Exercises: Classification of angles

Look at the following drawing and classify each of the indicated angles according to its type (acute, right or obtuse).

An analog clock shows exactly 3:00 o'clock. The hour hand is the short one (orange) and the minute hand is the long one (blue).

1. a) What type of angle do the hour hand and the minute hand form at 3:00?

1. b) If the clock advanced to 3:10, would the angle formed be acute, right or obtuse? Explain your answer.

State whether the following statements are true or false:

1. a) An angle that measures $180^\circ$ is called a full angle.
2. b) An acute angle is always smaller than a right angle.
3. c) If an angle measures $91^\circ$, then it is an obtuse angle.
4. d) A zero angle is one that measures $0^\circ$.

Look at the following figure formed by two intersecting lines. If angle $A$ measures $40^\circ$, how much does angle $B$ measure knowing that they are vertically opposite? And how much do angles $C$ and $D$ measure, which are supplementary to angle $A$?

Match each pair of angles with its corresponding definition:

1. Angles that add up to $90^\circ$.

2. Angles that share a vertex and one side, but do not overlap.

3. Angles that add up to a straight angle ($180^\circ$).

1. a) Adjacent angles.
2. b) Complementary angles.
3. c) Supplementary angles.

Work out quickly:

1. a) The complementary angle of $65^\circ$.
2. b) The supplementary angle of $110^\circ$.
3. c) The complementary angle of an angle of $45^\circ$.

Identify in the following chart which pair of angles are consecutive and which pair of angles are adjacent. Justify your answer based on whether or not they form a straight line.

Draw in your exercise book (or imagine) a straight angle. If we draw a ray that divides it into two angles and one of them is a right angle, what is the other angle like? What are these two angles called based on their sum relationship?

You can do it on paper and send a photo with your answer.

In the world of first-person shooter (FPS) video games, the "Field of View" (FOV) is the angle that the player can see on their screen. An eSports player is setting up their monitor and is deciding between three settings: $90^\circ$, $115^\circ$ and $180^\circ$.

Classify each of these angles according to its size. In addition, if a motion sensor covers the remaining reflex angle to complete the circle in the $115^\circ$ setting, what size would that angle be?

Look at the design of this skatepark ramp. To be safe, the entry angle $\alpha$ and the slope angle $\beta$ must be complementary. If we know that the entry angle measures $\alpha = 32^\circ 45'$, calculate the size of angle $\beta$ and classify it according to its type (acute, right or obtuse).

Two streets in a Dublin neighbourhood intersect forming four angles. If angle $\hat{1}$ and angle $\hat{2}$ are adjacent, and we know that angle $\hat{1}$ is twice angle $\hat{2}$, how much does each measure? Classify both angles and explain what relationship angles $\hat{1}$ and $\hat{3}$ have based on their position.

A wall clock shows exactly 10:00 in the morning.

1. a) Calculate the size of the smaller angle formed by the hands of the clock.
2. b) Classify this angle according to its type.
3. c) What is the angle formed by the hands "on the outside" (the larger angle) called? Classify it as well.

In a mobile solar panel, engineers need two angles to be supplementary. The first angle, which we will call $\hat{A}$, measures $72^\circ 15' 30''$.

Calculate the size of angle $\hat{B}$ so that the panel works correctly. Once calculated, state whether $\hat{B}$ is an acute, obtuse or straight angle.

Draw an angle of $50^\circ$ and call it $\hat{P}$. Next, draw its angle bisector.

1. a) What is the size of each of the two new angles formed?
2. b) Classify these new angles according to their size.
3. c) What is the relationship between these two new angles by position? (Choose from: vertically opposite, adjacent or consecutive).

You can do it on paper and send a photo with your answer.

State whether the following statements are true or false, justifying your answer with a short example or drawing:

1. Two acute angles are always complementary.

2. An obtuse angle cannot have a supplementary angle that is also obtuse.

3. Vertically opposite angles always have the same measure.

4. A straight angle is twice a right angle.

A satellite dish has shifted from its original position. The technician observes that the error angle with respect to the vertical is $\hat{E} = 15^\circ 20' 45''$.

1. a) If the optimal reception angle is the complementary angle of this error, how large should the reception angle be?
2. b) Classify the error angle $\hat{E}$ according to its size.
3. c) If we rotate the dish through a full turn from its original position, how many degrees will we have rotated it?

A video game designer is setting up a character's "view camera". The total viewing angle is a full angle ($360^\circ$). However, the character has a "blind spot" right behind them corresponding to a circular sector of $55^\circ 45' 20''$. Calculate the amplitude of the character's effective viewing angle and classify it according to its opening.

Three main avenues in a city intersect at a circular square as shown in the figure. We know that angle $\hat{A}$ and angle $\hat{B}$ are complementary. If angle $\hat{A}$ is half the size of an angle of $74^\circ 30'$, calculate the size of the supplementary angle of $\hat{B}$. Classify all the angles obtained ($\hat{A}$, $\hat{B}$ and the supplementary angle of $\hat{B}$) according to their type.

A drone turns to change direction. First it turns through a right angle, then it turns twice an angle of $35^\circ 15' 40''$ in the same direction, and finally it turns back by one third of a straight angle.

1. a) Calculate the total size of the resulting turn angle with respect to the initial position.
2. b) Classify the resulting angle as acute, obtuse or right.

In the design of a logo, two intersecting lines have been drawn. It is known that one of the angles formed measures $x$ degrees. The angle adjacent to this measures three times $x$ minus $20^\circ$.

1. a) Set up an equation to find the value of $x$.
2. b) Calculate the size of the four angles formed at the intersection of the two lines and classify them.

The sum of two consecutive angles is an obtuse angle of $145^\circ$. If the difference between them is $25^\circ 30'$, calculate the size of each angle. Then, calculate how much each angle needs to reach a straight angle and classify these new resulting angles.

An architect is designing an inclined ramp. The angle that the ramp makes with the ground (angle of elevation) is acute. It is known that its complementary angle is exactly four times the angle of elevation.

1. a) Find the size of this angle of elevation.
2. b) If the architect decides to double the slope of the ramp, what type of angle (acute, right or obtuse) will the ramp now make with the vertical (its new complementary angle)?

Look at the hands of an analogue clock when they show exactly 02:40.

1. a) Calculate the convex angle formed by the hour hand and the minute hand (keep in mind that the hour hand moves proportionally to the minutes).
2. b) Calculate its supplementary angle and classify it.
3. c) At what exact time between 02:00 and 03:00 will the hands form a zero angle? (Explain your approximate answer).

Two angles are vertically opposite. The size of one of them is given by the expression $2x + 15^\circ$ and the size of one of its adjacent angles is $x + 45^\circ$.

1. a) Calculate the value of $x$ knowing that adjacent angles are supplementary.
2. b) Determine the size of the four angles formed at the vertex and classify them all according to their size.