Criteria for Similarity of Triangles

🔄 Quick Recap

In the previous sections, we learned about similar figures and the Basic Proportionality Theorem. We know that two figures are similar when they have the same shape but possibly different sizes. This means their corresponding angles are equal and their corresponding sides are in the same ratio.

Now, let's focus specifically on triangles and learn some special rules that make it easier to check if two triangles are similar.

📚 What Makes Triangles Similar?

For two triangles to be similar, we need:

1. Their corresponding angles to be equal
2. Their corresponding sides to be in the same ratio

However, checking all six parts (three angles and three sides) would be time-consuming. Fortunately, we have some special criteria that make this easier!

🖼️ The AA Criterion (Angle-Angle)

The AA criterion states:

If two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar.

Why does this work? Because in a triangle, the three angles always add up to 180°. So if two angles are equal, the third angle must also be equal!

This means we only need to check two pairs of angles to determine if triangles are similar.

Example of AA Criterion:

In triangles ABC and PQR, if ∠A = 50°, ∠B = 60°, ∠P = 50°, and ∠Q = 60°, then triangles ABC and PQR are similar by the AA criterion.

🖼️ The SSS Criterion (Side-Side-Side)

The SSS criterion for similarity states:

If the corresponding sides of two triangles are proportional (in the same ratio), then the triangles are similar.

This means if:

AB/PQ = BC/QR = CA/RP

Then triangles ABC and PQR are similar.

Example of SSS Criterion:

In triangles ABC and PQR, if AB = 3 cm, BC = 4 cm, CA = 5 cm, PQ = 6 cm, QR = 8 cm, and RP = 10 cm, then:

AB/PQ = 3/6 = 1/2
BC/QR = 4/8 = 1/2
CA/RP = 5/10 = 1/2

Since all ratios are equal (1/2), triangles ABC and PQR are similar by the SSS criterion.

🖼️ The SAS Criterion (Side-Angle-Side)

The SAS criterion for similarity states:

If one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are proportional, then the triangles are similar.

If ∠A = ∠P, and:

AB/PQ = AC/PR

Then triangles ABC and PQR are similar.

Example of SAS Criterion:

In triangles ABC and PQR, if ∠A = 60°, ∠P = 60°, AB = 4 cm, AC = 6 cm, PQ = 8 cm, and PR = 12 cm, then:

AB/PQ = 4/8 = 1/2
AC/PR = 6/12 = 1/2

Since ∠A = ∠P and the ratios of the sides containing these angles are equal, triangles ABC and PQR are similar by the SAS criterion.

⚖️ Quick Comparison: Similarity vs. Congruence Criteria

Similarity Criteria	Congruence Criteria
AA: Two angles equal	ASA: Two angles and the included side equal
SSS: All sides proportional	SSS: All sides equal
SAS: One angle equal and sides including it proportional	SAS: One angle equal and sides including it equal

✅ Solved Examples

Example 1

In triangles ABC and DEF, ∠A = 50°, ∠B = 60°, ∠D = 50°, and ∠E = 60°. Are the triangles similar?

Solution: We have:

• ∠A = ∠D = 50°
• ∠B = ∠E = 60°

Since two angles of triangle ABC are equal to two angles of triangle DEF, by the AA criterion, triangles ABC and DEF are similar.

Example 2

In triangles PQR and XYZ, PQ = 3 cm, QR = 4 cm, RP = 5 cm, XY = 4.5 cm, YZ = 6 cm, and ZX = 7.5 cm. Are the triangles similar?

Solution: We need to check if the corresponding sides are in the same ratio:

PQ/XY = 3/4.5 = 2/3
QR/YZ = 4/6 = 2/3
RP/ZX = 5/7.5 = 2/3

Since all the ratios are equal (2/3), the corresponding sides are proportional. Therefore, by the SSS criterion, triangles PQR and XYZ are similar.

Example 3

In triangles ABC and DEF, ∠A = ∠D, AB = 5 cm, AC = 7 cm, DE = 10 cm, and DF = 14 cm. Are the triangles similar?

Solution: We have:

• ∠A = ∠D
• AB/DE = 5/10 = 1/2
• AC/DF = 7/14 = 1/2

Since one angle of triangle ABC is equal to one angle of triangle DEF, and the sides including these angles are proportional, by the SAS criterion, triangles ABC and DEF are similar.

🧪 Activity Time!

Let's verify the AA criterion with a simple hands-on activity:

1. Draw a triangle on a piece of paper and measure two of its angles (let's say 50° and 60°)
2. Draw another triangle with the same two angles but with sides of different lengths
3. Measure the third angle in both triangles - they should be equal!
4. Measure all the sides and calculate the ratios - they should all be equal
5. This confirms that if two angles of two triangles are equal, the triangles are similar

⚠️ Common Misconceptions

• Misconception: If one angle of a triangle equals one angle of another triangle, the triangles are similar. Truth: One equal angle is not enough. We need at least two equal angles or additional information about the sides.

• Misconception: Triangles with all three sides in proportion are similar only if the corresponding angles are also equal. Truth: If all three sides are proportional, the triangles are definitely similar. The equal angles follow as a result.

💡 Key Points to Remember

• The AA criterion: If two angles of one triangle equal two angles of another triangle, the triangles are similar.
• The SSS criterion: If all three sides of one triangle are proportional to the three sides of another triangle, the triangles are similar.
• The SAS criterion: If one angle of a triangle equals one angle of another triangle and the sides including these angles are proportional, the triangles are similar.
• For similar triangles, the ratio of corresponding sides equals the ratio of corresponding heights, medians, and angle bisectors.

🤔 Think About It!

1. Can we have an "ASA" (Angle-Side-Angle) similarity criterion like we have for congruence?
2. If two triangles have three pairs of corresponding sides proportional, must their corresponding angles be equal?
3. Can we have an "SSA" (Side-Side-Angle) similarity criterion? Why or why not?

In the next section, we'll look at applications of triangle similarity in solving real-world problems!