Congruent shapes vs similar shapes
Two shapes are considered congruent if they are exactly the same shape (each contains the exact same angles) and size (each have the exact same side measurements). One may be a rotation or reflection of the other, but they are still exactly the same as each other.
Two shapes are considered similar if they are exactly the same shape but can be of different sizes. This means that one shape has undergone a dilation (and also possibly a reflection or rotation). Thus the measurements of one shape will have all increased or decreased by the same scale factor to produce the new similar shape. So congruent shapes are similar shapes that have a scale factor of $1$1.
Similar shapes
When we have determined that two shapes are similar because they are exactly the same shape, we can then determine the scale factor of enlargement or reduction by dividing a pair of corresponding side lengths. Using the scale factor we can then determine unknown lengths.
Practice questions
Question 1
Are the two shapes similar?
Yes
ANo
B
Question 2
Given that the two trapeziums shown are similar.
Find the enlargement factor.
Calculate the value of $x$x
Similar triangles
A particular shape of interest when it comes to similarity is the triangle. Later we will see that similar triangles are very useful in a range of applications.
Two similar triangles will have corresponding angles equal and corresponding sides in the same ratio but we don't need to know all the angles and sides to determine if two triangles are similar. The following gives us ways to test if two triangles are similar with limited information.
Angle, angle, angle (AAA)
If two triangles each have the exact same corresponding angles, then they have the exact same shape and are therefore similar.
Note: Two angles are sufficient for this test since the remaining angle must also be equal due to the interior angle sum of a triangle.
If two triangle have the same three angles, they are similar.
Side, side, side (SSS)
If two triangles have three corresponding pairs of sides, all in the same ratio (that is, all with the same scale factor of enlargement or reduction) then they are also the exact same shape and are therefore similar.
Scale factor of $2$2.
Side, angle, side (SAS)
If two triangles have two corresponding pairs of sides in the same ratio, and the exact same included angle, they are similar triangles.
Scale factor of $3$3.
Right, hypotenuse, side (RHS)
If two triangles are both right-angled, the pair or hypotenuses are in the same ratio as a pair of other corresponding sides, they are similar triangles.
Note: This is in fact a special case of side-side-side, since using Pythagoras' theorem we can show given two corresponding pairs of sides of right-angled triangles in the same ratio, the third side must also be in the same ratio.
Scale factor of $\frac{1}{4}$14.
Practice questions
Question 3
Consider the two similar triangles.
By filling in the gaps, match the corresponding angles.
$\angle$∠$A$A corresponds to $\angle$∠$\editable{}$
$\angle$∠$B$B corresponds to $\angle$∠$\editable{}$
$\angle$∠$C$C corresponds to $\angle$∠$\editable{}$
$AB$AB corresponds to which side in $\triangle PQR$△PQR?
$QR$QR
A$RP$RP
B$PQ$PQ
C$BC$BC corresponds to which side in $\triangle PQR$△PQR?
$QR$QR
A$RP$RP
B$PQ$PQ
C
Question 4
Which of these triangles are similar?
A
B
C
DGive a suitable reason for your choice.
All matching angles are equal.
AAll sides are in the same ratio.
BAll the corresponding sides are equal.
C
Question 5
Consider the following triangles:
Which of these triangles are similar?
A
B
C
DGive the reason for their similarity.
Two sides are in the same ratio and the included angles are equal.
AAll corresponding angles are equal.
BAll corresponding sides are in the same ratio.
C
Proof of similarity
We can more formally prove that two triangles are similar by determining if they satisfy at least one of the similarity conditions listed above. Identify which similarity test to use and then state which angles are equal or sides are in equal proportion, giving a reason for each statement. Then conclude by stating the triangles that are similar and the test used.
Worked example
In the following triangles, $\frac{QZ}{SZ}=\frac{PZ}{RZ}$QZSZ=PZRZ. Prove that these triangles are similar.
Think: We have been given two pairs of corresponding side measures that are in the same proportion, and the angles between the relevant sides are vertically opposite angles. We should use SAS to prove these triangles are similar.
Do: We write down what we want to prove, followed by the information obtained from the information given to us:
| To prove: $\Delta PQZ\sim\Delta RSZ$ΔPQZ~ΔRSZ | |
| Statement | Reason |
| $\frac{QZ}{SZ}=\frac{PZ}{RZ}$QZSZ=PZRZ | (Given) |
| $\angle PZQ\cong\angle RZS$∠PZQ≅∠RZS | (Vertical opposite angles are congruent) |
| $\Delta PQZ\sim\Delta RSZ$ΔPQZ~ΔRSZ | (Two sides in the same ratio and the included angles equal - SAS) |
Commonly used notation to state two shapes are congruent are the symbols $\cong$≅ and $\equiv$≡. The first symbol abbreviates that the two shapes are similar and equal in size, the second is the symbol is commonly used to say two statements are equivalent.
Commonly used notation to state two shapes are similar are the symbols $\sim$~ and $///$///. We can see the first symbol relates to the first congruent symbol in saying the two shapes are similar but not necessarily equal in size. The second symbol is like the second congruent symbol on its side.
Practice questions
Question 6
Prove that $\triangle ABC$△ABC and $\triangle EGF$△EGF are similar.
In $\triangle ABC$△ABC and $\triangle EGF$△EGF we have:
Question 7
In the diagram, $AB=7$AB=7, $BC=10.5$BC=10.5 and $BE=6$BE=6.
Prove that the $\triangle ABE$△ABE and $\triangle ACD$△ACD are similar.
In the two triangles $\triangle ABE$△ABE and $\triangle ACD$△ACD we have:
Solve for the value of $f$f.