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Stage 5.1-2

7.01 Proving triangles congruent and similar

Lesson

Introduction

To prove that a pair of triangles are either congruent or similar, we need to find enough information to satisfy one of the  congruence  or  similarity  tests.

Prove triangles are similar of congruent

Finding enough information depends on whether we can find enough common features between the triangles. For this, we will need to use a variety of geometric properties that can relate angles or sides.

Features that can help us relate information between two triangles include, but are not limited to:

  • Parallel lines (giving us alternate or corresponding angles)

  • Vertically opposite angles at a point of intersection between lines

  • Sides or angles common to both triangles

  • Equal markings on pairs of sides or angles

  • Pairs of sides in a common ratio

  • Various properties of quadrilaterals

We can see a few of these being used to find common features in the triangles below.

Parallelogram K L M N where sides K L and M N are parallel and sides K N and M L are parallel. The diagonal L N is drawn.
\angle NKL and \angle LMN are opposite angles in a parallelogram, so they must be equal.
Triangles C D E and G F E where side C E equals side G E and side D E equals side F E. C E F and G E D are straight lines.
\angle DEC and \angle FEG are vertically opposite angles, so they must be equal.
Triangles P Q R and R S T where side P R equals side R S. P R S and Q R T are straight lines. P Q is parallel to T S.
\angle PQR and \angle STR are alternate angles on parallel lines, so they must be equal.
Kite P Q R S with diagonals Q S and P R intersecting at X. Side P Q equals side R Q and side P S equals side R S.
The long diagonal of a kite bisects the short diagonal, so PX and RX must be equal.

In addition to finding common features, we also need to give reasons for each piece of new information when writing a proof. The reason we give for a step of working is the feature or property we needed to know that the step was true.

In the triangles below we can determine that \angle ADB and \angle CDB are equal.

This image shows triangles C D E and G F E in which sides C E and G E are equal and sides D E and F E are equal.

We know this because the long diagonal of a kite bisects the opposite angles of the kite. As such, we would write our line of working as:

\displaystyle \angle ADB\displaystyle =\displaystyle \angle CDB(The longest diagonal of a kite bisects the angles through which it passes)

Other features that are more obvious still require some justification, but can be simpler to explain.

In that same pair of triangles we can also see that AD and CD are equal, as well as BD being a common side in both triangles. The lines of working with reasons for these common features would be:

\begin{array}{cll} AD=CD &\text{(Given)} \\ BD \text{ is common} & \\ \end{array}

When equal sides or angles are marked on the diagram, we say that these features are 'given'. For sides or angles that are in both triangles, it is enough to note that they are 'common'.

Once we have enough information with reasons, we can determine that two triangles are congruent or similar using one of the tests.

Examples

Example 1

Prove that \triangle KLN and \triangle MNL are congruent.

Parallelogram K L M N with diagonal N L and two pairs of opposite parallel sides K L and M N and sides K N and M L.
Worked Solution
Create a strategy

Show that the triangles satisfy one of the congruence tests: SSS, AAS, RHS, or SAS.

Apply the idea
To prove: \triangle KLN \equiv \triangle MNL
StatementsReasons
1.LN is common(Given)
2.KL = MN (Opposites sides, parallelogram KLMN)
3.\angle KLN = \angle MNL(Alternate angles, KL || MN)
4.\triangle KLN \equiv \triangle MNL(SAS)

Example 2

In the diagram below, XZ bisects \angle WZY.

2 triangles W X Z and Y X Z with equal angles at W and Y. The triangles have a common side X Z.

Prove that \triangle WXZ and \triangle YXZ are congruent.

Worked Solution
Create a strategy

Show that the triangles satisfy one of the congruence tests: SSS, AAS, RHS, or SAS.

Apply the idea
To prove: \triangle WXZ \equiv \triangle YXZ
StatementsReasons
1.\angle ZWX = \angle ZYX(Given)
2.XZ is common(Given)
3.\angle XZW = \angle XZY(XZ bisects \angle WZY)
4.\triangle WXZ \equiv \triangle YXZ(AAS)

Example 3

Prove that \triangle ABC and \triangle DFE are similar.

2 triangles A B C and D E F. Length of side A B is 8, A C is 6, and C B is 8, D E is 18, D F is 17 and E F is 24.
Worked Solution
Create a strategy

Show that the sides are in the same ratio.

Apply the idea
To prove: \triangle ABC ||| \triangle DFE
StatementsReasons
1.\dfrac{DE}{AC}=\dfrac{18}{6}=3\text{ }
2.\dfrac{DF}{AB}=\dfrac{27}{9}=3\text{ }
3.\dfrac{EF}{CB}=\dfrac{24}{8}=3\text{ }
4.\dfrac{DE}{AC}=\dfrac{DF}{AB}=\dfrac{EF}{CB}\text{ }
5.\triangle ABC ||| \triangle DFE(All pairs of matching sides are in the same ratio (SSS))
Idea summary

Features that can help us relate information between two triangles include, but are not limited to:

  • Alternate or corresponding angles on parallel lines

  • Vertically opposite angles

  • Sides or angles common to both triangles

  • Pairs of sides in a common ratio

  • Various properties of quadrilaterals

We also need to give reasons for each step of working which is the feature or property we needed to know that the step was true.

To prove that 2 triangles are congruent we need to prove that one of the following congruence tests holds: SSS, SAS, AAS, RHS.

To prove that 2 triangles are similar we need to prove that one of the following similarity tests holds:

  • Three pairs of equal angles (equiangular)

  • Three pairs of sides in the same ratio

  • Two pairs of sides in the same ratio and an equal included angle

  • Triangles are right angled and the hypotenuses and another pair of sides are in the same ratio

Outcomes

MA5.2-14MG

calculates the angle sum of any polygon and uses minimum conditions to prove triangles are congruent or similar

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