**Introduction**

In life, we encounter geometric concepts without even realizing that a process is modelled using geometry. For example, when cutting a cake into six or eight perfectly identical pieces, we need to determine the central angle of each piece of cake. When we look at a certain object, our eyes are directed along two rays, which converge on this object at one point and thus form an angle. Approaching the object, the angle increases, moving away from the object – the angle decreases. And what to say about such important parts of our life as architecture, physics or mechanics, where you can do almost nothing without angles. So, why not describe mathematically what we use in life?

**Definition of an acute angle**

An angle is defined as a figure formed when two rays meet at a common endpoint. An angle is represented by the symbol ∠.

Each angle consists of two arms and vertex:

- arms of the angle are those two rays that joining form the angle;
- vertex of the angle is a common endpoint at which two arms meet.

We can name an angle using the vertex, and a point on each of the angle’s arms. The name of the angle is the sequence of three letters representing those points, with the vertex in the middle. For example, the below diagram shows ∠AOB with arms $\overrightarrow{OA}$ and $\overrightarrow{OB}$ and vertex O.

We usually measure angles in degrees or radians.

Degrees: One full rotation around the point is 360°.

Full rotation{Picture taken from a free resource Pixabay}

You may ask why full rotation angle has 360°? Ancient Persian astronomers believed that a year consisted of 360 days. Observing the sun rotations, they noticed that it took the sun exactly 360 days to return to initial position. In other words, each day the sun revolves by 1°. Another historical reason suggests that the measure of full rotation angle comes from the Babylonians. They used number system with the base of 60 and it was easy to represent number 360 in this number system.

Radians: One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. A full circle has 2π radians, so there is a simple relationship between degrees and radians.

Angle | Radians | Degrees |

Full rotation | 2π | 360° |

Half of a full rotation | π | 180° |

Quarter of a full rotation | $\frac{π}{4}$ | 90° |

- If you want to convert radians to degrees, just remember that equals to 180°. For example, $\frac{π}{3}$ radians is $\frac{180°}{3}$=60°.
- If you want to convert degrees to radians, then find what fraction of 180° given angle forms. For example, 45° is $\frac{1}{4}$ of 180°, so 45° is $\frac{1}{4}π=\frac{π}{4}$.

We can measure angles with protractor. The normal protractor measures angles from 0° to 180°. For example, the measure of the angle below is 15° (right arm of the angle points at zero, and left arm of the angle points at 15°).

Using the normal protractor, we can measure angles in both directions: clockwise and anticlockwise.

There are also a full-circle protractors. For example, the diagram below shows the angle with the measure of 210°.

**Definition:** An acute angle is an angle with the degree measure greater than 0° and less than 90°.

**Examples of acute angles**

We already know that an acute angle measures less than 90°. The examples of acute angles are 12°, 35°, 61°, or 89°, but angles 0°, 90°, 146°, or 214° are not acute angles.

**EXAMPLE:** Finding the measure of each angle, determine which angle is acute and which is not.

**SOLUTION:**

a) In this case, angle has the measure of 60°, so it is acute angle (measure this angle in anticlockwise direction).

b) This angle is not acute angle because its measure is greater than 90° (measure this angle in clockwise direction).

c) The right arm of the angle points at 75°, the left arm of the angle points at 120°, so the measure of the angle is 120°-75°=45°. Since 45° is less than 90°, this angle is acute angle.

d) Shown angle is not acute angle, its measure 270° is even greater than 180°.

**Classification of angles**

Angles can be classified by their measures:

- if the degree measure of an angle is exactly 0°, the angle is zero angle;
- if the degree measure of an angle is greater than 0° and less than 90°, the angle is acute angle;
- if the degree measure of an angle is exactly 90°, the angle is right angle;
- if the degree measure of an angle is greater than 90° and less than 180°, the angle is obtuse angle;
- if the degree measure of an angle is exactly 180°, the angle is straight angle;
- if the degree measure of an angle is greater than 180° and less than 360°, the angle is reflex angle;
- if the degree measure of an angle is exactly 360°, the angle is full angle.

Let’s show it with the following table.

