What does congruent mean?

Congruent shapes are exactly the same shape and size.

Why is AAA not a congruence test?

Triangles with the same angles can be different sizes, like two equilateral triangles.

Why must I keep my construction arcs?

They show how you made the construction, and examiners award marks for them.

Year 8 constructions and congruence lesson to download as PDF

This Year 8 constructions and congruence lesson is written for pupils aged 12-13 and follows the National Curriculum. In short, it shows you how to construct triangles, bisectors and perpendiculars with compasses. Before you start, it helps to revise circles: circumference and area and volume and surface area of prisms, because this lesson builds on them.

Key points

A triangle can be constructed only if the two shorter sides add up to more than the longest side; construct with compasses and keep your arcs.
To construct a triangle from three sides (SSS), draw the base, then draw arcs from each end whose radii equal the other two sides.
The perpendicular bisector of a line is the set of points equidistant from both ends; draw equal arcs from each end and join the two crossings.
The angle bisector is the set of points equidistant from two lines; it halves the angle, so \(70^\circ\) becomes two angles of \(35^\circ\).
Two triangles are congruent if they match by SSS, SAS, ASA (or AAS) or RHS; AAA only shows they are similar.
Write a proof with reasons: list equal parts (given, common side, vertically opposite), name the test, then state what follows.

Year 8 constructions and congruence lesson: what you will learn

This lesson covers ruler and compass constructions, then congruent triangles and how to prove them. Keep your construction arcs visible, because they show the examiner how you made the shape.

Starter activity

Take a strip of card and cut three pieces of length 6 cm, 8 cm and 10 cm. Join them with paper fasteners. Can you push the joints to change the shape? Now try the same with four pieces of card joined in a rectangle. What is the difference, and why do builders use triangles in roofs and bridges?

Constructing triangles

Definition: construction

A construction is an accurate drawing made with only a pair of compasses, a straight edge (ruler) and a pencil. Every step is exact and the reason it works can be explained.

Method: constructing an SSS triangle

Draw the longest side as the base, for example \(AB=7\) cm.
Set the compasses to the length of the second side (6 cm), centre A, and draw an arc above AB.
Set the compasses to the third side (5 cm), centre B, and draw an arc that crosses the first one. Call the crossing point C.
Join AC and BC with the ruler.

Worked example 1: SSS triangle

Construct triangle ABC with \(AB=7\) cm, \(BC=5\) cm and \(CA=6\) cm.

Follow the four steps of the method above. The arcs cross at C, and joining A, B and C gives the triangle.

Key fact: when a triangle exists

Three lengths form a triangle only if the two shorter sides add up to more than the longest. So 5, 6 and 10 can make a triangle (\(5+6=11>10\)), but 3, 4 and 8 cannot (\(3+4=7<8\)), and the arcs would not cross.

The perpendicular bisector

Definition: perpendicular bisector

The perpendicular bisector of a line segment AB cuts it in half at right angles.

Method: perpendicular bisector

Open the compasses to more than half of AB. Put the point on A and draw an arc above and below the line.
With the same radius, put the point on B and draw arcs that cross the first two.
Join the two crossing points with a ruler. This line passes through the midpoint of AB and makes an angle of \(90^\circ\).

Worked example 2: bisecting a line of 8 cm

Construct the perpendicular bisector of \(AB=8\) cm.

Use the three steps above. The line meets AB at its midpoint, 4 cm from each end. Every point on the bisector is the same distance from A as from B, which makes it useful when you need a point equidistant from two places.

Bisecting an angle

Method: bisecting an angle

With centre at the vertex, draw an arc cutting both arms of the angle.
With the compasses centred on each of these two points in turn, and the same radius, draw arcs that cross inside the angle.
Join the vertex to the crossing point. This line splits the angle into two equal halves.

Worked example 3: bisecting angles

Bisect an angle of \(70^\circ\). What size is each half? What about a \(112^\circ\) angle?

A \(70^\circ\) angle becomes two angles of \(35^\circ\). A \(112^\circ\) angle becomes two angles of \(56^\circ\).

Perpendiculars and 60 degrees

Method: perpendicular from a point to a line

Draw an arc centred at P that crosses the line twice.
Bisect the line between those two crossing points using the perpendicular bisector method.
The line through P and the bisector point is at \(90^\circ\) to the original line.

To construct a perpendicular at a point on the line, mark equal distances either side of the point first and then bisect.

Note: constructing \(60^\circ\)

Draw a line and mark a point. Draw an arc from the point that crosses the line, then use the same radius from the crossing point to cross the arc. Joining the original point to this new point makes an angle of \(60^\circ\), because the three points form an equilateral triangle.

Congruent triangles

Definition: congruent

Two shapes are congruent if they are exactly the same shape and size. One can be turned, reflected or moved to fit exactly on the other. Matching sides and angles are called corresponding parts.

