Properties of triangles & quadrilaterals

The diagram shows a rectangle ABCD with diagonal AC.

Complete each statement to show that triangle ABC is congruent to triangle ADC.

Angle ADC = angle........ because both angles are 90°.

Side AD = side........ because opposite sides of a rectangle are equal.

Side CD = side........ because........

The above three statements show that triangle ABC is congruent to triangle ADC because

ABC is an equilateral triangle. BD is perpendicular to AC.

Prove that triangle ABD is congruent to triangle CBD.

In this question, all lengths are in centimetres.

The diagram shows an isosceles triangle ABC.

Find the perimeter of the triangle. You must show your working.

Measure angle x.

The angles of a triangle are 27°, 126° and 27°.

Explain how you know the triangle is isosceles.

Complete the drawing of the parallelogram.

Include notation to show that the drawing is a parallelogram.

Work out the area of the parallelogram.

The diagram shows an equilateral triangle and a rectangle.

The equilateral triangle has the same perimeter as the rectangle.

Find the value of a.

The diagram shows an equilateral triangle and a rectangle.

The equilateral triangle has the same perimeter as the rectangle.

Find the value of a.

The shape below is part of a circle, centre O and radius 12 cm. Angle OAB = 30°.

Work out the perimeter of the shape.

Write down the mathematical name of this solid. Choose from the list of names in the box.

Write down the mathematical name of this polygon.

On the grid below, sketch a quadrilateral with these properties.

• All angles are equal

and

• the diagonals are not at right angles to each other.

A regular polygon is being constructed inside a circle, centre O. Part of the construction is shown in this diagram.

Give a reason why OA = OB.

Write down the mathematical name of triangle OAB.

The regular polygon is completed.

Work out the sum of the interior angles of the regular polygon.

Work out the perimeter of this rectangle.

Finley draws a rectangle and says

The perimeter is 20 cm and the length is 10 cm.

Can Finley be correct? Show how you decide.

The diagram shows four triangles that are joined together.

• Points A, B, C and D lie on a straight line.

• Triangles ABF and CDE are equilateral triangles.

• Triangle BCF is an isosceles triangle with BF = BC.

Show that triangle CEF is a right-angled triangle. Give a reason for each stage of your working. Use the template below to help present your work. You may not need all of the lines.

Angle........ =........ ° because........

Angle........ =........ ° because........

Angle........ =........ ° because........

Angle........ =........ ° because........

Angle........ =........ ° because........

Angle........ =........ ° because........

Angle........ =........ ° because........

This shape is formed from a rectangle and two sectors of circles.

Points A, B and C lie on a straight line. Angle CB D = 3 5 °. DE = 5 t and EF = 2 t.

Explain why BC = 2 t. Give a reason for each step of your explanation.

Show that the perimeter of the shape is t + 14t.

The diagram shows a quadrilateral and an equilateral triangle. The perimeter of the quadrilateral is equal to the perimeter of the equilateral triangle.

Find an expression for the length of one side of the equilateral triangle. Give your answer in terms of x in its simplest form.

A rhombus is drawn on a coordinate grid. One diagonal of the rhombus has equation y = + 3. The other diagonal passes through the point (1, 7).

Find the equation of the other diagonal of the rhombus. Give your answer in the form y = mx + c.

Taylor designs a logo using isosceles triangles joined at a central point, P.

This is the start of Taylor’s design.

The completed design will have rotational symmetry, order 60 about point P.

Each triangle has base, b, and height, h, measured in mm.

Calculate h when b = 40 mm.

Give your answer correct to 1 decimal place.

The diagram shows an isosceles triangle, ABC.

AB = AC = 19 cm. Angle BAC = 54°.

Calculate the length of BC. You must show your working.

The diagram shows a right-angled triangle and a rectangle.

The triangle and rectangle have the same area.

Calculate the length, d cm, of the diagonal of the rectangle. You must show your working.

The graph shows a parallelogram ABCD.

A has coordinates (0, - 6), B has coordinates (3, 0) and C has coordinates (9, 0).

Find the equation of the line that passes through the points C and D, giving your answer in the form y = mx + c. You must show your working.

Here is a shape.

On the diagram, draw the shape’s two lines of symmetry.

Here is another shape.

Write down the order of rotation symmetry of the shape.

Sketch a quadrilateral that has exactly one line of symmetry.

Write down the mathematical name of your quadrilateral.

In this question, all lengths are in centimetres.

The diagram shows an equilateral triangle and a square.

The perimeter of each shape is 36 cm.

Find the value of b.

ABC is an equilateral triangle of side length 2 cm. M is the midpoint of AC.

Using this diagram, show that tan 30° = .

How many sides does a pentagon have?

Write down the mathematical name of this solid.

The angles in a triangle are 40°, 50° and 90°.

Write down the mathematical name for this type of triangle.

The diagram shows how a rhombus is made by joining two equilateral triangles.

Find the size of each interior angle of the rhombus.

(a)........ °,........ °,........ °,........ °

The same rhombus can be made by joining two copies of an isosceles triangle.

Find the size of each angle of the isosceles triangle.

The diagram shows rectangle ABCD.

DB = 13 cm and BC = 5 cm.