Type of angle | Diagram | Measure range |

zero | Measure =0° | |

acute | 0°< Measure <90° | |

right | Measure =90° | |

obtuse | 90°< Measure <180° | |

straight | Measure =180° | |

reflex | 180°< Measure <360° | |

full | Measure =360° |

To determine whether a given angle is an acute angle or not, we can compare this angle with right, straight or full angle:

- A full angle is a full rotation angle. If given angle limits less than a quarter of a full circle, then it is an acute angle.
- A straight angle is half a full angle. If given angle limits less than a half of a straight angle, then it is an acute angle. If given angle limits close to a half or more than a half of a straight angle, then it is not an acute angle.
- A right angle is a quarter of full angle. If given angle limits less than a right angle, then it is an acute angle.

For example, the diagram below shows angle that limits greater portion of a circle than one quarter, so this angle is not acute angle.

The following questions about angles often arise:

- “Is the angle with the measure 0° acute?”, “Is the angle with the measure 90° acute?” Now, you know that angles with measures 0° and 90° are not acute, they are zero and right angles respectively.
- “Is zero angle and full angle the same?” No, these angles have different measures.

**Real-life examples of acute angles**

I) “*Where in real life I can see zero angle?*”

You’ve probably seen a clock with a dial and hands. So, the hands of the clock pointing to a certain hour, form an angle with the vertex in the centre of the clock.

Now, consider an example, the clock shows the time 2 o’clock in the morning. What is the angle made by both hands of the clock? Is this angle acute?

First, find the angle between two consecutive numbers on the clock. There are 12 numbers on the clock, so the measure of the angle between two consecutive numbers is $\frac{360°}{12}$=30°.

Now, if the clock shows that time is 2 o’clock, then the longer hand points at 12 and the shorter hand points at 2. Between these two numbers are two angles of 30°, therefore, the measure of unknown angle is 2⋅30°=60°. Since 60°<90°, angle with the degree measure 60° is an acute angle.

II) “*Where in real life I can see full angle?*”

Winding the thread on the spool each time when we return to the starting point, we pass a full angle.

III) “*Where in real life I can see straight angle?*”

Whenever you go for a drive, you can see the straight angle at the road. Moreover, at the T-junction you can see two right angles.

IV) “*Where in real life I can see acute angle?*”

In real life, acute angles are very common: the flanks of open scissors, a pizza slice, angle between two fingers, clothes hanger, house roof and even crocodile mouth. In everyday life we see a lot of angles – we just need to have a little bit of imagination to see them and understand “Oh, I see the angle and it’s acute!”

**Pairs of angles**

- Adjacent angles: angles that have the same vertex and one common arm.
- Linear pair of angles: two adjacent angles that together form a straight angle;
- Complementary angles: a pair of angles which together form a right angle, in other words they add up to 90º.
- Supplementary angles: a pair of angles which together form a straight angle, in other words the sum of these angles is equal to 180º.
- Vertical angles: vertical angles are two opposite angles formed when two lines intersect.
- Same-side interior angles: same-side interior angles are a pair of angles formed when a line intersects two parallel lines. Same-side interior angles add up to 180°.
- Alternate interior angles: alternate interior angles are a pair of angles formed when a line intersects two parallel lines. Alternate interior angles are always equal to each other.
- Alternate exterior angles: alternate exterior angles are a pair of angles formed when a line intersects two parallel lines. Alternate exterior angles are simply vertical angles of the alternate interior angles. Alternate exterior angles are congruent.
- Same-side exterior angles: same-side interior angles are a pair of angles formed when a line intersects two parallel lines. Same-side exterior angles are simply vertical angles of the same-side interior angles. Same-side exterior angles add up to 180°.
- Corresponding angles: corresponding angles are a pair of angles formed when a line intersects a pair of parallel lines. Corresponding angles are also equal to each other.

Let parallel lines a and b be cut by transversal c. Here is an example of each pair of angles:

- adjacent angles: ∠1 and ∠2;
- supplementary angles: ∠1 and ∠2;
- vertical angles: ∠1 and ∠3;
- same-side interior angles: ∠3 and ∠5;
- alternate interior angles: ∠3 and ∠6;
- same-side exterior angles: ∠1 and ∠7;
- alternate exterior angles: ∠1 and ∠8;
- corresponding angles: ∠1 and ∠6.

