Lesson

Two triangles can be closely related even though they're not congruent. The most important relationship after congruence is called similarity, and we say two triangles are similar when one can be **uniformly scaled** into a triangle that is congruent to the second:

*All of these six triangles are similar - each is just an enlarged version of the leftmost triangle.*

*This spiral was built from many congruent triangles by enlarging them the right amount and rotating them into place.*

Two triangles that are similar have three pairs of congruent angles, one from each in each pair. This is because shrinking or growing a triangle doesn't change the relative arrangement of the three sides to each other. Here is an illustration:

*Each of the sides is parallel to the others, so we can use what we know about parallel lines to reason that their angles are all congruent.*

It is enough to know that there are only **two pairs of congruent angles** since the other pair must also be congruent. We concluded that the other pair of angles in the two triangles must also be congruent because the sum of the measures of the interior angles of any triangle is always equal to 180 degrees.

Another feature of similar triangles is that **the length of each side is in the same proportion**. Without knowing any angles, we know these two triangles are similar:

Each side of $\Delta XYZ$Δ`X``Y``Z` is three times as long as a side from $\Delta ACB$Δ`A``C``B`, so by shrinking $\Delta XYZ$Δ`X``Y``Z` by a factor of three we would produce a congruent triangle. We write $\Delta XYZ\sim\Delta ACB$Δ`X``Y``Z`~Δ`A``C``B` to indicate that they are similar.

In fact, each of our congruence tests may be used, though modified to account for the scaling factor. Since only two angles are required, some congruence tests are redundant.

Proving similarity in triangles

Two triangles are similar if they satisfy one of the following criteria:

- Angle-angle similarity, or AA~
- The two triangles have two pairs of congruent angles

- Side-side-side similarity, or SSS~
- The two triangles have three pairs of sides whose lengths are in the same proportion

- Side-angle-side similarity, or SAS~
- The two triangles have two pairs of sides whose lengths are in the same proportion, and the angles between these sides are also congruent

Consider the two triangles given in the diagram below:

Are the two given triangles similar?

**Think**: We can tell right away that $\angle QPR\cong\angle FHG$∠`Q``P``R`≅∠`F``H``G`, but the other two angles don't match up. We will have to use another property to identify the missing angles.

**Do**: Since $\Delta PQR$Δ`P``Q``R` is a triangle, the sum of the measures of its angles must be $180^\circ$180°. Expressing this as an equation and by substitution we have $m\angle PRQ+92^\circ+45^\circ=180^\circ$`m`∠`P``R``Q`+92°+45°=180°, so $m\angle PRQ=43^\circ$`m`∠`P``R``Q`=43° after rearranging. This means $\angle PRQ\cong\angle HGF$∠`P``R``Q`≅∠`H``G``F`, and since $\angle QPR\cong\angle FHG$∠`Q``P``R`≅∠`F``H``G` we conclude that they are similar by AA~ and write $\Delta PQR\sim\Delta HFG$Δ`P``Q``R`~Δ`H``F``G`.

Consider the two triangles in the diagram below:

Are the two given triangles similar?

**Think**: We have been provided with two pairs of sides and the angle between them, so we should use the similarity version of SAS to test whether or not they are similar.

**Do**: We are given that $\angle BAC\cong\angle XYZ$∠`B``A``C`≅∠`X``Y``Z`, and we notice that the lengths $AB$`A``B` and $YZ$`Y``Z` are in the ratio $3:5$3:5. However, the lengths $AC$`A``C` and $YX$`Y``X` are in the ratio $7:12$7:12. Since the measures of these sides are not in the same proportion, we conclude that these triangles are not similar.

**Reflect**: How can we conclude that they are **not** similar from this information? How can we know that the sides and angles don't match up in some other way?

Select the two triangles that are similar, if possible.

- ABC
There are no similar triangles.

DABCThere are no similar triangles.

D

Consider the diagram below.

Is there enough information to show whether or not the triangles are similar?

Yes, the diagram has enough information to prove the triangles are similar.

AYes, the diagram has enough information to prove the triangles are

**not**similar.BNo, there is not enough information in the diagram to show whether the triangles are similar or not.

CYes, the diagram has enough information to prove the triangles are similar.

AYes, the diagram has enough information to prove the triangles are

**not**similar.BNo, there is not enough information in the diagram to show whether the triangles are similar or not.

CGive a reason for the triangles' similarity.

Side-angle-side similarity (SAS$\sim$~)

ASide-side-side similarity (SSS$\sim$~)

BAngle-angle similarity (AA$\sim$~)

CSide-angle-side similarity (SAS$\sim$~)

ASide-side-side similarity (SSS$\sim$~)

BAngle-angle similarity (AA$\sim$~)

C

Consider the diagram below.

Is there enough information to show whether or not the triangles are similar?

Yes, the diagram has enough information to prove the triangles are similar.

AYes, the diagram has enough information to prove the triangles are

**not**similar.BNo, there is not enough information in the diagram to show whether the triangles are similar or not.

CYes, the diagram has enough information to prove the triangles are similar.

AYes, the diagram has enough information to prove the triangles are

**not**similar.BNo, there is not enough information in the diagram to show whether the triangles are similar or not.

CWhich of the following reasons explains why the triangles are not similar.

The sides are not in the same proportion.

AThe angles are not in the same proportion.

BThe sides are not congruent.

CThe angles are not congruent.

DThe sides are not in the same proportion.

AThe angles are not in the same proportion.

BThe sides are not congruent.

CThe angles are not congruent.

D