Triangles
About 6 min · Mathematical Reasoning · English
Time worked out from the lesson's 438 words at 150 words a minute, plus a minute for each question.
Triangles form the foundation of geometry. You'll face problems on congruence (proving triangles are identical), properties of special triangles (especially isosceles), and triangle inequalities (which side lengths are possible). This topic connects directly to quadrilaterals, constructions, and coordinate geometry, so mastering it unlocks several downstream chapters.
Key ideas
- Congruent triangles are identical in shape and size. All corresponding sides and angles are equal. Symbol: △ABC ≅ △PQR means AB = PQ, BC = QR, CA = RP and all corresponding angles match.
- Four congruence criteria: SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), and RHS (right angle, hypotenuse, one side). No AAA criterion—similar triangles can have different sizes.
- CPCT principle: Corresponding Parts of Congruent Triangles are equal. After proving congruence, you can state any corresponding element is equal without further proof.
- Isosceles triangle: Two sides equal implies two base angles equal. Conversely, two angles equal implies opposite sides equal. The altitude from the apex bisects the base and is perpendicular to it.
- Equilateral triangle: All sides equal, all angles 60°. Every altitude is also a median, angle bisector, and perpendicular bisector.
Watch out for
- Confusing SAS with SSA → SSA (two sides and a non-included angle) is NOT a congruence criterion. The angle must be between the two sides. Fix: Check that the angle is sandwiched between the given sides.
- Using AAA to prove congruence → Three equal angles mean similar triangles, not necessarily congruent (they could be different sizes). Fix: You need at least one side length to prove congruence.
- Forgetting the strict inequality → Writing a + b ≥ c instead of a + b > c. A degenerate triangle (collinear points) where a + b = c is not a valid triangle. Fix: Always use > not ≥.
Quick reference
- Congruence shorthand: SSS, SAS, ASA, RHS (no AAA or SSA).
- Isosceles rule: Equal sides → equal opposite angles; altitude from apex bisects base at 90°.
- Equilateral triangle: All sides equal, all angles 60°.
- Triangle inequality: Sum of two sides > third side; difference of two sides < third side.
- Exterior angle = sum of remote interior angles (and > either one alone).
Check yourself: 3 questions
1. In triangle ABC, AB = AC and angle A = 40°. What is the measure of angle B?
2. In triangles PQR and XYZ, PQ = XY, QR = YZ, and angle Q = angle Y. By which congruence criterion are the two triangles congruent?
3. In triangle DEF, DE = 5 cm, EF = 12 cm, and DF = 14 cm. Which angle is the largest?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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