SAT-73: Triangle Inequality and Similarity (AA, SAS)

SAT-73: Triangle Inequality and Similarity (AA, SAS)

Description: This lesson covers two ideas: the triangle inequality (which side lengths can actually form a triangle) and similarity (when two triangles have the same shape but different size). Both are common SAT geometry topics.

The triangle inequality

For any triangle, the sum of any two sides must be greater than the third side.

The fastest test: the third side must be between the difference and the sum of the other two. (Oʻzbekcha: uchburchak hosil boʻlishi uchun ixtiyoriy ikki tomon yigʻindisi uchinchisidan katta boʻlishi kerak.)

Worked Example 1 — can these be sides?

Can 3, 4, and 8 form a triangle?

• Check 3 + 4 = 7, which is not greater than 8. The inequality fails → no triangle.

Worked Example 2 — range for the third side

Two sides are 6 and 10. What lengths can the third side be?

• It must be greater than 10 − 6 = 4 and less than 10 + 6 = 16.
• So the third side is between 4 and 16 (4 < side < 16).

(Oʻzbekcha: uchinchi tomon ikki tomon ayirmasi bilan yigʻindisi orasida boʻladi.)

Similar triangles — same shape, scaled size

Two triangles are similar if their angles match and their sides are in the same proportion. You can prove similarity without checking everything:

• AA (Angle-Angle): if two angles of one triangle equal two angles of another, they are similar (the third angle must match too).
• SAS (Side-Angle-Side): if two pairs of sides are in proportion and the angle between them is equal, they are similar.

(Oʻzbekcha: oʻxshash uchburchaklarda burchaklar teng, tomonlar esa bir xil nisbatda boʻladi.)

Worked Example 3 — solve with proportions

Two triangles are similar. The small one has sides 3 and 5; the large one's side matching the 3 is 9. Find the side matching the 5.

• Scale factor = 9 ÷ 3 = 3 (each large side is 3× the small one).
• Matching side = 5 × 3 = 15. (Or solve 3/9 = 5/x → x = 15.)

Tip: in similar triangles, set up matching sides as a proportion and cross-multiply (review SAT-49). (Oʻzbekcha: mos tomonlarni proporsiya qilib, krestga koʻpaytiramiz.)

Worked Example 4 — a triangle inside a triangle

A common SAT figure has a small triangle sitting in the corner of a big one, sharing the same top angle, with their bases parallel. The small triangle has height 2 and base 3; the big one has height 6. Find the big base.

• Parallel bases make the two triangles similar (AA: shared angle + equal corresponding angles).
• Scale factor = 6 ÷ 2 = 3, so big base = 3 × 3 = 9.

(Oʻzbekcha: parallel asoslar uchburchaklarni oʻxshash qiladi, soʻng nisbatdan foydalanamiz.)

Why AA is usually enough

Of the similarity tests, AA is the one you reach for most, because angles are easy to spot in SAT figures — parallel lines, shared angles, and right angles hand them to you. As soon as two angles match, the third must too (they all add to 180°), so the triangles are guaranteed similar without checking any sides. Train your eye to hunt for two equal angles first. (Oʻzbekcha: AA — eng koʻp ishlatiladigan belgi, chunki ikki teng burchak topilsa, uchburchaklar oʻxshash boʻladi.)

Common mistake — matching the wrong sides

In a proportion, each side must be paired with the side in the same position (opposite the same angle). Pairing a long side with a short side flips the ratio and gives a wrong answer. Label the corresponding vertices first, then build the proportion. (Oʻzbekcha: proporsiyada har bir tomonni oʻziga mos (bir xil burchak qarshisidagi) tomon bilan juftlang.)

Practice 1

Two sides of a triangle are 5 and 9. Between what values must the third side lie?

Practice 2

Triangle ABC ~ triangle DEF with AB = 4 matching DE = 12. If BC = 7, find EF.

Key words — Kalit soʻzlar

• Triangle inequality — uchburchak tengsizligi
• Similar triangles — oʻxshash uchburchaklar
• Proportion — proporsiya
• Scale factor — oʻlchov koeffitsiyenti
• AA (Angle-Angle) — burchak-burchak
• SAS (Side-Angle-Side) — tomon-burchak-tomon
• Corresponding sides — mos tomonlar
• Sum / Difference — yigʻindi / ayirma
• Shape — shakl

Summary

• Triangle inequality: each side must be between the difference and sum of the other two.
• Similar triangles have equal angles and proportional sides.
• Prove similarity with AA or SAS, then use proportions to find missing sides.