6.01 Proving triangles congruent and similar

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.

\angle NKL and \angle LMN are opposite angles in a parallelogram, so they must be equal.

\angle DEC and \angle FEG are vertically opposite angles, so they must be equal.

\angle PQR and \angle STR are alternate angles on parallel lines, so they must be equal.

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.

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:

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.

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

	Statements	Reasons
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)

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Example 2

In the diagram below, XZ bisects \angle WZY.

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

	Statements	Reasons
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)

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Example 3

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

Worked Solution

Create a strategy

Show that the sides are in the same ratio.

Apply the idea

To prove: \triangle ABC ||| \triangle DFE

	Statements	Reasons
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))

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