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5.03 Proving triangles congruent

Lesson

Now that we know the criteria for identifying congruent triangles, the next step is to find a way to communicate that knowledge in the form of a proof. Recall two commonly used proof forms, the two-column proof and the paragraph proof.

Two-column proofs

A two-column proof might be considered the most structured form, where every line of work is placed on a row with an accompanying reason. A generic shell for the proof looks like this:

To prove: $\left[\text{Statement to be proved}\right]$[Statement to be proved]
Statement Reason
Statement 1 Reason 1
Statement 2 Reason 2
Statement 3 Reason 3
$\ldots$ $\ldots$
$\left[\text{Statement to be proved}\right]$[Statement to be proved] Reason $n$n

We use as many lines of reasoning as we need, and make sure the last line always has the statement we want to prove at the end. Any information provided to us should have the reason Given written next to it, to indicate that no reasoning actually took place to arrive at the statement. Let's try a few examples where we prove triangles congruent using a two-column proof.

Worked examples

Question 1

Show the following two triangles are congruent using a two-column proof:

Think: Before writing anything down, it's a good idea for us to think for a minute about which congruence test we will use. Since none of the angles have been marked, we will have to rely on the sides. The next step (even if you haven't decided on the test yet!) will be to translate the information provided to us into the start of a two-column proof.

Do: We write down what we want to prove, followed by the information obtained from the markings on the diagram:

To prove: $\Delta ADC\cong\Delta CBA$ΔADCΔCBA
Statement Reason
$\overline{AD}\cong\overline{CB}$ADCB Given
$\overline{DC}\cong\overline{BA}$DCBA Given

We have two sides, and by the SSS rule, we only need to show one more pair of sides (one from each triangle) are congruent. In this case the two triangles share a side in common, so we use the reflexive property of congruence to note this:

To prove: $\Delta ADC\cong\Delta CBA$ΔADCΔCBA
Statement Reason
$\overline{AD}\cong\overline{CB}$ADCB Given
$\overline{DC}\cong\overline{BA}$DCBA Given
$\overline{AC}\cong\overline{AC}$ACAC Reflexive property of congruence

Now we have demonstrated that three sides are the same, we can write our conclusion at the end:

To prove: $\Delta ADC\cong\Delta CBA$ΔADCΔCBA
Statement Reason
$\overline{AD}\cong\overline{CB}$ADCB Given
$\overline{DC}\cong\overline{BA}$DCBA Given
$\overline{AC}\cong\overline{AC}$ACAC Reflexive property of congruence
$\Delta ADC\cong\Delta CBA$ΔADCΔCBA Side-side-side congruency (SSS)
Question 2

Show the following two triangles are congruent using a two-column proof:

Think: Make sure to take note of every marking on the diagram, and think about what rule to use.

Do: Once again we translate the information provided to us, both what we want to show and what the markings tell us:

To prove: $\Delta AMB\cong\Delta CMD$ΔAMBΔCMD
Statement Reason
$\overline{AM}\cong\overline{CM}$AMCM Given
$\overline{BM}\cong\overline{DM}$BMDM Given

We don't know anything about the third pair of sides, so we will have to use an angle instead. To use the SAS rule, we need to show that the angles lying between the two pairs of congruent sides are also congruent. Here we can use the fact that the angles are vertical, and reach our conclusion:

To prove: $\Delta AMB\cong\Delta CMD$ΔAMBΔCMD
Statement Reason
$\overline{AM}\cong\overline{CM}$AMCM Given
$\overline{BM}\cong\overline{DM}$BMDM Given
$\angle AMB\cong\angle CMD$AMBCMD Vertical angles
$\Delta AMB\cong\Delta CMD$ΔAMBΔCMD Side-angle-side congruency (SAS)
Question 3

Show the following two triangles are congruent using a two-column proof:

Think: As usual we make sure to take note of every marking on the diagram, and think about what test would be best to use. Since we are only shown information about a single pair of sides, we need to concentrate on finding two pairs of congruent angles.