Given all the described definitions and properties of pairs of angles, let’s see which of them can be acute.

- In a pair of adjacent angles can be two acute, one acute or no acute angles.
- Each pair of complementary angles consists of two acute angles.
- Each pair of not right angles consists of one acute and one obtuse angle.
- When two not perpendicular lines intersect, one pair of vertical angles consists of two congruent acute angles and another pair of vertical angles consists of two congruent obtuse angles.
- When two parallel lines are cut by not perpendicular transversal, one of same-side interior angles is always acute.
- When two parallel lines are cut by not perpendicular transversal, one pair of alternate interior angles consists of two congruent acute angles and another pair of alternate interior angles consists of two congruent obtuse angles.
- When two parallel lines are cut by not perpendicular transversal, one of same-side exterior angles is always acute.
- When two parallel lines are cut by not perpendicular transversal, one pair of alternate exterior angles consists of two congruent acute angles and another pair of alternate exterior angles consists of two congruent obtuse angles.
- When two parallel lines are cut by not perpendicular transversal, four pairs of corresponding angles are formed. Two pairs consist of two congruent acute angles and other two pairs consist of two congruent obtuse angles.

**Triangles with acute angles**

All triangles can be classified by angles or by sides.

Classification of triangles by angles:

- acute triangle – all angles of the triangle are acute;
- right triangle – there is one right angle;
- obtuse triangle – there is one obtuse angle.

In acute triangles all three angles are acute, at the same time in right and obtuse triangles there are two acute angles.

Classification of triangles by sides:

- scalene triangle – all three sides of different length;
- isosceles triangle – two sides are congruent;
- equilateral triangle – all three sides are congruent.

Scalene and isosceles triangles could be acute, right and obtuse. However, an isosceles triangle can never have two obtuse base angles. An equilateral triangle is also an equiangular triangle, all its acute angles are congruent and have the measures of 60°.

**In the acute triangle the sum of the squares of the two sides of a triangle is greater than the square of the largest side,**

a^{2}+b^{2}>c^{2}

**EXAMPLE: **In the acute triangle ABC, the lengths of two shorter sides are 5 cm and 12 cm. What is the range of all possible lengths of the longest side in triangle ABC?

**SOLUTION:** In the acute triangle ABC, the length of the longest side c satisfies the inequality

5^{2}+12^{2}>c^{2}

c^{2}<25+144

c^{2}<169

c<13

We found the upper boundary of lengths c. To find the smallest possible length c, apply the triangle’s inequality:

c+5>12

c>7

Therefore, the range of all possible lengths of the longest side in the acute triangle ABC is

7 cm<c<13 cm

**FAQs**

**1. Why the full rotation is 360° and not 100° or 1000° for example?**

Ancient Persian astronomers believed that a year consisted of 360 days. Observing the sun rotations, they noticed that it took the sun exactly 360 days to return to its initial position. In other words, each day the sun revolves by degree. Another historical reason suggests that the measure of full rotation angle comes from the Babylonians. They used a number system with a base of 60 and it was easy to represent the number 360 in this number system.

**2. How to convert degrees to radians and radians to degrees?**

- If you want to convert radians to degrees, just remember that π equals $\frac{π}{3}$. For example, radians is $\frac{180^o}{3}=60^o$.
- If you want to convert degrees to radians, then find what fraction of 180
^{o}given angle forms. For example, 45^{o}is $\frac{1}{4}$ of 180^{o}, so 45^{o}is $\frac{1}{4}π=\frac{π}{4}$.

**3. Does a triangle with one acute angle exist?**

No, if a triangle has only one acute angle, then the other two angles have measures at least 90^{o} and the sum of the measures of all interior angles of the triangle will be greater than 180^{o}.

**4. What is the greatest and what is the smallest number of acute angles in an arbitrary triangle? **

The greatest number of acute angles in a triangle is three, the smallest number of acute angles in a triangle is two.

**5. Is the angle with the measure 0° acute?**

No, this angle is known as zero angle.

