Criteria for Similarity of Triangles

Similar Triangles: Two triangles are similar if (i) their corresponding angles are equal, and (ii) their corresponding sides are in the same ratio. The notation triangle ABC ~ triangle DEF means A corresponds to D, B corresponds to E, C corresponds to F.

The Three Similarity Criteria:

1. AA (Angle-Angle) Criterion: If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

If Angle A = Angle D and Angle B = Angle E, then triangle ABC ~ triangle DEF.

Note: Since the sum of angles of a triangle is 180 degrees, if two pairs of angles are equal, the third pair must also be equal. Therefore, the AA criterion is sometimes stated as AAA (Angle-Angle-Angle), but checking two angles is sufficient.

2. SSS (Side-Side-Side) Similarity Criterion: If in two triangles, the corresponding sides are in the same ratio (i.e., all three pairs of sides are proportional), then the two triangles are similar.

If AB/DE = BC/EF = CA/FD, then triangle ABC ~ triangle DEF.

3. SAS (Side-Angle-Side) Similarity Criterion: If one angle of a triangle is equal to one angle of the other triangle, and the sides including these angles are proportional, then the two triangles are similar.

If Angle A = Angle D and AB/DE = AC/DF, then triangle ABC ~ triangle DEF.

Note: The equal angle must be the included angle between the two proportional sides. If the equal angle is not the included angle, the criterion does not apply.

Comparison with Congruence Criteria:

Proof of the AA Similarity Criterion:

Given: Two triangles ABC and DEF such that Angle A = Angle D and Angle B = Angle E.

To Prove: Triangle ABC ~ Triangle DEF, that is, AB/DE = BC/EF = CA/FD.

Construction: On ray DE, mark a point P such that DP = AB. On ray DF, mark a point Q such that DQ = AC. Join PQ.

Step 1: In triangles ABC and DPQ: AB = DP (construction), Angle A = Angle D (given), AC = DQ (construction). By SAS congruence, triangle ABC is congruent to triangle DPQ.

Step 2: From the congruence: Angle B = Angle DPQ. But Angle B = Angle E (given). Therefore Angle DPQ = Angle E.

Step 3: In triangle DEF, since Angle DPQ = Angle DEF, and these are corresponding angles with PQ as a transversal, PQ is parallel to EF.

Step 4: By BPT applied to triangle DEF with PQ || EF: DP/DE = DQ/DF.

Step 5: Since DP = AB and DQ = AC: AB/DE = AC/DF. ... (i)

Step 6: Similarly, by taking construction on other sides, we get AB/DE = BC/EF. ... (ii)

Step 7: From (i) and (ii): AB/DE = BC/EF = CA/FD. Combined with equal angles, triangle ABC ~ triangle DEF. QED.

Proof of SSS Similarity Criterion:

Given: In triangles ABC and DEF, AB/DE = BC/EF = CA/FD.

To Prove: Triangle ABC ~ Triangle DEF.

Construction: On ray DE, mark P such that DP = AB. On ray DF, mark Q such that DQ = AC. Join PQ.

Step 1: AB/DE = AC/DF (given). Since DP = AB and DQ = AC: DP/DE = DQ/DF.

Step 2: By the converse of BPT, PQ || EF in triangle DEF.

Step 3: Since PQ || EF, triangle DPQ ~ triangle DEF (by AA: angle D common, angle DPQ = angle DEF).

Step 4: So DP/DE = PQ/EF. Since DP = AB: AB/DE = PQ/EF.

Step 5: But AB/DE = BC/EF (given). So PQ/EF = BC/EF, giving PQ = BC.

Step 6: In triangles ABC and DPQ: AB = DP, AC = DQ, BC = PQ. By SSS congruence, triangle ABC is congruent to triangle DPQ.

Step 7: Since triangle DPQ ~ triangle DEF, and triangle ABC is congruent to triangle DPQ, therefore triangle ABC ~ triangle DEF. QED.

Proof of SAS Similarity Criterion:

Given: In triangles ABC and DEF, AB/DE = AC/DF and Angle A = Angle D.

To Prove: Triangle ABC ~ Triangle DEF.

Construction: Same as above, mark P on DE with DP = AB and Q on DF with DQ = AC.

Step 1: In triangles ABC and DPQ: AB = DP, Angle A = Angle D, AC = DQ. By SAS congruence, triangle ABC is congruent to triangle DPQ.

Step 2: Since DP/DE = DQ/DF (from given ratio and construction), by converse of BPT, PQ || EF.

Step 3: Therefore triangle DPQ ~ triangle DEF (by AA). Since triangle ABC is congruent to triangle DPQ, we get triangle ABC ~ triangle DEF. QED.

The three similarity criteria lead to different types of problems and geometric configurations:

1. AA Criterion Problems:

These are the most common. You identify two pairs of equal angles using properties like vertically opposite angles, alternate interior angles (with parallel lines), corresponding angles, common angles, or angles in the same segment of a circle.

