An Obtuse Angle Has No Complement at All

No. A complement must add to $90°$, so an angle of $120°$ would need a partner of $-30°$ — and a negative angle does not exist. Only angles **below $90°$** have complements.

That boundary is exactly the kind of detail this chapter is tested on. This page covers everything in the ICSE Class 7 Mathematics chapter's first part: lines, segments and rays, types of angles, complements and supplements, and the angle pairs formed at a point and on a line.

All three are straight, and they differ in how many ends they have.

- A line extends endlessly in both directions and has no endpoints. It is written $\overleftrightarrow{AB}$, and its length cannot be measured.
- A line segment has two endpoints and a definite, measurable length. It is written $\overline{AB}$.
- A ray has one endpoint and extends endlessly in the other direction. It is written $\overrightarrow{AB}$, where A is the endpoint.

A taut thread cut to 15 cm is a segment; a torch beam is a ray, starting at the bulb and travelling on.

An angle is formed when two rays meet at a common endpoint. The rays are its arms and the common endpoint is its vertex. An angle with vertex B and arms BA and BC is written $\angle ABC$ — or $\angle B$ when no confusion can arise.

The middle letter is always the vertex, and that is what the naming convention exists for. So $\angle ABC$ and $\angle CBA$ are the same angle, but $\angle BAC$ is a different one — its vertex is A.

Angles are named by size:

- Acute — less than $90°$
- Right — exactly $90°$
- Obtuse — more than $90°$ but less than $180°$
- Straight — exactly $180°$
- Reflex — more than $180°$ but less than $360°$
- Complete — exactly $360°$

Two angles are complementary if they add to $90°$, and supplementary if they add to $180°$.

$$\text{complement of }x=90-x,\text{supplement of }x=180-x$$

Worked examples. For $35°$:

$$\text{complement}=90-35=55°,\text{supplement}=180-35=145°$$

For $72°$: complement $18°$, supplement $108°$.

And running it backwards — find the angle whose supplement is four times itself:

$$x+4x=180\Rightarrow 5x=180\Rightarrow x=36°$$

The hands of a clock at 3 o'clock make a right angle, and at 4 o'clock an obtuse one.

The limits are where the marks sit. An obtuse angle has no complement, since $90-x$ would be negative. A reflex angle has neither a complement nor a supplement. And two right angles are supplementary but never complementary, since $90+90=180$.

Adjacent angles share a common vertex and a common arm, and lie on opposite sides of that common arm without overlapping.

A linear pair is a pair of adjacent angles whose non-common arms form a straight line. Their sum is therefore $180°$ — they are supplementary.

Vertically opposite angles are the pairs formed opposite each other when two lines cross. They are always equal.

When two lines intersect, four angles are formed: two pairs of vertically opposite angles, and four linear pairs around the crossing.

So if one angle at a crossing is $70°$, the angle vertically opposite is also $70°$, and each of the two neighbouring angles is $180-70=110°$. The four angles are $70°$, $110°$, $70°$, $110°$, adding to $360°$.

A pair of scissors shows it directly — opening the blades changes both opposite angles together, always keeping them equal.

The reason vertically opposite angles must be equal follows from linear pairs rather than being a separate fact. If $a$ and $b$ form a linear pair then $a+b=180$, and if $b$ and $c$ also form one then $b+c=180$ — so $a$ and $c$ are both $180-b$, and therefore equal.

One caution: adjacent angles are not always supplementary. They form a linear pair only when their outer arms make a straight line.

Two totals do all the work: angles on a straight line add to $180°$, and angles at a point add to $360°$.

On a straight line. Three angles $x$, $2x$ and $60°$ lie on a line:

$$x+2x+60=180\Rightarrow 3x=120\Rightarrow x=40°$$

So the angles are $40°$, $80°$ and $60°$, and $40+80+60=180$. Correct.

At a point. Four angles $x$, $90°$, $130°$ and $x$ meet at a point:

$$2x+220=360\Rightarrow 2x=140\Rightarrow x=70°$$

Using a linear pair. If one angle of a linear pair is $3x$ and the other is $2x$:

$$5x=180\Rightarrow x=36°$$

so the angles are $108°$ and $72°$.

Using vertically opposite angles. Two lines cross and one angle is $(2x+10)°$ while the angle vertically opposite is $(3x-20)°$. Since they are equal:

$$2x+10=3x-20\Rightarrow x=30$$

so each of those angles is $70°$, and the other two are $110°$.

Choosing which total applies is the decision that matters. A straight line gives 180, a full turn around a point gives 360 — and reading a diagram as a point when it is actually a line is what doubles an answer.