symmetry
- Match the description to the word. The equal sides are next to each other, not opposite.The answer is kite.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Geometry words must be used exactly. In an exam, a correct word gains a mark and a vague word does not.
Method. Recall what the word means, then match it to the description or the picture.
- Recall the meaning of the word. Geometry has its own vocabulary. Each word has one exact meaning.rectangle: a quadrilateral with four right angles (opposite sides equal)
- Match the description to the word. All angles are 90° but the sides are not all equal.The answer is rectangle.
Check. Say the definition out loud in your own words and see if it fits the question. ✓
Watch out. Learn the exact meaning. Many words sound similar (for example perpendicular and parallel).
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. In a parallelogram, angles that share a side are co-interior angles between parallel sides, so they add up to 180°.
- Spot the relationship. Angles A and B are at the two ends of side AB. The sides AD and BC are parallel, so A and B are co-interior angles.A and B are co-interior angles
- State the fact. Co-interior angles add up to 180°.A + B = 180°
- Subtract. x = 180 − 68 = 112°
Check. The angles of the parallelogram: 68 + 112 + 68 + 112 = 360°. ✓
Watch out. Neighbouring angles of a parallelogram add to 180°; opposite angles are equal.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. The opposite angles of a parallelogram are equal.
- Spot the relationship. Angle C is opposite angle A, across the parallelogram.C is opposite A
- State the fact. Opposite angles of a parallelogram are equal.Reason: opposite angles of a parallelogram are equal
- Write the answer. y = 68°
Check. Using co-interior angles instead: y + 112 = 180, so y = 68°. ✓
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. The opposite angles of a parallelogram are equal.
- Spot the relationship. Angle D is opposite angle B.D is opposite B, and B = 112°
- State the fact. Opposite angles of a parallelogram are equal.Reason: opposite angles of a parallelogram are equal
- Write the answer. z = 112°
Check. Angles A and D are co-interior: 68 + 112 = 180°. ✓
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. In a parallelogram, angles that share a side are co-interior angles between parallel sides, so they add up to 180°.
- Spot the relationship. Angles A and B are at the two ends of side AB. The sides AD and BC are parallel, so A and B are co-interior angles.A and B are co-interior angles
- State the fact. Co-interior angles add up to 180°.A + B = 180°
- Subtract. x = 180 − 125 = 55°
Check. The angles of the parallelogram: 125 + 55 + 125 + 55 = 360°. ✓
Watch out. Neighbouring angles of a parallelogram add to 180°; opposite angles are equal.
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. The opposite angles of a parallelogram are equal.
- Spot the relationship. Angle C is opposite angle A, across the parallelogram.C is opposite A
- State the fact. Opposite angles of a parallelogram are equal.Reason: opposite angles of a parallelogram are equal
- Write the answer. y = 125°
Check. Using co-interior angles instead: y + 55 = 180, so y = 125°. ✓
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. The opposite angles of a parallelogram are equal.
- Spot the relationship. Angle D is opposite angle B.D is opposite B, and B = 55°
- State the fact. Opposite angles of a parallelogram are equal.Reason: opposite angles of a parallelogram are equal
- Write the answer. z = 55°
Check. Angles A and D are co-interior: 125 + 55 = 180°. ✓
Method. The two angles at B and D are equal. Take the known angles from 360° and share what is left.
- Say which angles are equal. In a kite the two angles between the unequal sides are equal.angle ABC = angle ADC
Reason: a kite has one pair of equal angles - Use the angle sum. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.110 + 50 + x + x = 360
- Subtract. 110 + 50 = 160.2x = 360 − 160 = 200
- Divide by 2. x = 100°
Check. 110 + 100 + 50 + 100 = 360°. ✓
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Angles B and C are at the ends of the side BC, which joins the parallel sides, so they are co-interior angles.
- Spot the pair. AB and DC are parallel and BC crosses them.angle B and angle C are co-interior
- State the fact. Co-interior angles add up to 180°.70 + x = 180
- Subtract. x = 180 − 70 = 110°
Check. The angles at C and B look obtuse and acute, adding to a straight line. ✓
Key words. Parallel lines always stay the same distance apart and never meet (they are marked with arrows). A transversal is a line that crosses them. It makes alternate angles (a Z shape), corresponding angles (an F shape) and co-interior angles (a C or U shape).