**6. Is the angle with the measure 90° acute?**

No, this angle is a right angle.

**7. Is zero angle and full angle the same?**

No, they have different measures.

**8. Where in real life can I see zero angle?**

On the clock or scissors.

**Quiz**

- In each case, determine whether the given angle is acute. If it is not acute angle, what is its type? Explain your reasoning.

a) 13°;

b) 125°;

c) 90°;

d) 78°;

e) 180°;

f) 66°;

g) 0°;

h) 330°.

i) 360°.

**SOLUTION**:

a) 0°<13°<90°, then angle with the measure 13° is an acute angle.

b) Since 90°<125°<180°, angle with the measure 125° is not an acute angle, this angle is obtuse.

c) 90° angle is a right angle, so this angle is not acute.

d) 0°<78°<90°, then angle with the measure 78° is an acute angle.

e) 180° angle is a straight angle, so this angle is not acute.

f) 0°<66°<90°, then angle with the measure 66° is an acute angle.

g) 0° angle is a zero angle, so this angle is not acute.

h) Since 180°<330°<360°, angle with the measure 330° is not an acute angle, this angle is reflex.

i) 360° angle is a full angle, so this angle is not acute.

- Finding the measure of each angle, determine which angle is acute and which is not.

**SOLUTION:** a) The right arm of the angle points at 45°, the left arm of the angle points at 150°, so the measure of the angle is 150°-45°=105°. Since 105° is greater than 90°, this angle is not acute angle.

b) The right arm of the angle points at 15°, the left arm of the angle points at 75°, so the measure of the angle is 75°-15°=60°. Since 60° is less than 90°, this angle is acute angle.

**ANSWER:** a) not acute angle b) acute angle

- In each case, comparing given angle with right, straight or full angle determine whether it is an acute angle.

**SOLUTION**: a) The diagram shows angle that limits greater angle than right angle, so this angle is not acute angle.

b) The diagram shows angle that is closer to straight angle than to right angle. This means given angle is not acute angle.

c) The diagram shows angle that limits smaller angle than right angle, so this angle is acute angle.

**ANSWER:** a) not acute b) not acute c) acute

- Which of the following statements is true? Explain why.

a) When two parallel lines are cut by not perpendicular transversal, same-side interior angles are congruent.

b) When two parallel lines are cut by perpendicular transversal, one of alternate exterior angles is always acute.

c) When two parallel lines are cut by not perpendicular transversal, one pair of the corresponding angles is always acute.

d) When two lines intersect, there is always a pair of congruent vertical acute angles.

**SOLUTION**: a) When two parallel lines are cut by transversal, same-side interior angles are supplementary. Two supplementary angles are congruent only when they are both right angles. Since transversal is not perpendicular, same-side interior angles are not right angles, therefore, they are not congruent and this statement is false.

b) When two parallel lines are cut by perpendicular transversal, alternate exterior angles are always right angles. Right angle is not an acute angle, so this statement is false.

c) When two parallel lines are cut by not perpendicular transversal, four pairs of corresponding angles are formed. Two pairs consist of two congruent acute angles and other two pairs consist of two congruent obtuse angles. So, this statement is true.

d) When two perpendicular lines intersect, they form two pairs of vertical right angles. If intersecting lines are not perpendicular, this statement is true. But in general, the statement is false.

**ANSWER**: a) False b) False c) True d) False

- In the acute triangle ABC, the lengths of two shorter sides are 8 cm and 15 cm. What is greatest possible natural length of the longest side in triangle ABC?

**SOLUTION:** In the acute triangle ABC, the length of the longest side c satisfies the inequality

8^{2}+15^{2}>c^{2}

c^{2}<64+225

c^{2}<289

c<17

The greatest possible natural length of the longest side is 16 cm.

**ANSWER:** 16 cm

**Conclusions**

- Each acute angle has the measure less than 90°.
- We can find angles measures using normal or full-circle protractors.
- Visually, we can compare angles with right angle to determine whether it is acute or not.
- An angle with the measure of 0° is not acute angle.
- We can find acute angles everywhere in real life.
- Each triangle contains at least two acute angles.

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