Property 1: Equal Corresponding Angles

When two triangles are similar, each angle in the first triangle equals the corresponding angle in the second triangle. The order of vertices tells us which angles correspond:

• If ΔABC ~ ΔPQR, then:
  • ∠CAB = ∠QPR
  • ∠ABC = ∠PQR
  • ∠BCA = ∠QRP

Property 2: Proportional Corresponding Sides

The ratios of all corresponding sides are equal. This common ratio is called the constant of proportionality or scale factor.

If ΔABC ~ ΔPQR, then:

$\frac{AB}{PQ} = \frac{BC}{QR} = \frac{CA}{RP}$

This value is the constant of proportionality.

Given: ΔSLK ~ ΔNFR

Find: All corresponding angles and corresponding sides

Solution:

From the order of vertices:

• Angles: ∠SLK corresponds to ∠NFR, ∠KSL corresponds to ∠RNF, ∠LKS corresponds to ∠FRN
• Sides: SL corresponds to FN, SK corresponds to NR, KL corresponds to RF

Given: ΔABC ~ ΔPQR, with ∠BAC = 120° and ∠PRQ = 25° Find: ∠ABC

Solution:

Since corresponding angles are equal: ∠BCA = ∠PRQ = 25°

In ΔABC, the sum of interior angles is 180°:

∠ABC + ∠BCA + ∠BAC = 180° ∠ABC + 25° + 120° = 180° ∠ABC = 180° - 145° = 35°

Therefore, ∠ABC = 35°

Given: ΔABC ~ ΔPQR, where AB = 4 cm, AC = 4.8 cm, and PQ = 9 cm Find: PR

Solution:

From the definition of similar triangles:

$\frac{AB}{PQ} = \frac{AC}{PR}$

$\frac{4 \text{ cm}}{9 \text{ cm}} = \frac{4.8 \text{ cm}}{PR}$

$4 \times PR = 9 \times 4.8$

$PR = \frac{43.2}{4} = 10.8 \text{ cm}$

You can prove triangles are similar using any one of these three theorems:

1. AA Similarity Theorem

If two pairs of corresponding angles are equal, the triangles are similar.

2. SSS Similarity Theorem

If three pairs of corresponding sides are proportional, the triangles are similar.

3. SAS Similarity Theorem

If two pairs of corresponding sides are proportional AND the included angles are equal, the triangles are similar.

• Similar triangles have equal angles and proportional sides
• The constant of proportionality is the ratio of any pair of corresponding sides
• Similar triangles are named by matching the order of vertices: ΔABC ~ ΔPQR means A ↔ P, B ↔ Q, C ↔ R
• Similar figures maintain the same shape but differ in size
• The symbol ~ means "similar to"

In Tanzania, similar triangles are used to measure heights of objects that are difficult to measure directly, such as trees, buildings, or poles. For example, if Adolfina wants to find the height of a tall tree at her school, she can plant a vertical stick of known height (like 2 metres), measure its shadow and the tree's shadow at the same time, then use the proportion between the triangles formed to calculate the tree's height. This method is commonly used by surveyors and farmers when determining the height of structures or trees without needing ladders or climbing equipment.