Method. Angles A and D are at the ends of the side AD, which joins the parallel sides, so they are co-interior angles.
- Spot the pair. AB and DC are parallel and AD crosses them.angle A and angle D are co-interior
- State the fact. Co-interior angles add up to 180°.75 + y = 180
- Subtract. y = 180 − 75 = 105°
Check. 75 + 70 + 110 + 105 = 360°. ✓
Method. Triangle ABC is isosceles because the sides of a rhombus are equal. Use the angle sum.
- Spot the isosceles triangle. All four sides of a rhombus are equal, so AB = BC.AB = BC, so triangle ABC is isosceles
- Say which angles are equal. Base angles of an isosceles triangle are equal.angle BAC = angle BCA
- Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.180 − 110 = 70° for the two base angles
- Divide by 2. 70 ÷ 2 = 35°
Check. The diagonal of a rhombus bisects the angle: angle BAD = 180 − 110 = 70°, and 70 ÷ 2 = 35°. ✓
Watch out. The angle between a side and a diagonal of a rhombus is half the corner angle.
Method. Triangle ABC has a right angle at B, so use the angle sum of a triangle.
- Find the right angle. A rectangle has four right angles, so angle ABC = 90°.angle ABC = 90°
- Use the angle sum of a triangle. If you tear off the three corners of any triangle and place them side by side they form a straight line, so the three angles make 180°.37 + 90 + x = 180
- Subtract. 37 + 90 = 127.x = 180 − 127 = 53°
Check. The two acute angles of a right-angled triangle add to 90°: 37 + 53 = 90. ✓
Method. Imagine folding the shape so that one half lies exactly on the other. Count the different fold lines.
- Picture the shape. A line of symmetry is a mirror line: the two halves are reflections of each other.square
- Count the fold lines. Two diagonals, and two lines joining the middle of opposite sides.4
Check. Lines of symmetry of a square: 4. ✓
Watch out. The diagonals of a rectangle are not lines of symmetry.
Method. Imagine folding the shape so that one half lies exactly on the other. Count the different fold lines.
- Picture the shape. A line of symmetry is a mirror line: the two halves are reflections of each other.rectangle
- Count the fold lines. Two lines joining the middle of opposite sides. The diagonals do not work.2
Check. Lines of symmetry of a rectangle: 2. ✓
Watch out. The diagonals of a rectangle are not lines of symmetry.
Method. Imagine folding the shape so that one half lies exactly on the other. Count the different fold lines.
- Picture the shape. A line of symmetry is a mirror line: the two halves are reflections of each other.parallelogram
- Count the fold lines. Slanted sides stop any fold from matching. (It does have rotational symmetry.)0
Check. Lines of symmetry of a parallelogram: 0. ✓
Watch out. The diagonals of a rectangle are not lines of symmetry.
Method. Imagine folding the shape so that one half lies exactly on the other. Count the different fold lines.
- Picture the shape. A line of symmetry is a mirror line: the two halves are reflections of each other.rhombus
- Count the fold lines. The two diagonals. Lines joining the middle of opposite sides do not work.2
Check. Lines of symmetry of a rhombus: 2. ✓
Watch out. The diagonals of a rectangle are not lines of symmetry.
Method. Imagine folding the shape so that one half lies exactly on the other. Count the different fold lines.
- Picture the shape. A line of symmetry is a mirror line: the two halves are reflections of each other.kite
- Count the fold lines. Only the diagonal between the two pairs of equal sides.1
Check. Lines of symmetry of a kite: 1. ✓
Watch out. The diagonals of a rectangle are not lines of symmetry.
Method. Imagine folding the shape so that one half lies exactly on the other. Count the different fold lines.
- Picture the shape. A line of symmetry is a mirror line: the two halves are reflections of each other.isosceles trapezium
- Count the fold lines. One line through the middle of the two parallel sides.1
Check. Lines of symmetry of a isosceles trapezium: 1. ✓
Watch out. The diagonals of a rectangle are not lines of symmetry.
