Triangles — scalene, isosceles, equilateral; acute, right, obtuse

What is a Triangle?

Why 180°? Imagine walking along the perimeter of a triangle, turning at each vertex. You make three turns that bring you back to your starting direction. Those three exterior angles sum to 360° (one full rotation). Since each interior angle and its corresponding exterior angle are supplementary (sum to 180°), we have: $3 \times 180° - 360° = 540° - 360° = 180°$ for the sum of interior angles.

Classification by Side Lengths

Derivation of equilateral angles: Let each angle be $\theta$. Since the sum of angles is 180°: $\theta + \theta + \theta = 180°$ $3\theta = 180°$ $\theta = 60°$

Example 1: Scalene Triangle with sides 3 cm, 4 cm, 5 cm.

• All sides different → scalene.
• Why this matters: No symmetry, no shortcut formulas. Must use Heron's formula or $\frac{1}{2} \times \text{base} \times \text{height}$ for area.

Example 2: Isosceles Triangle with sides 5 cm, 5 cm, 6 cm.

• Two sides equal (5 cm each) → isosceles.
• The angles opposite the 5 cm sides are equal. Let's call them $\alpha$.
• If the base angles are $\alpha$ each, and the apex angle is $\beta$: $\alpha + \alpha + \beta = 180°$
• This symmetry lets us drop a perpendicular from the apex to bisect the base, creating two congruent right triangles.

Example 3: Equilateral Triangle with sides 7 cm, 7 cm, 7 cm.

• All sides equal → equilateral.
• All angles = 60°.
• Highly symmetric: any median, altitude, angle bisector, and perpendicular bisector coincide.
• Area formula derived from scratch:

For side $a$, drop an altitude $h$ to split the base. The altitude creates two 30-60-90 right triangles. Using the Pythagorean theorem on half the triangle: $h^2 + \left(\frac{a}{2}\right)^2 = a^2$ $h^2 = a^2 - \frac{a^2}{4} = \frac{3a^2}{4}$ $h = \frac{a\sqrt{3}}{2}$ Area: $A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times \frac{a\sqrt{3}}{2} = \frac{a^2\sqrt{3}}{4}$

Classification by Angles

Why can't a triangle have two obtuse angles? Suppose two angles are obtuse, each > 90°. Their sum alone exceeds 180°, leaving no room for a third positive angle. Contradiction.

Why can't a triangle have two right angles? Two 90° angles sum to 180°, leaving 0° for the third angle—not a valid triangle.

Example 4: Acute Triangle Triangle with angles 60°, 70°, 50°.

• All angles < 90° → acute.
• Check: $60° + 70° + 50° = 180°$. ✓
• The longest side is opposite the largest angle (70°), but all sides are "reasonably" sized relative to each other.

Example 5: Right Triangle Triangle with angles 90°, 60°, 30°.

• One angle = 90° → right triangle.
• Check: $90° + 60° + 30° = 180°$. ✓
• Special case: This is a 30-60-90 triangle. If the shortest side (opposite 30°) has length $x$:
  • Side opposite 60°: $x\sqrt{3}$
  • Hypotenuse (opposite 90°): $2x$

Derivation: Start with an equilateral triangle with side $2x$. Cut it in half vertically. Each half is a 30-60-90 triangle with hypotenuse $2x$, shortest side $x$ (half the base), and height $x\sqrt{3}$ (from the equilateral area formula).

Example 6: Obtuse Triangle Triangle with angles 120°, 40°, 20°.

• One angle > 90° (120°) → obtuse.
• Check: $120° + 40° + 20° = 180°$. ✓
• The side opposite 120° is the longest. In fact, in any obtuse triangle, the square of the longest side is greater than the sum of squares of the other two sides (opposite of Pythagorean theorem).

Derivation from scratch: Draw a right triangle with legs $a$, $b$ and hypotenuse $c$. Construct a square of side $(a+b)$. Inside, arrange four copies of the triangle at the corners, leaving a tilted square of side $c$ in the center.

Outer square area: $(a+b)^2 = a^2 + 2ab + b^2$

This equals: 4 triangles + inner square = $4 \times \frac{1}{2}ab + c^2 = 2ab + c^2$

Equating: $a^2 + 2ab + b^2 = 2ab + c^2$ $a^2 + b^2 = c^2$

Combined Classification

A triangle can be classified by both side and angle properties simultaneously:

• Equilateral acute: All sides equal, all angles 60° (always acute).
• Isosceles right: Two sides equal, one 90° angle (the 45-45-90 triangle).
• Scalene obtuse: All sides different, one angle > 90°.

Sides: $a$, $a$, $a\sqrt{2}$ where the two legs are equal. Angles: 45°, 45°, 90°.

Derivation of hypotenuse: Using Pythagorean theorem on legs of length $a$: $c^2 = a^2 + a^2 = 2a^2$ $c = a\sqrt{2}$

Check angles: If legs are equal, base angles are equal. Let each be $\theta$. $\theta + \theta + 90° = 180°$ $2\theta = 90°$ $\theta = 45°$

Common Mistakes

Why it's wrong: "Isosceles" means "two legs equal" (from Greek isos = equal, skelos = leg). Requiring all three equal describes equilateral, which is a special case of isosceles (at least two equal), not the general meaning.

The fix: Isosceles = at least two sides equal. Equilateral triangles are a subset of isosceles, but we use the name "equilateral" specifically when all three are equal.

Why it's wrong: The moment you connect the third side to close the triangle, the third angle is determined by $180° - (\text{angle}_1 + \text{angle}_2)$. If two angles already sum to more than 180°, the third "angle" would be negative—impossible.

The fix: Angles must sum to exactly 180°. At most one angle can be ≥ 90°.

Why it's wrong: By the Pythagorean theorem, $c^2 = a^2 + b^2$, so $c > a$ and $c > b$. The hypotenuse (opposite the right angle) is always the longest side.

The fix: Longest side = hypotenuse. It's opposite the largest angle (90°). Always.

Active Recall Practice

By side lengths: If all three sticks are the same length, you get an equilateral triangle—super symmetric, like a perfect tripod. If two sticks are the same and one is different, that's isosceles—it has a line of symmetry down the middle. If all three sticks are different lengths, it's scalene—no symmetry at all, each side does its own thing.

By angles: Now look at the corners where the sticks meet. If all three corners are "sharp" (each less than a right angle from a square corner), it's acute—like a narrow slice of pizza. If one corner is exactly square (90°), it's a right triangle—like half a rectangle. If one corner is "wide" (more than 90°), it's obtuse—like a door slightly open past square.

You can combine these! A triangle might be isosceles AND right (two equal sides meeting at a square corner), or scalene AND acute (all different sides, all sharp corners). The classifications tell you what shortcuts you can use and what the triangle will look like before you even draw it.

Angles — "ARO":

• Acute = All angles small (< 90°)
• Right = Right angle present (= 90°)
• Obtuse = One big angle (> 90°)

Connections

• Interior Angles of Polygons — Triangle angle sum is the base case ($n=3$)
• Pythagorean Theorem — Defines right triangles, fails for acute/obtuse
• Congruence Criteria (SSS, SAS, ASA) — Classification determines which criteria apply easily
• Triangle Inequality — Constrains possible side lengths for each type
• Area of Triangles — Different formulas optimize for equilateral, right, or general triangles
• Trigonometric Ratios — Defined using right triangles, extended via any triangle
• Symmetry in Geometry — Isosceles has 1 line, equilateral has 3 lines of symmetry

#flashcards/maths

What are the three classifications of triangles by side length? :: Scalene (all sides different), Isosceles (at least two sides equal), Equilateral (all sides equal)

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