Summary: Angles Triangles

Table of Contents
1. What is a Triangle?
2. Classification by Sides
3. Classification by Angles
4. Basic Properties of a Triangle
5. Key Centers and Median Facts
6. Other Important Centers
7. Equilateral Triangle Formulas
8. Congruency
9. Tests for Congruency
10. Mid-Point Theorem
11. Similar Triangles
12. Tests for Similarity
13. Areas and Corresponding Elements in Similar Triangles
14. Basic Proportionality Theorem (Thales)
15. Angle Bisector Theorem
16. Intercept Theorem
17. Pythagoras Theorem
18. Other Theorems
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What is a Triangle?

• Triangle: a polygon with three sides.
• Classified by sides and by angles.

Classification by Sides

• Scalene: all three sides different.
• Isosceles: two equal sides; angles opposite equal sides are equal.
• Equilateral: all sides equal; all angles equal (each 60°).

Classification by Angles

• Obtuse-angled: one angle greater than 90°.
• Acute-angled: all angles less than 90°.
• Right-angled: one angle exactly 90°; side opposite it is the hypotenuse, others are the legs (base and height).

Basic Properties of a Triangle

• Sum of interior angles = 180°.
• Sum of exterior angles = 360°.
• An exterior angle = sum of the two opposite interior angles.
• Triangle inequality: sum of any two sides > third side; difference of any two sides < third side.
• Longest side is opposite largest angle; shortest side is opposite smallest angle.

Key Centers and Median Facts

• Centroid: intersection of medians; a median joins a vertex to midpoint of opposite side.
• Centroid divides each median in ratio 2 : 1 (vertex to centroid : centroid to midpoint).
• Apollonius theorem for median AD: AB² + AC² = 2(AD² + BD²).
• Medians bisect the triangle's area.
• If medians are x, y, z and sides are a, b, c: 4(x² + y² + z²) = 3(a² + b² + c²).

Other Important Centers

• Orthocentre: intersection of altitudes; in a right triangle it is at the right-angle vertex.
• Circumcentre: intersection of perpendicular bisectors; centre of circumcircle and equidistant from vertices. Relation: abc = 4RΔ (R = circumradius, Δ = area).
• Incentre: intersection of internal angle bisectors; centre of incircle. Relation: Δ = r·s (r = inradius, s = semi-perimeter). Also BF = BD = s - b, CE = CD = s - c, AF = AD = s - a. Angle between internal and external bisector = 90°.

Equilateral Triangle Formulas

• All sides equal and all angles equal.
• Altitude, area, inradius and circumradius have fixed standard expressions for side length (formulas shown in the original content).

Congruency

• Congruent shapes have same size and shape; corresponding sides and angles equal (notation: ≅ or ≡).

Tests for Congruency

• SAS: two sides and included angle equal.
• SSS: three sides equal.
• ASA: two angles and included side equal.
• RHS: for right triangles, hypotenuse and one side equal.

Mid-Point Theorem

• A segment joining midpoints of two sides of a triangle is parallel to the third side and equals half its length.

Similar Triangles

• Similar shapes have same shape; corresponding angles equal and corresponding sides in proportion.

Tests for Similarity

• AAA: three angles equal.
• SAS: two side ratios equal and included angle equal.
• SSS: three corresponding side ratios equal.

Areas and Corresponding Elements in Similar Triangles

• Ratio of areas = square of ratio of corresponding sides.
• Area ratio also equals square of ratio of corresponding medians, altitudes, or angle bisectors.
• Ratios of medians, altitudes, and internal bisectors equal ratio of corresponding sides.

Basic Proportionality Theorem (Thales)

• A line drawn parallel to one side of a triangle divides the other two sides proportionally.

Angle Bisector Theorem

• An internal angle bisector divides the opposite side internally in the ratio of the adjacent sides (AD bisects ∠A ⇒ BD/DC = AB/AC).

Intercept Theorem

• Intercepts made by two transversals on three or more parallel lines are proportional.

Pythagoras Theorem

• In a right triangle, square of hypotenuse = sum of squares of other two sides (c² = a² + b²).
• Pythagorean triplets: integer triples (a, b, c) satisfying a² + b² = c².

Other Theorems

• Acute-angled theorem: relations involving altitude in an acute triangle (formula shown in original content).
• Obtuse-angled theorem: relation for an obtuse triangle with altitude produced (formula shown in original content).
• Apollonius' Theorem: combined result giving median length: AB² + AC² = 2(AD² + BD²) (formula shown in original content).