Typical configurations: parallel lines cutting two sides of a triangle, X-shaped figures with vertically opposite angles, right triangles sharing an acute angle, tangent-radius configurations.

2. SSS Similarity Problems:

You are given all six side lengths of two triangles and must check if the three ratios are equal. Arrange the sides of each triangle in ascending (or descending) order, pair them up, and check ratios.

Typical configurations: two triangles with all measurements given, problems asking to identify similar triangles from a set of given side lengths.

3. SAS Similarity Problems:

You are given two sides and the included angle for each triangle. Check if the two sides are proportional and the included angles are equal.

Typical configurations: two triangles sharing a common angle (like an angle of a parallelogram), isosceles triangles with the vertex angle given, configurations where one angle is clearly identified as the included angle.

Common Geometric Configurations for Similarity:

Method 1: Establishing Similarity Using AA

Step 1: Identify two pairs of equal angles using geometric properties (parallel lines, vertically opposite angles, common angles, right angles). Step 2: State the AA criterion. Step 3: Write the similarity statement with correct vertex correspondence.

Example: In triangle ABC, DE || BC with D on AB and E on AC. Angle ADE = Angle ABC (corresponding angles). Angle A is common. By AA, triangle ADE ~ triangle ABC.

Method 2: Establishing Similarity Using SSS

Step 1: List all six sides. Step 2: Arrange each triangle's sides in order. Step 3: Compute the three ratios of corresponding sides. Step 4: If all ratios are equal, state SSS similarity with correct correspondence.

Example: Triangle ABC has sides 3, 4, 5. Triangle DEF has sides 6, 8, 10. Ratios: 3/6 = 4/8 = 5/10 = 1/2. By SSS, triangle ABC ~ triangle DEF.

Method 3: Establishing Similarity Using SAS

Step 1: Identify one pair of equal angles. Step 2: Check that the sides including this angle are proportional. Step 3: State SAS similarity.

Example: In triangles ABC and DEF, Angle B = Angle E = 70 degrees. AB/DE = 6/9 = 2/3, BC/EF = 8/12 = 2/3. Since AB/DE = BC/EF and the included angle B = angle E, by SAS, triangle ABC ~ triangle DEF.

Method 4: Finding Unknown Sides Using Similarity

Once similarity is established, use the equal ratios of corresponding sides to find unknowns. Set up the proportion and solve.

Method 5: Finding Unknown Angles Using Similarity

Once similarity is established, equate corresponding angles. Use the angle sum property (180 degrees) if needed.

Important Tips:

• Always write the similarity statement with the correct order of vertices. A corresponds to D, B to E, C to F in triangle ABC ~ triangle DEF.
• For SSS, arrange sides in ascending order in both triangles before computing ratios to ensure correct pairing.
• For SAS, verify that the equal angle is the INCLUDED angle between the two proportional sides. A non-included equal angle does not guarantee similarity.
• The AA criterion is the most commonly used because angles are often easier to identify than side lengths in geometric figures.

The criteria for similarity of triangles have extensive applications in mathematics and the real world.

Indirect Height Measurement: One of the oldest practical applications of similar triangles is measuring the height of tall objects like buildings, towers, trees, and mountains. By observing the shadow cast by a stick of known height and comparing it to the shadow of the tall object (both measured at the same time so that the sun's angle is identical), similar triangles are formed. The AA criterion applies because the sun's rays are parallel, creating equal angles of elevation. The unknown height is then found by proportion.

River Width Measurement: Surveyors standing on one bank of a river can measure its width without crossing it. By setting up a triangle with a known baseline on their bank and measuring angles to a point on the opposite bank, they create a triangle similar to a smaller reference triangle. The width is then calculated using proportionality.

Scale Models and Maps: Every architectural model and every map is a practical application of similar triangles. The scale factor (like 1:100 for a model or 1:50000 for a map) means that every triangle on the model or map is similar to the corresponding triangle in reality. The SSS similarity criterion guarantees that if all distances are scaled by the same factor, the shapes are preserved.

Photography and Lens Optics: When a camera lens forms an image, the object and the image form similar triangles. The magnification ratio equals the ratio of image distance to object distance, which is derived from similar triangle relationships. This is fundamental in optics, telescope design, and microscopy.

Engineering and Machine Design: Gear ratios, pulley systems, and lever mechanisms often involve proportional relationships derivable from similar triangles. The SAS criterion is particularly useful in structural engineering where angles and adjacent member lengths are the primary design parameters.

Computer Vision and Image Recognition: Algorithms for detecting objects in images use the properties of similar triangles. When an object moves closer to or further from a camera, its image on the sensor changes size proportionally. By recognising this similarity transformation, software can estimate distance and object size, enabling applications from autonomous vehicles to facial recognition.

Astronomy: The distance to nearby stars is measured using parallax, which involves similar triangles. The baseline is the diameter of Earth's orbit, and the tiny angle shift of the star creates a very elongated similar triangle whose proportions reveal the stellar distance.