Angles, polygons and properties of shapes – Year 10 GCSE maths worksheet with answers to download as PDF

Non-calculator. Work out the size of each lettered angle. Give a reason for each answer.

1. Angles on a straight line are \(128^\circ\) and \(a\).
2. Angles round a point are \(95^\circ\), \(110^\circ\), \(80^\circ\) and \(d\).
3. Two angles of a triangle are \(47^\circ\) and \(68^\circ\). Find the third angle \(e\).
4. Two straight lines cross. One angle is \(112^\circ\). Find the angle \(f\) next to it.

In the diagram the two horizontal lines are parallel. The angle marked \(64^\circ\) is above the top line, to the right of the crossing line. Find \(a\), \(b\) and \(c\) and give the special name of each pair of angles.

1. Angle \(a\) is in the matching position at the lower line.
2. Angle \(b\) is the alternate angle to \(64^\circ\).
3. Angle \(c\) is between the lines on the left, and together with the vertically opposite angle of \(64^\circ\) forms a pair of co-interior angles.

Non-calculator. Work out the size of one interior angle of each regular polygon by finding the exterior angle first.

1. a regular pentagon
2. a regular octagon
3. a regular 12-sided polygon
4. A regular polygon has exterior angles of \(15^\circ\). How many sides has it, and what is each interior angle?

The sum of the interior angles of an \(n\)-sided polygon is \((n-2)\times180^\circ\). Use it to answer these.

1. Find the angle sum of a hexagon.
2. Find the angle sum of a decagon (10 sides).
3. Find the angle sum of a 15-sided polygon.
4. The angles of a polygon add up to \(2880^\circ\). How many sides has it?

Name the quadrilateral described each time, then answer the last part.

1. Four equal sides, with diagonals that cross at right angles but with angles that are not \(90^\circ\).
2. Exactly one pair of parallel sides and the other two sides equal in length.
3. Two pairs of adjacent sides equal and one line of symmetry.
4. A parallelogram has one angle of \(70^\circ\). Find the other three angles.

Work out the lettered angles, and give a reason for each.

1. An isosceles triangle has an apex angle of \(40^\circ\). Find each base angle.
2. An isosceles triangle has a base angle of \(55^\circ\). Find the apex angle.
3. A triangle has interior angles of \(50^\circ\) and \(65^\circ\) at two corners. Find the exterior angle at the third corner.
4. An exterior angle of a triangle is \(130^\circ\) and one of the opposite interior angles is \(55^\circ\). Find the other opposite interior angle.

A roof truss is an isosceles triangle ABC with AB = AC and apex angle \(A=110^\circ\). A horizontal tie beam DE joins the two sloping rafters and is parallel to the base BC. A vertical strut runs from A to the midpoint M of BC.

1. Find the angle the rafter makes with the base, angle \(ABC\).
2. What is the size of angle \(ADE\) (D on AB, E on AC)? Give the reason.
3. Find angle \(BAM\).
4. Check that the angles in triangle \(ABM\) add up to \(180^\circ\).

Tiles of different regular polygons can fit together round a point only if their interior angles add up to exactly \(360^\circ\).

1. Show that three hexagons fit together at a point.
2. Show that two octagons and one square fit together at a point.
3. Show that three pentagons leave a gap and find the size of the gap.
4. Find how many equilateral triangles and how many squares can fit round a point if there are 3 triangles. (Hint: \(3\times60=180\).)

A road sign is a regular octagon with sides of 25 cm. Calculator allowed.

1. Find the size of each exterior angle and each interior angle.
2. Find the perimeter of the sign.
3. The sign is split into triangles by drawing lines from one corner to every other corner. How many triangles are there, and what is the sum of their angles?
4. Show that this sum equals the interior angle sum of an octagon.

A boat sails from a harbour A on a bearing of \(050^\circ\) to a buoy B. It then turns and sails on a bearing of \(130^\circ\) to a lighthouse C. The distances AB and BC are both 12 km. Bearings are measured clockwise from north.

1. Work out the back bearing of A from B.
2. Find angle \(ABC\).
3. Use the fact that triangle ABC is isosceles to find angle \(BAC\).
4. Find the bearing of C from A, and the bearing of A from C.

Two straight roads AB and CD are parallel. A cycle path crosses both of them. At the crossing with AB the path makes an angle of \(2x+10\) degrees with the road on one side, and at the crossing with CD it makes an angle of \(x+20\) degrees on the matching co-interior side.