Method. Compare the definitions.
- Recall a square. A square has four equal sides and four right angles.square: equal sides + right angles
- Recall a rectangle. A rectangle only needs four right angles.rectangle: right angles
- Decide. A 3 cm by 5 cm rectangle has right angles but unequal sides.False
Check. A square is a rectangle, but a rectangle is not always a square. ✓
Method. Compare the definitions.
- Recall a parallelogram. Two pairs of parallel sides.opposite sides parallel
- Check a rhombus. The opposite sides of a rhombus are parallel.it meets the definition
- Decide. True
Check. A rhombus is a parallelogram with equal sides. ✓
Method. Recall the definition.
- Recall a trapezium. It has exactly one pair of parallel sides.one pair
- Compare. Two pairs of parallel sides is a parallelogram.False
Check. A trapezium has only one pair of parallel sides. ✓
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 360°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Write an equation. The angles must add up to 360°, so we add all the expressions and put the answer equal to 360.x° + 2x° + 3x° + 4x° = 360°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.10x = 360
- Divide both sides by 10. 10x means 10 × x, so dividing by 10 leaves x on its own.x = 360 ÷ 10 = 36
- Substitute x = 36 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.x° = 1 × 36 = 36°
2x° = 2 × 36 = 72°
3x° = 3 × 36 = 108°
4x° = 4 × 36 = 144°
Check. Add the angles: 36 + 72 + 108 + 144 = 360°, which is 360°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 360°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Write an equation. The angles must add up to 360°, so we add all the expressions and put the answer equal to 360.(2x + 10°) + 3x° + (x + 20°) + (4x − 10°) = 360°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.10x + 20 = 360
- Get the number on the other side. Subtract 20 from both sides.10x = 340
- Divide both sides by 10. 10x means 10 × x, so dividing by 10 leaves x on its own.x = 340 ÷ 10 = 34
- Substitute x = 34 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.2x + 10° = 2 × 34 + 10 = 78°
3x° = 3 × 34 = 102°
x + 20° = 1 × 34 + 20 = 54°
4x − 10° = 4 × 34 − 10 = 126°
Check. Add the angles: 78 + 102 + 54 + 126 = 360°, which is 360°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Method. Use the angle sum 360° for the fourth angle, then think about which quadrilateral has four right angles.
- Add the three known angles. We need their total.90 + 90 + 90 = 270°
- Subtract from 360°. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.360 − 270 = 90°
- Name the quadrilateral. A quadrilateral with four right angles is a rectangle (or a square).rectangle
Check. 4 × 90 = 360°. ✓
Watch out. We cannot say it is a square: the sides might not be equal.
Method. Use co-interior angles twice and compare.
- Angle A and angle B. AD is parallel to BC, so A and B are co-interior angles.a + b = 180° (co-interior angles)
- Angle B and angle C. AB is parallel to DC, so B and C are co-interior angles.b + c = 180° (co-interior angles)
- Compare. Both a and c equal 180° − b.a = 180° − b = c. So opposite angles are equal. ■
Check. Test with b = 112°: a = c = 68°. ✓
Watch out. In a proof, give a reason on every line.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 360°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. A quadrilateral can be cut into two triangles by a diagonal. Each triangle has angles totalling 180°, so the quadrilateral has 2 × 180° = 360°.Angles in a quadrilateral add up to 360°.
- Write an equation. The angles must add up to 360°, so we add all the expressions and put the answer equal to 360.x° + (x + 30°) + 2x° + (3x − 20°) = 360°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.7x + 10 = 360
- Get the number on the other side. Subtract 10 from both sides.7x = 350
- Divide both sides by 7. 7x means 7 × x, so dividing by 7 leaves x on its own.x = 350 ÷ 7 = 50
- Substitute x = 50 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.x° = 1 × 50 = 50°
x + 30° = 1 × 50 + 30 = 80°
2x° = 2 × 50 = 100°
3x − 20° = 3 × 50 − 20 = 130°
Check. Add the angles: 50 + 80 + 100 + 130 = 360°, which is 360°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.