Equal sides face equal angles. Most of this lesson follows from that one sentence.
Find the angle
An isosceles triangle has a vertex angle of 40°. What is each base angle?
The three angles add to 180°, so the two base angles share 180 − 40 = 140° between them. Since they are equal, each is 140 ÷ 2 = 70°. C is the pair's total, not one angle — the halving step is the one most often skipped. Check: 70 + 70 + 40 = 180 ✓
1
The isosceles triangle theorem, both ways.
Equal sides → equal opposite angles, and the converse holds too.
Theoremtwo equal sides→ the angles opposite them are equal
Conversetwo equal angles→ the sides opposite them are equal
Equilateralthree equal sides→ three 60° angles
This is one of the few places where the converse is also true, which makes it doubly useful: equal marks on the sides tell you about the angles, and equal angle marks tell you about the sides.
2
Finding the angles of an isosceles triangle.
The three angles of every triangle add to 180°. A triangle with angles 90° and 25° has a third angle of 180 − 90 − 25 = 65°.
vertex 36°
find the base angles
180 − 36 = 144
shared between two
each = 72°
halve it
base angle 55°
find the vertex
both base angles are 55°
= 110 together
vertex = 180 − 110 = 70°
no halving this time
Which angle were you given?
Given the vertex, you subtract then halve. Given a base
angle, you double then subtract. Doing the wrong pair of steps is the
usual error.
3
Two bisector theorems.
If a point is on…
Then it is…
the perpendicular bisector of a segment
the same distance from both endpoints
the bisector of an angle
the same distance from both sides of the angle
Both converses hold as well: a point equidistant from two endpoints must lie on the
perpendicular bisector.
This is why the perpendicular bisector is how you find a spot equally far from two towns — every point on it qualifies, and nothing off it does.
4
A triangle has four centres.
Centre
Where three… meet
It is
Centroid
medians (vertex to opposite midpoint)
the balance point
Circumcentre
perpendicular bisectors of the sides
equidistant from the vertices
Incentre
angle bisectors
equidistant from the sides
Orthocentre
altitudes
—
The centroid sits two thirds of the way along each median from the vertex —
the one numerical fact worth memorising here.
The circumcentre can sit outside the triangle
For an obtuse triangle it lies beyond the longest side. Only the incentre
and centroid are always inside.
5
Using the theorems with algebra.
isosceles: legs (3x − 4) and (x + 8)
find each leg
3x − 4 = x + 8
the legs are equal
2x = 12, x = 6
solve
each leg = 14
3(6) − 4 = 14 ✓
Remember the flow
Equal sides face equal angles · Vertex given? subtract then halve · Base given? double then subtract · Check the three add to 180
The check costs nothing and catches the halving error every time.
6
The exterior angle theorem.
An exterior angle equals the sum of the two interior angles NOT next to it (the remote interior angles).
An exterior angle of a triangle is 125°, and one remote interior angle is 48°. Find the other.
48 + x = 125
remote interiors add to the exterior
x = 77°
subtract
third angle = 180 − 125 = 55°
the one next to the exterior angle
48 + 77 + 55 = 180 ✓
the check
Putting the exterior angle into the 180
Writing 180 − 125 − 48 = 7° treats the exterior angle as if it were inside the triangle. It is not: it sits on a straight line with the 55° angle, which is why the two remote angles must add to 125°.
7
Variations: an exterior angle in x, a centroid length, an isosceles triangle in x.
The theorems again, with the unknown somewhere new.
An exterior angle is (3x + 10)°; its remote interior angles are (x + 20)° and (x + 30)°.
3x + 10 = 2x + 50
exterior = sum of the remote two
x = 40
solve
exterior 130°
check: 60 + 70 = 130 ✓
The equation has x on both sides; the 180° is not needed at all.
G is the centroid of △ABC and median AM = 18. Find AG and GM.
AG = (23) × 18 = 12
two thirds from the vertex
GM = 18 − 12 = 6
the remaining third
AG : GM = 12 : 6 = 2 : 1. The longer piece is always the one touching the vertex.
An isosceles triangle has base angles of (2x)° each and a vertex angle of (x + 15)°.
2x + 2x + x + 15 = 180
TWO base angles
5x = 165, so x = 33
solve
66°, 66° and 48°
check: 66 + 66 + 48 = 180 ✓
Write the base angle twice in the sum; counting it once is the halving error in reverse.
On the exam
Expect the base or vertex angle of an isosceles triangle, a leg written in x, an exterior angle with one remote interior angle, and the names and facts of the triangle centres (centroid 2 : 1). The fastest route: equal sides face equal angles, exterior = sum of the two remote interiors, and write each base angle twice in the 180° sum. The 10-second check is always the same — the three angles you found must add to exactly 180°.
Now you try
Practice problem
Find the vertex angle
An isosceles triangle has base angles of 48° each. What is the vertex angle?
Both base angles are 48°, totalling 96°, so the vertex is 180 − 96 = 84°. Check: 48 + 48 + 84 = 180 ✓ B halves after subtracting, which is the routine for when the vertex is given — here it is the base angle, so there is nothing to halve.
Hint: you were given a base angle, so there are two of them. Double
first, then subtract from 180.