1. Explain why \((2x+10)+(x+20)=180\).
2. Solve the equation to find \(x\).
3. Find the size of both angles.
4. The council plans a third road parallel to AB. How many angles of \(70^\circ\) will there be where the cycle path crosses all three roads (counting the 4 angles round each crossing)?

A window frame is a regular hexagon with side 40 cm. Lines from the centre O to every corner split it into six identical triangles. Calculator allowed.

1. Find the angle at O in each triangle.
2. Show that each triangle is equilateral.
3. Find the perimeter of the window.
4. Each triangle has height \(20\sqrt3\) cm. Work out the area of the window to the nearest cm².

Use algebra to find the angles in each shape. Show your equation each time.

1. The angles of a triangle are \(x\), \(2x+10\) and \(3x-10\).
2. The angles of a quadrilateral are \(x\), \(x+30\), \(2x\) and \(2x+30\).
3. The angles of a pentagon are \(x\), \(2x\), \(3x\), \(2x-10\) and \(110\).
4. Which of the shapes has an angle bigger than \(150^\circ\)?

Kite ABCD has AB = AD and CB = CD. Angle \(A=76^\circ\) and angle \(C=112^\circ\). The diagonals AC and BD cross at E.

1. Find angles \(B\) and \(D\).
2. Find angle \(BAC\) and angle \(BCA\), using the line of symmetry AC.
3. Find angle \(ABE\) and angle \(CBE\), given that the diagonals cross at right angles.
4. Check that angle \(ABE\) + angle \(CBE\) = angle \(B\).

Triangle PQR has interior angles \(p\) at P, \(q\) at Q and \(r\) at R. A line is drawn through P parallel to QR. The line makes angles \(x\) and \(y\) with PQ and PR as shown (one each side of P, above the triangle).

1. Explain why \(x=q\) and \(y=r\).
2. Use angles on a straight line to prove that \(p+q+r=180^\circ\).
3. Side QR is extended beyond R. Prove that the exterior angle at R equals \(p+q\).
4. Check with \(p=55^\circ\) and \(q=75^\circ\).

Non-calculator. Each question is about a regular polygon.

1. Each interior angle is 8 times the exterior angle. Find the number of sides.
2. Each interior angle is \(156^\circ\). Find the number of sides.
3. The interior angle is \(100^\circ\) bigger than the exterior angle. How many sides has the polygon?
4. Explain why no regular polygon has an interior angle of \(100^\circ\).

Triangle ABC is isosceles with AB = AC. M is the midpoint of BC. The line AM is drawn.

1. Prove that triangles ABM and ACM are congruent. State the condition (SSS, SAS, ASA or RHS) you use.
2. Hence prove that angle \(ABC\) = angle \(ACB\) and that angle \(AMB=90^\circ\).
3. Which pair of triangles is congruent? Give the condition. (i) 5 cm, 7 cm, 9 cm and 9 cm, 5 cm, 7 cm; (ii) 6 cm, 8 cm with an angle of \(40^\circ\) between them, and 6 cm, 8 cm with an angle of \(50^\circ\) between them.
4. Explain why two triangles with all three angles equal need not be congruent.

O is the centre of a circle. TA and TB are tangents from a point T outside the circle, touching it at A and B. Angle \(ATB=40^\circ\).

1. Explain why angles \(OAT\) and \(OBT\) are both \(90^\circ\).
2. Triangle TAB is isosceles (TA = TB). Find angle \(TAB\).
3. Find angle \(OAB\).
4. Find angle \(AOB\) and explain how you can check it using the quadrilateral OATB.

Higher tier. A, B, C and D lie on a circle with centre O. Use circle theorems and give a reason for each answer.

1. Angle \(AOB=108^\circ\) at the centre. Find angle \(ACB\) at the circumference on the same arc.
2. Angle \(ABC=78^\circ\) in cyclic quadrilateral ABCD. Find angle \(ADC\).
3. AB is a diameter and C is on the circle. Angle \(CAB=34^\circ\). Find angle \(ABC\).
4. A tangent at A makes an angle of \(52^\circ\) with the chord AB. Find the angle in the alternate segment.

A regular pentagon has each side extended until the lines meet, forming a five-pointed star (a pentagram) with a pentagon in the middle.

1. Show that the interior angle of the regular pentagon is \(108^\circ\).
2. Each point of the star is an isosceles triangle on a side of the pentagon. Show that its base angles are \(72^\circ\).
3. Find the angle at the tip of each point.
4. Show that the five tip angles add up to \(180^\circ\).