SAS Similarity Criterion

SAS Similarity Criterion:

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

Formal statement: If in triangles ABC and DEF:

• Angle A = Angle D
• AB/DE = AC/DF (sides including the equal angles are proportional)

then triangle ABC is similar to triangle DEF.

Key points:

• The equal angle must be the angle included between the two proportional sides.
• If the angle is NOT between the proportional sides, SAS similarity does NOT apply.
• SAS similarity is different from SAS congruence — congruence requires sides to be equal, similarity requires proportionality.

Comparison of the three similarity criteria:

Comparison of all three similarity criteria:

Important distinction: In SAS congruence, the sides must be equal. In SAS similarity, the sides must be proportional. This is a common source of confusion that must be avoided in examinations.

Proof of SAS Similarity Criterion:

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

To prove: Triangle ABC is similar to triangle DEF.

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

Proof:

1. In triangles ABC and DPQ:
   • AB = DP (by construction)
   • AC = DQ (by construction)
   • Angle A = Angle D (given)
2. By SAS congruence: triangle ABC is congruent to triangle DPQ.
3. Therefore: angle B = angle DPQ and angle C = angle DQP.
4. Now, from the given: AB/DE = AC/DF, and AB = DP, AC = DQ.
5. So: DP/DE = DQ/DF.
6. By the converse of BPT: PQ is parallel to EF.
7. Since PQ is parallel to EF: angle DPQ = angle E and angle DQP = angle F (corresponding angles).
8. From step 3: angle B = angle DPQ = angle E, and angle C = angle DQP = angle F.
9. Therefore: angle A = angle D, angle B = angle E, angle C = angle F.
10. All corresponding angles are equal, so triangle ABC is similar to triangle DEF.

QED.

Types of SAS similarity problems:

Type 1: Verifying SAS similarity

• Given one pair of equal angles and two pairs of sides, check whether the angle is included and the sides are proportional.

Type 2: Finding unknown sides

• Given SAS similarity, find an unknown side using proportionality.

Type 3: Proving triangles similar

• In a geometric figure, identify the equal angle and show that the including sides are proportional.

Type 4: Application with other results

• Combining SAS similarity with BPT, Pythagoras theorem, or angle properties.

Type 5: Area and perimeter ratios

• Using SAS similarity to establish the scale factor, then computing area/perimeter ratios.

Step-by-step method:

1. Identify the equal angle in both triangles.
2. Identify the sides that include (form) this angle in each triangle.
3. Compute the ratios of these including sides.
4. If the ratios are equal, SAS similarity applies.
5. Write the similarity statement with correct vertex correspondence.

Identifying the included angle:

• If the equal angle is A in triangle ABC and D in triangle DEF, then the including sides are AB, AC in the first and DE, DF in the second.
• The ratio to check is AB/DE and AC/DF (NOT BC/EF).

Common mistakes:

• Using a non-included angle — the angle MUST be between the two proportional sides.
• Confusing SAS similarity with SAS congruence.
• Not checking that the angle is actually equal (just assumed).
• Wrong correspondence of vertices.

Method 4: Finding Ratios of Areas, Perimeters, and Altitudes

Once two triangles are established as similar with scale factor k:

• Ratio of corresponding sides = k
• Ratio of corresponding altitudes = k
• Ratio of corresponding medians = k
• Ratio of corresponding angle bisectors = k
• Ratio of perimeters = k
• Ratio of areas = k^2
• Ratio of circumradii = k

Method 5: Identifying the Included Angle in Complex Figures

In complex geometric figures, identify the included angle by tracing the two sides from a common vertex. The angle at that vertex, formed between the two proportional sides, is the included angle. Common situations:

• In overlapping triangles: the common angle at the shared vertex is the included angle.
• In intersecting lines: vertically opposite angles serve as equal included angles.
• In cyclic quadrilaterals: angles subtended by the same arc may provide the equal included angles.

Applications of SAS Similarity:

• Height and distance problems — finding heights of buildings or trees using shadow measurements (angle of elevation is common).
• Mirror reflections — angle of incidence equals angle of reflection, creating similar triangles.
• Navigation — determining distances using triangulation with a known angle.
• Architecture — scaling designs while preserving angles.
• Optics — lens and mirror problems where light rays form similar triangles.
• Surveying — measuring inaccessible distances with known angles and proportional sides.

Detailed applications:

• Shadow problems: At a given time, the sun's rays make equal angles with the ground. If a person of height h1 casts a shadow s1, and a building of height h2 casts a shadow s2, the triangles formed are similar by SAS (equal angle of elevation, proportional sides). This gives h1/h2 = s1/s2.
• Photography: When a camera captures an image, the triangle formed by the lens, the object, and the image sensor is similar to the triangle formed by the lens, the sensor, and the image — by SAS similarity.
• Model building: Architects build scale models maintaining the same angles but proportional lengths. SAS similarity ensures that any cross-section through the model corresponds proportionally to the actual structure.