Rule: the four congruence tests

SSS: three sides equal.
SAS: two sides and the included angle (the angle between them) equal.
ASA (or AAS): two angles and a corresponding side equal.
RHS: a right angle, the hypotenuse and one other side equal.

Watch out: AAA is not enough

Three equal angles (AAA) is not a congruence test: two equilateral triangles of different sizes both have angles of \(60^\circ\) but are not congruent. They are similar, not congruent.

Worked example 4: choosing a test

Triangle PQR has \(PQ=6\), \(QR=8\) and angle \(Q=40^\circ\). Triangle XYZ has \(XY=6\), \(YZ=8\) and angle \(Y=40^\circ\). Are they congruent?

The two given sides surround the given angle in both triangles, so the triangles are congruent by SAS.

Worked example 5: finding missing values

Triangle ABC is congruent to triangle PQR, with A matching P, B matching Q and C matching R. Angle \(A=40^\circ\) and angle \(B=65^\circ\). Find angle R.

Angle \(C=180-40-65=75^\circ\), so angle \(R=75^\circ\). The order of the letters tells you which parts match.

Using congruence in reasoning

Property: base angles of an isosceles triangle

In an isosceles triangle ABC with \(AB=AC\), draw a line from A to the midpoint M of BC. Triangles ABM and ACM have \(AB=AC\), \(BM=CM\) and the shared side AM, so they are congruent by SSS. It follows that angle B equals angle C: the base angles of an isosceles triangle are equal.

Worked example 6: right-angled triangles

Two right-angled triangles each have hypotenuse 13 cm and one side 5 cm. Are they congruent?

The third side of each is \(\sqrt{169-25}=12\) cm. They also match by RHS (right angle, hypotenuse, one side), so they are congruent.

Watch out: common mistakes

Changing the compass setting between the two arcs on a bisector.
Rubbing out the construction arcs.
Using SSA or AAA as if they proved congruence.
Writing the letters in the wrong order, so corresponding parts do not match.
Measuring angles with a protractor when a construction is requested.

Worked example 7: a triangle that fails

A pupil says that sides of 4 cm, 6 cm and 11 cm make a triangle. Show that the pupil is wrong.

The two shorter sides add up to \(4+6=10\) cm. This is less than the longest side, 11 cm. When you draw arcs of radius 4 cm and 6 cm from the two ends of the 11 cm line, they cannot meet. So no triangle exists.

Worked example 8: angle bisector problem

Two paths meet at a corner with an angle of \(64^\circ\). A bench is to be placed the same distance from both paths. Describe where it goes and find the angle it makes with each path.

The set of points the same distance from two lines is the angle bisector. The bisector splits \(64^\circ\) into two angles of \(64\div2=32^\circ\). So the bench is on the line from the corner at \(32^\circ\) to each path.

Worked example 9: proving congruence

In the diagram \(AB=DC\) and \(AD=BC\). Prove that triangles \(ABC\) and \(CDA\) are congruent.

\(AB=DC\) (given). \(BC=DA\) (given). \(AC\) is a common side to both triangles. The triangles have three pairs of equal sides, so they are congruent by SSS. Always write the reason beside each fact: given, common side, vertically opposite angles.

Note: writing a proof

A good congruence proof has three parts. First list each pair of equal parts with a reason. Second, state the test (SSS, SAS, ASA or RHS) in capital letters. Third, say what the congruence tells you, such as so angle \(B\) = angle \(D\). Write the vertices in matching order, so that \(ABC\) and \(PQR\) mean that \(A\) corresponds to \(P\).

Watch out: congruent or similar?

Congruent triangles are identical in shape and size. Similar triangles have the same shape but may be different sizes.
Equal angles alone (AAA) never prove congruence.
For SAS the angle must be between the two sides. Two sides and a non-included angle (SSA) can give two different triangles.

Did you know?

Surveyors have used congruent triangles for thousands of years. The Egyptians re-measured field boundaries after each flood of the River Nile, and today satellites use triangulation to find your position: your phone measures its distance from at least three satellites, much as you used arcs from three points to find where they cross.

Note

When you write a congruence proof, list the three pairs of equal sides or angles, give a reason for each one, and finish by naming the condition (SSS, SAS, ASA or RHS) that proves the triangles congruent. Keep the letters of the vertices in matching order so that corresponding points are easy to see.

Otto’s tip

Check a construction by measuring. The perpendicular bisector should meet the line at 90 degrees, and each half should be the same length.

How to use this Year 8 constructions and congruence lesson

First, read the key points so that you know where the Year 8 constructions and congruence lesson is going. Then work through each definition and worked example, and cover the solution before you read it. After that, practise with the constructions and congruence worksheet and check every question with the worksheet answers. When you feel ready, test yourself with the multiple choice questions or the quick quiz. Finally, sit the constructions and congruence test and mark it with the test answers.

Also, the starters are a good warm-up, so you can keep the key facts fresh. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday. In short, this Year 8 constructions and congruence lesson gives you everything you need for the chapter.