Do: Translating what we want to prove and each of the markings in the diagram into the two-column format gives us the following:

To prove: $\Delta PXQ\cong\Delta RXS$ΔPXQΔRXS
Statement Reason
$\overline{PQ}\cong\overline{RS}$PQRS Given
$\overleftrightarrow{PQ}\parallel\overleftrightarrow{SR}$PQSR Given

Having established that $\overleftrightarrow{PQ}$PQ and $\overleftrightarrow{SR}$SR are parallel, we can use the alternate interior angles theorem to show that the angles lying on either end of $\overline{PQ}$PQ and $\overline{RS}$RS are congruent, and reach our conclusion:

To prove: $\Delta PXQ\cong\Delta RXS$ΔPXQΔRXS
Statement Reason
$\overline{PQ}\cong\overline{RS}$PQRS Given
$\overleftrightarrow{PQ}\parallel\overleftrightarrow{SR}$PQSR Given
$\angle XPQ\cong\angle XRS$XPQXRS Alternate interior angles theorem
$\angle XQP\cong\angle XSR$XQPXSR Alternate interior angles theorem
$\Delta PXQ\cong\Delta RXS$ΔPXQΔRXS Angle-side-angle congruency (ASA)

 

Paragraph proofs

The paragraph proof describes the same information, and goes through the same process, as a two-column proof, but we write it all out in more natural language and style. Here are some examples.

Worked examples

question 4

Show the following two triangles are congruent using a paragraph proof:

Think: Just like in a two-column proof we want to make sure we know what we want to prove, know what each marking means, and have an idea for a congruence test we want to use in the proof.

Do: We want to prove that $\Delta XAY\cong\Delta XBY$ΔXAYΔXBY. First we can see that $\angle XAY$XAY and $\angle XBY$XBY are both given as right-angles, so both $\Delta XAY$ΔXAY and $\Delta XBY$ΔXBY are right triangles. We can also see that $\overline{XY}$XY is the hypotenuse of both triangles, so by the reflexive property of congruence, the hypotenuses of both triangles are congruent. Finally, the segments $\overline{XA}$XA and $\overline{XB}$XB are congruent legs, as they are both radii of the circle centered at $X$X. We have therefore shown by Hypotenuse-leg congruency (HL) that $\Delta XAY\cong\Delta XBY$ΔXAYΔXBY, QED.

Did you know?

You may have seen proofs conclude with the letters QED, an abbreviation for the Latin phrase "quod erat demonstrandum". This roughly translates to "I have proved what I set out to prove", but it really means "I delivered on my promise and no-one can argue with me".

It is commonly mispronounced as "quod erat demonstraTum", and modern mathematicians usually use the symbol  in its place. Still, it is a powerful phrase and powerful idea, which is why it shows up outside of mathematics - in other academic learning areas, as well as books, films, and other popular media.

Question 5

Show the following two triangles are congruent using a paragraph proof:

Think: If you're more comfortable with two-column proofs, you might want to write out a list of statements first in that style before writing it in full sentences.

Do: We wish to demonstrate that $\Delta PAC\cong\Delta QBD$ΔPACΔQBD. We are given that $\overline{PA}$PA and $\overline{QB}$QB are parallel, that $\angle APC$APC and $\angle BQD$BQD are congruent, and that the segments $\overline{AB}$AB$\overline{BC}$BC, and $\overline{CD}$CD are all congruent. Since congruent segments have equal measure, we know that $AB=CD$AB=CD, and by the addition property of equality we can conclude that $AB+BC=BC+CD$AB+BC=BC+CD. This then means that the segments $\overline{AC}$AC and $\overline{BD}$BD have equal measure, so they must be congruent. The angles $\angle PAC$PAC and $\angle QBD$QBD are congruent since they are corresponding angles in the parallel lines $\overleftrightarrow{PA}$PA and $\overleftrightarrow{QB}$QB. We have therefore shown using angle-angle-side congruency (AAS) that $\Delta PAC$ΔPAC and $\Delta QBD$ΔQBD are congruent, QED.

No matter which method of proof you use, showing that two triangles are congruent involves preparing carefully justified statements that line up with one of the tests for congruency. Make sure you know them all and are ready to use them before trying to learn these two new proof methods.

 

Practice questions

Question 6

This two-column proof shows that $\Delta ABC\cong\Delta XYZ$ΔABCΔXYZ in the attached diagram, but it is incomplete.

Statements Reasons
$\overline{AC}\cong\overline{XZ}$ACXZ Given
$\angle CBA\cong\angle ZYX$CBAZYX

Given

$\angle CAB\cong\angle ZXY$CABZXY

Given

$\Delta ABC\cong\Delta XYZ$ΔABCΔXYZ

$\left[\text{____}\right]$[____]
  1. Select the correct reason to complete the proof.