Calculate the area of the rectangle. You must show your working.

The region R is shown on this grid.

Region R is defined by four inequalities. One of the inequalities is x ⩾ 0.

Use the symbols ⩽ and ⩾ to complete the other three inequalities.

x ⩾ 0

x + 2 y........ 24

The inequality x ⩾ 0 is replaced by a new inequality. Region R is then a kite.

Write down the new inequality.

ABC is an isosceles triangle. The sides of the triangle ABC are all tangents to a circle of radius 6 cm, centre O.

Angle BAC = 70° and BA = BC.

Show that length BO is 17.54 cm, correct to 2 decimal places.

Find the area of triangle ABC. You must show your working.

Write down the mathematical name of each of these shapes.

A triangle with 3 equal sides.

(a)(i)........ triangle

A quadrilateral with 4 equal sides and no right angles.

Here is a rectangle.

On the diagram, draw the rectangle’s two lines of symmetry.

The rectangle has rotation symmetry of order 2.

Amaya says

A rectangle is the only quadrilateral that has rotation symmetry of order 2.

Is she correct? Show how you decide.

Add the correct symbols to this diagram to show that line AB is parallel to line DC.

Point A is plotted at (4, - 2) on this one-centimetre square grid.

Point A is a corner of a square with area 36 cm². The other corners of the square have integer coordinates and lie on the grid.

Find the coordinates for the corner of the square that is diagonally opposite point A. You may use the grid above to help you.

Write down the mathematical name of this quadrilateral.

Write down the mathematical name of this solid.

Three lines meet at a point.

Work out the size of angle a.

XY and CW are parallel lines. AB = CB. Angle CAX = 130°.

Complete this sentence.

Angle CAB = 50° because........

Work out angle BCW. Give a reason for each angle you work out.

The diagram shows a kite, ABCD. AFE and CEF are equilateral triangles.

Write down a mathematical name for quadrilateral AFCE.

The ratio of angle DAE: angle EAF = 1: 4.

Work out angle x. Write on the diagram the values of any other angles you use in your working.

In the diagram,

• A, B and C are points on the circumference of a circle

• O is the centre of the circle

• angle OAB = 15°

• angle BCO = 10°.

Calculate the acute angle AOC.

In the diagram,

• E, F and D are points on the circumference of the circle

• G, D and H lie on a tangent to the circle

• angle EFD = 55°

• angle FDE = 43°.

Explain why angle HDF is 82°.

The diagram shows a regular hexagon made from six equilateral triangles. Each side is 10 cm. The angle ACB is a right angle.

Show that AC = 8.66 cm, correct to 3 significant figures.

Show that the area of triangle ACB is 21.7 cm², correct to 3 significant figures.

Find the area of the hexagon, giving your answer to an appropriate degree of accuracy.

An interior angle of an isosceles triangle is p ° and an exterior angle is q °.

It is given that q = 5 p.

Write the ratio p: q in its simplest form.

Work out the two different possible sets of angles for the isosceles triangle.

(b) Triangle 1:........ °,........ °,........ °

Triangle 2:........ °,........ °,........ °

A, B, C and D are points on the circumference of a circle, centre O.

Angle BAD = 112° and angle DCO = 33°.

Show that angle y = 35°.

Give reasons for each stage of your working.

Work out angle z. Give reasons for your answer.

Angle z =........ ° because........

Complete the statement using a term from the list.

These are the names of some special quadrilaterals.

Choose a quadrilateral from the list that satisfies each set of conditions.

• All four sides are the same length.

• Opposite angles are equal.

• All four angles are right angles.

• Opposite sides are equal.

This is a square based pyramid.

Complete the following.

A square based pyramid has........ faces and........ edges.

Here are two identical tiles.

Sketch two quadrilaterals that can be made by joining the tiles. Write the mathematical name of each quadrilateral.

Quadrilateral 1:
Quadrilateral 2:

The two tiles can also be joined to make two triangles.

Work out the interior angles of each triangle.

(b) Triangle 1:........ °........ °........ °

Triangle 2:........ °........ °........ °

The diagram shows a circle, centre O.

Points A, B, C and D lie on the circumference of the circle. EDF is a tangent to the circle. Angle ABC = 82 ° and angle ODC = 57 °.

Work out the value of x.

Work out the value of y.

Write down the mathematical name of this triangle. Choose from the list in the box.

Write down the order of rotation symmetry of this regular octagon.

The diagram shows a straight line that passes through points A and B, and a curve that passes through points P and Q.

Find the equation of the straight line.

The equation of the curve is y = x² + kx + 8.

Find the value of k.

Diann draws line BQ. She says

Triangle ABQ is isosceles.

Is Diann correct? You must show all your working.

Write down the number of lines of symmetry of this hexagon.

Write down the order of rotation symmetry of this shape.

A triangle has just one line of symmetry.

Write down the mathematical name of this type of triangle.

Sara says

All parallelograms have 2 lines of symmetry and rotation symmetry of order 2.

Explain why Sara is not correct.

Two congruent, isosceles triangles are joined, as shown, to form a parallelogram. The largest angle of the parallelogram is 110°.

Write two equations. Solve them to find the value of a and the value of b.