International Baccalaureate · PYP 4

Turns and Angles

⌚ ~58 min ▢ 5 blocks ✓ 23 exercises ➣ Prerequisites : Division, Fractions, Turns and Angles

I Degrees

A full turn, or complete circle, can be divided into 360 equal parts. The angle corresponding to one of these parts measures one degree ($1^\circ$).

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Definition — Degree Unit

A degree, written with the symbol $^\circ$, is a unit of angle measurement. A full turn measures $360^\circ$.

Definition — Measure of an Angle in Degrees

The measure of an angle in degrees counts how many of the 360 equal parts of a full turn the angle covers: a quarter of a turn is a quarter of $360^\circ$, that is $90^\circ$.

Proposition — The Right Angle Measures 90°

A right angle — the corner of the set square, the quarter turn — measures $90^\circ$.
A half turn is two right angles: $2 \times 90^\circ = 180^\circ$.
A full turn is four right angles: $4 \times 90^\circ = 360^\circ$.

Example

Calculate the measure of an angle that represents one-third of a full turn.

Answer

$$\begin{aligned} \text{Angle} &= \frac{1}{3} \text{ of } 360^\circ \\ &= 360^\circ \div 3 \\ &= 120^\circ \end{aligned}$$

Skills to practice

Dividing the Full Turn
Explaining Degrees with Right Angles

II Classification of Angles

An acute angle is more closed than a right angle, an obtuse angle more open. Since a right angle measures $90^\circ$, the measure is enough to classify them: the table below gives four types of angles, acute, right, obtuse and straight.

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Definition — Classification of Angles

Name	Fraction of a Full Turn	Angle Measure	Figure
Acute Angle	Less than $\dfrac{1}{4}$ of a full turn	More than $0^{\circ}$, less than $90^{\circ}$
Right Angle	$\dfrac{1}{4}$ of a full turn	Exactly $90^{\circ}$
Obtuse Angle	Between $\dfrac{1}{4}$ and $\dfrac{1}{2}$ of a full turn	More than $90^{\circ}$, less than $180^{\circ}$
Straight Angle	$\dfrac{1}{2}$ of a full turn	Exactly $180^{\circ}$

Method — Checking an Angle with a Set Square

First estimate by eye: does the angle look more closed or more open than a right angle? Then check with the set square: place its corner on the vertex, one edge along one arm of the angle.

The other arm lines up with the set square's edge: the angle is right.
The other arm is inside the corner: the angle is acute.
The other arm goes past the corner, without making a straight line with the first arm: the angle is obtuse.

Method — Estimating an Angle with Benchmark Angles

Without a protractor, an angle is estimated by comparing it with a few angles we know well, the benchmark angles:

the right angle, $90^\circ$ — the corner of the set square;
half a right angle, $45^\circ$ — the corner of the set square cut into two equal openings;
the straight angle, $180^\circ$ — two right angles side by side, the two arms in a straight line.

To estimate an angle: first compare it with the right angle (the set square): more closed, it is acute and measures less than $90^\circ$; more open, but less open than a straight angle, it is obtuse and measures between $90^\circ$ and $180^\circ$. Then compare it with the nearest other benchmark: an acute angle about half as open as the corner measures about $45^\circ$; an obtuse angle halfway between the right angle and the straight angle measures about $90^\circ + 45^\circ = 135^\circ$; an angle between two benchmarks measures between their two measures — for example, more open than half the corner but acute: between $45^\circ$ and $90^\circ$. The estimate must fit the type: an acute angle always measures less than $90^\circ$, an obtuse angle between $90^\circ$ and $180^\circ$. To draw an angle of about $45^\circ$ or $135^\circ$, start from a right angle drawn with the set square, then close it, or open it further, by about half a right angle.

Example

Estimate the measure of each angle with the benchmark angles, and give its type.

Answer

[(a)] With the set square (dashed), the second arm stays inside the corner: the angle is acute, less than $90^\circ$. It is more open than half the corner ($45^\circ$) but clearly more closed than the corner: it lies between $45^\circ$ and $90^\circ$, about $60^\circ$.
[(b)] The second arm goes past the corner: the angle is obtuse, more than $90^\circ$. It lies about halfway between the right angle and the straight angle: it measures about $90^\circ + 45^\circ$, that is about $135^\circ$.

Skills to practice

Identifying Angle Types by Measure
Identifying Angle Types
Drawing and Estimating Angles

III Review & Quizz

Skills to practice

Quiz