Three Angles in the Ratio 2 to 3 to 4 Are Fixed Already

Yes — because the three must total $180°$, which supplies the missing information. Angles in the ratio $2:3:4$ give $9$ parts in all, so one part is $\frac{180}{9}=20°$ and the angles are $40°$, $60°$ and $80°$.

The ratio fixes the shape; the angle sum fixes the size. This page covers everything in the ICSE Class 7 Mathematics chapter's first part: classifying triangles, the angle sum property, the exterior angle property, and isosceles and equilateral triangles.

Every triangle carries one label from each of two lists.

By sides:

- Scalene — all three sides different, so all three angles differ.
- Isosceles — exactly two sides equal, so the two angles opposite them are equal.
- Equilateral — all three sides equal, so all three angles are $60°$.

By angles:

- Acute-angled — all three angles below $90°$.
- Right-angled — one angle exactly $90°$.
- Obtuse-angled — one angle above $90°$.

So a triangle can be isosceles and right-angled at once — the set-square with two $45°$ angles is exactly that.

Two combinations are impossible, and knowing why earns marks. An equilateral triangle is always acute-angled, since its angles are each $60°$, so "equilateral and right-angled" describes nothing. And a triangle can have at most one angle of $90°$ or more, because two would already use up the whole $180°$ before the third angle existed.

A scalene triangle may still be right-angled — the 3, 4, 5 triangle has three different sides and a right angle.

The three angles of any triangle add to $180°$:

$$\angle A+\angle B+\angle C=180°$$

Two angles known. If two angles are $55°$ and $65°$:

$$\angle C=180-(55+65)=60°$$

Angles in a ratio. Three angles in the ratio $2:3:4$. Let them be $2x$, $3x$ and $4x$:

$$2x+3x+4x=180\Rightarrow 9x=180\Rightarrow x=20$$

So the angles are $40°$, $60°$ and $80°$, and $40+60+80=180$. Correct — and since all three are under $90°$, the triangle is acute-angled.

Another ratio: $1:2:3$ gives $6x=180$, so $x=30$ and the angles are $30°$, $60°$ and $90°$ — a right-angled triangle.

Angles described in words. *One angle is twice the second, and the third is $30°$ more than the second.* Let the second be $x$:

$$2x+x+(x+30)=180\Rightarrow 4x=150\Rightarrow x=37.5°$$

so the angles are $75°$, $37.5°$ and $67.5°$.

The ratio method works because a ratio has no units — the $2:3:4$ says nothing about size on its own. It is the $180°$ that converts "9 parts" into "$20°$ per part", which is why you must always add the parts first rather than guessing a scale.

Extend one side of a triangle and the angle formed outside — the exterior angle — equals the sum of the two interior opposite angles.

$$\text{exterior angle}=\text{sum of the two interior opposite angles}$$

"Interior opposite" means the two angles not adjacent to the exterior angle.

Worked example. The two interior opposite angles are $45°$ and $60°$, so the exterior angle is

$$45+60=105°$$

Verify with the angle sum: the third interior angle is $180-105=75°$, and $45+60+75=180$. Also $75+105=180$, a linear pair — exactly as expected.

Worked example in reverse. An exterior angle is $120°$ and one interior opposite angle is $70°$, so the other is

$$120-70=50°$$

With algebra. An exterior angle is $(4x)°$ and the interior opposite angles are $(2x+10)°$ and $(x+20)°$:

$$4x=2x+10+x+20\Rightarrow x=30$$

so the exterior angle is $120°$.

The property is not an extra fact but a consequence of the angle sum and the straight line, which is why either can check the other. It also means an exterior angle is always greater than either interior opposite angle on its own, since it equals their sum — a quick sanity test on any answer.

The common error is adding the exterior angle to the adjacent interior angle instead of using the two opposite ones.

In an isosceles triangle the angles opposite the equal sides are equal — these are the base angles. In an equilateral triangle all three angles are $60°$.

Vertex angle given. An isosceles triangle has a vertex angle of $40°$. The two base angles share the rest:

$$\frac{180-40}{2}=\frac{140}{2}=70°\text{ each}$$

Base angle given. A base angle is $50°$, so both base angles are $50°$ and the vertex angle is

$$180-(50+50)=80°$$

Equilateral. Each angle is $\frac{180}{3}=60°$.

Isosceles and right-angled. The right angle must be the vertex angle, so the base angles are

$$\frac{180-90}{2}=45°\text{ each}$$

The converse is true as well, and questions use it: if two angles of a triangle are equal, the sides opposite them are equal, so the triangle is isosceles.

The wording is where care is needed. "A base angle is $50°$" means both base angles are $50°$, so you subtract $100$ from $180$. But "the vertex angle is $50°$" means you subtract $50$ and then halve. Reading which angle was given decides whether you halve or not, and mixing the two is the standard mistake in this section.