Intuition The one core ideaA triangle is just three straight sticks joined tip-to-tip, and everything we say about triangles is a game of comparing two things: how the side lengths relate to each other, and how big the corner angles are. Master those two comparisons and every name — scalene, isosceles, right, obtuse — becomes obvious.
This page assumes you know nothing . Before we can talk about "isosceles" or "the angle sum is 180°", we must earn every word and symbol the parent note throws at you. We build them one at a time, each on top of the last.
A point is a single dot — a location with no size at all. We name points with capital letters: A B C
A line segment is the straight path connecting two points. The segment between A B A B length (how long it is) is written A B
The bar means "the stick itself"; no bar means "how long the stick is". That tiny difference matters later.
Three points are non-collinear if they do not all sit on one straight line. If they did lie on a line, the "triangle" would flatten into a stick with zero area — no triangle at all.
Look at the figure: the three dots are spread out, so joining them makes a real, bulging shape.
Intuition What an angle really isStand at a corner. One stick points one way, the other stick points another way. The angle is the amount of turn between those two directions — how far you'd have to swing one stick to line it up with the other.
An angle is the amount of opening between two segments that share a vertex. We write the angle at vertex B ∠ B ∠
A degree , written with the small circle ∘ 36 0 ∘ 18 0 ∘ 9 0 ∘
Why 360 and not 100? History and convenience — 360 divides evenly by lots of numbers. What matters is the picture : think of a clock hand sweeping around.
A right angle is exactly 9 0 ∘
Everything in "acute / right / obtuse" is measured against this 9 0 ∘
less than 9 0 ∘ sharp equal to 9 0 ∘ more than 9 0 ∘ blunt / spread open
Classification is all about comparing. So we need the three comparison signs:
Definition Comparison signs
a = b a equal to b a < b a less than b a > b a greater than b
We use these constantly: equal sides make isosceles/equilateral, unequal sides make scalene; angles < 9 0 ∘ = 9 0 ∘ > 9 0 ∘
A variable is a letter that stands in for a number we haven't fixed yet. Latin letters a , b , c side lengths . Greek letters stand for angles :
θ α β
Why Greek for angles? Pure convention — it lets you glance at a formula and instantly know "letter = side" vs "Greek = angle" without re-reading.
Worked example Reading a mixed expressionWhen the parent writes θ + θ + β = 18 0 ∘ θ β 18 0 ∘
× ⋅ × ⋅ multiply . When two letters sit next to each other with nothing between them — like 3 θ ab also means multiply: 3 θ 3 θ ab a b
So θ + θ + θ 3 θ 3 θ = 18 0 ∘ ⇒ θ = 3 18 0 ∘ = 6 0 ∘
Definition The fraction bar and2 a fraction 2 a a 2 2 a half of side a — which is exactly what you get when you drop a line from the tip of a triangle straight down to cut the base in two.
The parent uses a 2 c 2 3
a 2 a 2 a a × a The picture: it is literally the area of a square whose side is a why it's called "squared".
n n "which number, times itself, gives n Since 2 × 2 = 4 4 = 2 3 ≈ 1.732
Squaring builds areas; the square root undoes squaring to recover a length. That pair is the whole engine behind the Pythagorean rule c 2 = a 2 + b 2 Pythagorean Theorem ).
π π ≈ 3.14159 18 0 ∘ π radian is just a different unit for the same turn, where half a spin is called π 18 0 ∘ π = 18 0 ∘
Definition Supplementary anglesTwo angles are supplementary if they add up to 18 0 ∘ 18 0 ∘
Two shapes are congruent if they are identical in size and shape — you could slide/flip one onto the other perfectly. When you cut an isosceles triangle down the middle, the two halves are congruent. (The rules for proving this live in Congruence Criteria (SSS, SAS, ASA) .)
Symmetry means a shape looks the same after a flip or turn. An isosceles triangle has one mirror line; an equilateral has three. More in Symmetry in Geometry .
compare signs less equal greater
The parent topic sits at K : it only makes sense once points/angles (left branch), comparison (middle), the angle-sum fact, and squaring/roots (right branch) are all in place.
Cover the right side and answer each before revealing.
What does the bar in A B A B A B A B
What makes three points non-collinear , and why do we need it? They don't all lie on one line; otherwise the triangle flattens to zero area.
How many degrees is a full turn? A half turn? A square corner? 36 0 ∘ 18 0 ∘ 9 0 ∘
Which way does the mouth of > Toward the larger number.
Rewrite θ + θ + θ 3 θ = 18 0 ∘ 3 θ θ = 6 0 ∘
What does a 2 picture ? The area of a square with side a
What question does 3 "Which number times itself gives 3 1.732
What does "supplementary" mean? Two angles that add to 18 0 ∘
What does "congruent" mean? Same size and shape — one fits exactly onto the other.
In an equilateral triangle, which special lines all coincide? Altitude, median, angle bisector, perpendicular bisector.
You now hold every symbol the parent note uses. Head back to Triangles — scalene, isosceles, equilateral; acute, right, obtuse and it should read like plain English. Related tools you'll meet next: Pythagorean Theorem , Area of Triangles , Trigonometric Ratios , Triangle Inequality , and Interior Angles of Polygons .
