Classifying Angles

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

State whether the following statements are true or false:

1. a) An angle that measures $180^\circ$ is called a full turn.
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 $0^\circ$.

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

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

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

Match each pair of angles with their corresponding definition:

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

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

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

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

Calculate 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 $45^\circ$.

Draw a straight angle in your notebook (or imagine one). 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 due to their sum relationship?

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

Two streets in a Chicago neighborhood intersect forming four angles. If angle $\hat{1}$ and angle $\hat{2}$ are adjacent, and we know that angle $\hat{1}$ measures twice as much as angle $\hat{2}$, what is the size of each one? Classify both angles and explain the relationship between angles $\hat{1}$ and $\hat{3}$ due to their position.

Look at the design of this skatepark ramp. For it to be safe, the entry angle $\alpha$ and the angle of incline $\beta$ 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 size (acute, right or obtuse).

In a mobile solar energy 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 $\hat{B}$ for the panel to work correctly. Once calculated, state whether $\hat{B}$ is an acute, obtuse or straight angle.

A satellite dish has moved from its original position. The technician observes that the angle of error 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, what must the reception angle measure?
2. b) Classify the angle of error $\hat{E}$ according to its size.
3. c) If we rotate the dish a full straight angle from its original position, how many degrees will we have rotated?

A video game designer is setting up a character's "camera view". The total viewing angle is reflex and measures exactly three times a straight angle. However, the character has a "blind spot" right behind them corresponding to a circular sector of $55^\circ 45' 20''$. Calculate the size of the character's effective viewing angle and classify it according to its size.

A drone makes a turn to change direction. First it turns a right angle, then it turns double an angle of $35^\circ 15' 40''$ in the same direction, and finally it reverses a turn equivalent to a third of a straight angle.

1. a) Calculate the size of the resulting total angle of turn 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 it 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.

An architect is designing an inclined ramp. The angle 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 incline of the ramp, what type of angle (acute, right or obtuse) will the ramp now make with the vertical (its new complementary angle)?

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 on a straight line are supplementary.
2. b) Determine the size of the four angles formed at the vertex and classify them all according to their size.

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

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 $\hat{A}$ measures half of an angle of $74^\circ 30'$, calculate the size of the supplementary angle of $\hat{B}$. Classify all the obtained angles ($\hat{A}$, $\hat{B}$ and the supplementary angle of $\hat{B}$) according to their type.

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 due to their position? (Choose from: vertically opposite, adjacent on a straight line or adjacent).

State whether the following statements are true or false, justifying your answer with a small 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 are always equal in size.

4. A straight angle is double a right angle.

An analog clock shows exactly 3:00 o'clock.

1. What type of angle is formed by the hour hand and the minute hand?

2. If the clock advanced to 3:10, would the angle formed be acute, right or obtuse?

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

1. a) Calculate the size of the minor angle formed by the hands of the clock.
2. b) Classify this angle according to its size.
3. c) What is the angle formed by the hands "on the outside" (the major angle) called? Classify it too.

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

1. a) Calculate the convex angle formed by the hour hand and the minute hand (bear 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? (Reason your approximate answer).