NSW Stage 5.2 - 2020 Edition

7.02 Further triangle proofs

Lesson

One of the main advantages of being able to prove things is that we can then use what we have proved to prove something else. By starting from some basic information, we can often follow a chain of proofs to discover complex and interesting properties that we would never have seen before.

Specifically for triangles, if we know how to prove triangles congruent or similar then we can use the properties of congruent or similar triangles to discover other properties in a diagram.

After proving that two triangles are similar or congruent, we gain access to all the properties of similar or congruent triangles.

If we can prove that two triangles are congruent using any of the congruence tests, we have also proved that and angle or side in one triangle must be equal to the corresponding angle or side in the other.

The same applies if we can prove two triangles are similar, except instead of equal sides we get sides in a common ratio.

In particular, knowing which angles are equal can help us find relationships between the lines that the angles lie between, since there are many line properties relating to equal angles.

In the diagram below, $XZ=WY$`X``Z`=`W``Y`.

Prove that $AX$`A``X` and $BY$`B``Y` are parallel.

**Think:** Consider what must be true if $AX$`A``X` and $BY$`B``Y` were parallel. If $AX$`A``X` and $BY$`B``Y` were parallel then $\angle AXY$∠`A``X``Y` and $\angle BYX$∠`B``Y``X` would be equal since they are alternate angles on the two lines.

One way to prove that these angles are equal is to show that they are corresponding angles in congruent or similar triangles. So let's start by proving that $\triangle AXZ$△`A``X``Z` and $\triangle BYW$△`B``Y``W` are congruent or similar.

**Do:** Using the information we are given, we can prove that:

- $XZ=WY$
`X``Z`=`W``Y`(Given) - $\angle AZX=\angle BWY$∠
`A``Z``X`=∠`B``W``Y`(Alternate angles in parallel lines are equal) - $AZ=BW$
`A``Z`=`B``W`(Opposite sides in a parallelogram are equal)

With these three facts, we have established enough to prove that:

$\triangle AZX$△`A``Z``X`$\equiv$≡$\triangle BYW$△`B``Y``W`

(SAS: Two pairs of corresponding sides and the pair of included angles are equal)

Since we have proved that the two triangles are congruent, we now know for certain that:

$\angle AXY=\angle BYX$∠`A``X``Y`=∠`B``Y``X`

(Corresponding angles in congruent triangles are equal)

Since these two angles are equal, we can finally prove that:

$AX\parallel BX$`A``X`∥`B``X`

(Alternate angles in parallel lines are equal)

**Reflect:** We started by considering what else we would know about the diagram if we knew what we were trying to prove. We then found a different way to find that information, allowing us to prove what we wanted to.

Consider the diagram.

Why is $BE$

`B``E`parallel to $CD$`C``D`?$\angle ABE$∠

`A``B``E`and $\angle ACD$∠`A``C``D`form a pair of equal corresponding angles.A$\angle ABE$∠

`A``B``E`and $\angle ACD$∠`A``C``D`form a pair of supplementary cointerior angles.B$\angle ABE$∠

`A``B``E`and $\angle ACD$∠`A``C``D`form a pair of equal alternate angles.C$\angle ABE$∠

`A``B``E`and $\angle ACD$∠`A``C``D`form a pair of equal corresponding angles.A$\angle ABE$∠

`A``B``E`and $\angle ACD$∠`A``C``D`form a pair of supplementary cointerior angles.B$\angle ABE$∠

`A``B``E`and $\angle ACD$∠`A``C``D`form a pair of equal alternate angles.CWhich angle is equal to $\angle BEA$∠

`B``E``A`?$\angle BEA=\editable{}$∠

`B``E``A`=How do we know that $\triangle ABE\simeq\triangle ACD$△

`A``B``E`⫻△`A``C``D`?Two right-angled triangles have the ratio of their hypotenuses equal to the ratio of another pair of matching sides

ATwo pairs of matching sides are in the same ratio and the included angles are equal

BAll pairs of matching sides are in the same ratio

CAll three pairs of corresponding angles are equal.

DTwo right-angled triangles have the ratio of their hypotenuses equal to the ratio of another pair of matching sides

ATwo pairs of matching sides are in the same ratio and the included angles are equal

BAll pairs of matching sides are in the same ratio

CAll three pairs of corresponding angles are equal.

DWhat is the scale factor relating $\triangle ABE$△

`A``B``E`to $\triangle ACD$△`A``C``D`?Solve for the value of $f$

`f`.

Consider the diagram below:

Prove that $\triangle ABE$△

`A``B``E`is similar to $\triangle BCD$△`B``C``D`.Prove that $\triangle EDB$△

`E``D``B`is similar to $\triangle BCD$△`B``C``D`.Can we conclude that $\triangle ABE$△

`A``B``E`is similar to $\triangle EDB$△`E``D``B`?Yes

ANo

BYes

ANo

B

Consider the diagram below:

Prove that $CE=EB$

`C``E`=`E``B`.

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