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Stage 5.1-3

5.01 Proving triangles congruent and similar

Geometry (reasoning) questions

The questions in this topic are quite different from other Mathspace questions. You will need to enter complete proofs one step at a time, and provide reasons for the steps you enter.

A tutorial for this kind of question is provided at the end of this lesson.

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.


Finding common features

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.

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

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

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

The long diagonal of a kite bisects the short diagonal, so $PX$PX and $RX$RX must be equal.


Giving reasons

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 $\angle ADB$ADB and $\angle CDB$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:

$\angle ADB=\angle CDB$ADB=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 $AD$AD and $CD$CD are equal, as well as $BD$BD being a common side in both triangles. The lines of working with reasons for these common features would be:

$AD=CD$AD=CD (Given)

$BD$BD is common

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'.


Proving triangles similar or congruent

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

Worked example

Prove that $\triangle JKL$JKL and $\triangle MNJ$MNJ are similar.

Think: Looking at the diagram, we can see that both triangles have a right angle and that we have a pair of parallel lines to work with.

Do: We can start by using the right angles to say that:

$\angle JKL=\angle MNJ$JKL=MNJ (Given)

Then, using the parallel lines with $JL$JL as the transversal, we can also say that:

$\angle MJN=\angle JLK$MJN=JLK (Alternate angles in parallel lines are equal)

Since we have shown that the triangles have two pairs of matching angles, the third angle in each triangle must also be matching. Therefore, we can prove that:

$\triangle JKL\simeq\triangle MNJ$JKLMNJ (All three pairs of corresponding angles are equal (AAA))

Reflect: Using the features of the diagram, we found the common features between the triangles, giving reasons. Once we had enough information we used one of the similarity tests to prove that the triangles were similar.

It is worth noting that we could also have proved that $JK$JK and $NM$NM are parallel (both perpendicular to parallel lines) which could then have been used to prove that $\angle JMN=\angle LJK$JMN=LJK (alternate angles on parallel lines). This shows us that there can be multiple ways to prove things.


Practice questions

Question 1

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

  1. Show all working and reasoning.

Question 2

In the diagram below, $XZ$XZ bisects $\angle WZY$WZY.

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

  1. Show all working and reasoning.

Question 3

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

  1. Enter two statements of the form $\frac{\editable{}}{\editable{}}=\frac{\editable{}}{\editable{}}$= before stating your conclusion.


Geometry (reasoning) tutorial

There are two stages to this tutorial. The first stage will show you how to enter geometrical objects and expressions, and the second stage will show you how to enter geometrical proofs.

Click or tap on "Try this problem!" to launch each stage.

First stage

Welcome to the first of two stages in the geometry tutorial.

This stage will show you how to enter geometrical objects and expressions.

Use the hints to help you if you get stuck.

  1. The toolbar appears when you click the input box (where you enter your answers). The toolbar has new symbols for geometry questions.

    Try inputting the following statement using the toolbar:

    $\angle RAD$RAD$=$=$\angle SAX$SAX

  2. You may have noticed that the toolbar also provides keyboard shortcuts. For example, you can type angle to create the angle symbol $\angle$ rather than clicking on the toolbar button.

    The solution for this part of the tutorial is the same as the previous question:

    $\angle RAD$RAD$=$=$\angle SAX$SAX.

    Try typing angleRAD=angleSAX rather than using the toolbar.

    Check the hints for more tips.

  3. Whenever you hover over a symbol on the toolbar, the keyboard shortcut will appear.

    You can type this into the input box instead of clicking on that button each time.

    Try inputting the statement using only your keyboard:

    $\angle ABC$ABC$\text{is common}$is common

  4. The geometry question type has many more ways to relate two objects than equality (the $=$= sign).

    Click the right-angled triangle symbol on the toolbar to reveal these symbols.

    To complete this part, enter the following fact:

    $AB$AB is parallel to $CD$CD

  5. Let's try a few more of the symbols under the right-angled triangle symbol on the toolbar.

    Enter the following expressions, one step at a time:

    $\triangle ABC$ABC$\simeq$$\triangle LMN$LMN

    $\triangle ABC$ABC$\equiv$$\triangle DEF$DEF

    Every time you use a new symbol, check its keyboard shortcut by hovering over it.

  6. Sometimes you will also enter numbers as part of your answer.

    Consider this diagram:

    To enter the value of $\angle DEF$DEF we would enter the equation $\angle DEF=42$DEF=42.

    Note that in geometry questions we never use the degree symbol ($^\circ$°).

    Instead of $\angle DEF$DEF, enter the value of $\angle DFE$DFE in the form $\angle DFE=\editable{}$DFE= to continue.


Second stage

Welcome to the second of two stages in the geometry tutorial.

This final stage will show you how to enter geometrical proofs.

Use the hints to help you if you get stuck.

  1. The geometry question type will ask you to input your reasoning after entering a step.

    Warning! After entering a step you will not be able to request further hints. Make sure to open all available hints before entering a step!

    Open all hints now before continuing.

    Now, consider this triangle:

    To find the value of $x$x, first enter $x=12$x=12. You will then be prompted to give a reason.

    The reason we want to use is "Sides opposite base angles are equal in an isosceles triangle".

    Type the words in the reason into the reasoning box to narrow down the list.

  2. When completing proofs you may receive the message "This step is correct, but you need to show some more working first".

    This will show up when your step is correct, but you haven't entered a crucial step that lets you make that conclusion.

    In this question we want to prove that $x=40$x=40.

    Try entering the steps of working in the order listed below:

    Step of working Reason
    $OP=OQ$OP=OQ Radii of a circle are equal
    $x=40$x=40 Base angles are equal in an isosceles triangle

    Remember to open all hints before entering each step.

  3. You may also receive the message "This step is not necessary, there is a better way".

    This will show up when you enter a line of working that is technically true, but doesn't directly help you to prove the result.

    In this question, we will prove that $y=100$y=100.

    Try entering the steps of working in order listed below to complete the proof.

    Step of working Reason
    $\angle OPQ+\angle OQP+\angle POQ=180$OPQ+OQP+POQ=180 The interior angle sum of a triangle is $180^\circ$180°

    Remember to open all hints before entering each step.

  4. When we want to state that triangles are similar or congruent, we have to be careful of the order we enter the vertices.

    The diagram below shows the six different ways we can order the vertices in a triangle:

    If we wanted to show that $\triangle ABC$ABC was similar or congruent to another triangle, we must ensure that the matching points in the other triangle appear in a matching order. Let's do that now by looking at these two triangles:

    Match up the vertices and enter the fact that they are congruent, together with the reason:

    "SSS: All sides are equal"

    Remember to open all hints before entering each step.

  5. If you are proving that two triangles are similar or congruent, you must enter each fact before stating your conclusion - even if it seems obvious.

    If the fact is explicitly provided to you in the question, or clearly marked on the diagram, you should enter "Given" for your reason.

    When a side or angle is common to both triangles, we must state that they are common using the "$\text{is common}$is common" statement. You will not be prompted for a reason in this case.

    Try proving that $\triangle TUV\equiv\triangle WVU$TUVWVU by entering the steps below:

    Fact Reason
    $UT=VW$UT=VW Given
    $UV$UV is common
    $\angle TUV=\angle WVU$TUV=WVU Given
    $\triangle TUV\equiv\triangle WVU$TUVWVU SAS: Two pairs of corresponding sides and the pair of included angles are equal

    Remember to open all hints before entering each step.




applies trigonometry to solve problems, including problems involving bearings


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

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