Triangles

Triangles

Triangles are three-sided polygons that form the foundation of geometry. Here's what you need to know:

Basic Properties

Angles

• The sum of angles in a triangle is 180°
• An exterior angle equals the sum of the two non-adjacent interior angles

Sides

• Triangle Inequality: The sum of the lengths of any two sides must be greater than the length of the third side
• The longest side is opposite the largest angle

Types of Triangles

By Angles

• Acute triangle: All angles are less than 90°
• Right triangle: One angle is exactly 90°
• Obtuse triangle: One angle is greater than 90°

By Sides

• Equilateral triangle: All sides are equal (all angles are 60°)
• Isosceles triangle: Two sides are equal (angles opposite the equal sides are equal)
• Scalene triangle: No sides are equal

Right Triangles

Pythagorean Theorem

In a right triangle with legs a and b and hypotenuse c:

$$a^2 + b^2 = c^2$$

Special Right Triangles

45°-45°-90° Triangle:

• Isosceles right triangle
• If legs = a, then hypotenuse = $$a\sqrt{2}$$

30°-60°-90° Triangle:

• If shortest leg = a, then:
• Hypotenuse = 2a
• Longer leg = $$a\sqrt{3}$$

Trigonometric Ratios

In a right triangle with angle θ:

• $$\sin(\theta) = \frac{opposite}{hypotenuse}$$
• $$\cos(\theta) = \frac{adjacent}{hypotenuse}$$
• $$\tan(\theta) = \frac{opposite}{adjacent} = \frac{\sin(\theta)}{\cos(\theta)}$$

Triangle Congruence

Two triangles are congruent if they have exactly the same shape and size.

Congruence Criteria

• SSS (Side-Side-Side): All three pairs of corresponding sides are equal
• SAS (Side-Angle-Side): Two pairs of sides and the included angle are equal
• ASA (Angle-Side-Angle): Two pairs of angles and the included side are equal
• AAS (Angle-Angle-Side): Two pairs of angles and a non-included side are equal
• HL (Hypotenuse-Leg): For right triangles, the hypotenuse and one leg are equal

Triangle Similarity

Two triangles are similar if they have the same shape but not necessarily the same size.

Similarity Criteria

• AAA (Angle-Angle-Angle): All three pairs of corresponding angles are equal
• SAS (Side-Angle-Side): Two pairs of sides are proportional and the included angles are equal
• SSS (Side-Side-Side): All three pairs of corresponding sides are proportional

Properties of Similar Triangles

• Corresponding angles are equal
• Corresponding sides are proportional
• The ratio of areas equals the square of the ratio of corresponding sides

Area and Perimeter

Perimeter

Perimeter = sum of all sides

P = a + b + c

Area

Using base and height: $$A = \frac{1}{2} \times base \times height$$

Using sides (Heron's formula):

If s = (a + b + c)/2 (semi-perimeter), then:

$$A = \sqrt{s(s-a)(s-b)(s-c)}$$

Using trigonometry:

$$A = \frac{1}{2} \times ab \times \sin(C)$$

where C is the angle between sides a and b

Centers of a Triangle

Centroid

• Intersection of the three medians (lines from vertices to midpoints of opposite sides)
• Divides each median in a 2:1 ratio
• Center of mass of the triangle

Circumcenter

• Intersection of the three perpendicular bisectors of the sides
• Center of the circumscribed circle (circle passing through all vertices)

Incenter

• Intersection of the three angle bisectors
• Center of the inscribed circle (circle tangent to all sides)

Orthocenter

• Intersection of the three altitudes (perpendicular lines from vertices to opposite sides)

Common Pitfalls and Tips

• Remember that congruent triangles have the same size and shape, while similar triangles have the same shape but not necessarily the same size
• When using the Pythagorean theorem, make sure you're working with a right triangle
• Be careful with the triangle inequality—not all combinations of three lengths form a triangle
• In similar triangles, the ratio of areas is the square of the ratio of corresponding sides
• When finding the area using Heron's formula, calculate the semi-perimeter first