    Angle-angle-side congruence (AAS)

    A

    Side-side-side congruence (SSS)

    B

    Side-angle-side congruence (SAS)

    C

    Side-side-angle congruence (SSA)

    D

    Angle-side-angle congruence (ASA)

    E

    Angle-angle-side congruence (AAS)

    A

    Side-side-side congruence (SSS)

    B

    Side-angle-side congruence (SAS)

    C

    Side-side-angle congruence (SSA)

    D

    Angle-side-angle congruence (ASA)

    E

Question 7

This two-column proof shows that $\Delta PQR\cong\Delta RSP$ΔPQRΔRSP in the attached diagram, but it is incomplete.

To prove: $\Delta PQR\cong\Delta RSP$ΔPQRΔRSP
Statements Reasons
$\left[\text{__________}\right]$[__________] $\left[\text{__________}\right]$[__________]
$\overline{PR}\cong\overline{RP}$PRRP Reflexive property of congruence

$\angle QRP\cong\angle SPR$QRPSPR

Given

$\Delta PQR\cong\Delta RSP$ΔPQRΔRSP

Side-angle-side congruence (SAS)
  1. Select the correct reason to complete the proof.

    $\overline{PS}\cong\overline{RQ}$PSRQ Transitive property of congruence
    A
    $\overline{SR}\cong\overline{QP}$SRQP Given
    B
    $\overline{PS}\cong\overline{RQ}$PSRQ Given
    C
    $\overline{SR}\cong\overline{QP}$SRQP Transitive property of congruence
    D
    $\overline{PS}\cong\overline{RQ}$PSRQ Transitive property of congruence
    A
    $\overline{SR}\cong\overline{QP}$SRQP Given
    B
    $\overline{PS}\cong\overline{RQ}$PSRQ Given
    C
    $\overline{SR}\cong\overline{QP}$SRQP Transitive property of congruence
    D

Question 8

This two-column proof shows that $\Delta DEH\cong\Delta FEG$ΔDEHΔFEG in the attached diagram, but it is incomplete.

Statements Reasons
$E$E is the midpoint of $\overline{DF}$DF Given
$\overline{DH}\cong\overline{FG}$DHFG Given

$\overline{EH}\cong\overline{EG}$EHEG

Given
$\left[\text{___}\right]$[___] $\left[\text{___}\right]$[___]

$\Delta DEH\cong\Delta FEG$ΔDEHΔFEG

$\left[\text{___}\right]$[___]
  1. Select the correct pair of reasons to complete the proof.

    $\angle EDH\cong\angle FEG$EDHFEG Corresponding angles theorem
    and
    $\Delta DEH\cong\Delta FEG$ΔDEHΔFEG Side-side-angle congruence (SSA)
    A
    $\overline{DE}\cong\overline{EF}$DEEF Reflexive property of congruence
    and
    $\Delta DEH\cong\Delta FEG$ΔDEHΔFEG Side-side-side congruence (SSS)
    B
    $\angle DHE\cong\angle FGE$DHEFGE Triangle hinge theorem
    and
    $\Delta DEH\cong\Delta FEG$ΔDEHΔFEG Side-angle-side congruence (SAS)
    C
    $\overline{DE}\cong\overline{EF}$DEEF Properties of a midpoint
    and
    $\Delta DEH\cong\Delta FEG$ΔDEHΔFEG Side-side-side congruence theorem (SSS)
    D
    $\angle EDH\cong\angle FEG$EDHFEG Corresponding angles theorem
    and
    $\Delta DEH\cong\Delta FEG$ΔDEHΔFEG Side-side-angle congruence (SSA)
    A
    $\overline{DE}\cong\overline{EF}$DEEF Reflexive property of congruence
    and
    $\Delta DEH\cong\Delta FEG$ΔDEHΔFEG Side-side-side congruence (SSS)
    B
    $\angle DHE\cong\angle FGE$DHEFGE Triangle hinge theorem
    and
    $\Delta DEH\cong\Delta FEG$ΔDEHΔFEG Side-angle-side congruence (SAS)
    C
    $\overline{DE}\cong\overline{EF}$DEEF Properties of a midpoint
    and
    $\Delta DEH\cong\Delta FEG$ΔDEHΔFEG Side-side-side congruence theorem (SSS)
    D

Outcomes

G.G-SRT.B.5

Use congruence and similarity criteria to prove relationships in geometric figures and solve problems utilizing real-world context.

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