Lines and Angles

In the following drawing of a straight road, identify the name of each of the marked elements.

Indicate which is a line segment, which is a ray, and which represents a complete line.

Look at the map of these streets. Indicate the relative position (parallel, intersecting, or perpendicular) of the following pairs of streets:

• Street A and Street B.

• Street B and Street C.

• Street A and Street C.

A group of friends is designing a new skatepark on a vacant lot. To do this, they have drawn several lines on the ground representing the edges of the ramps. Look at the plan they have drawn and answer the questions about the relative position of the lines:

1. a) Indicate a pair of lines that are parallel to each other.
2. b) Find two lines that are intersecting but not perpendicular.
3. c) Identify a line that is perpendicular to line $s$.
4. d) If line $t$ and line $u$ never cross no matter how much we extend them, what geometric relationship do they have?

Classify the following angles according to their size. Indicate whether they are acute, right, or obtuse:

• $\alpha = 45^\circ$

• $\beta = 120^\circ$

• $\gamma = 90^\circ$

Two angles are complementary if their sum is $90^\circ$. If angle $\alpha$ measures $35^\circ$, what is the measure of angle $\beta$ for them to be complementary?

Find the value of angle $\hat{X}$ in the following figure knowing that the two angles are supplementary (they add up to $180^\circ$).

When two lines intersect, the vertical angles are equal. Look at the figure and, without using a protractor, indicate the size of angle $\hat{A}$ and angle $\hat{B}$.

A navigation radar rotates a total of $5^\circ$ every second. Express this movement in arcminutes ($'$). Remember that $1^\circ = 60'$.

A security camera in a shopping center has a total viewing angle of $110^\circ$. The maintenance technician decides to divide this angle into two zones: Zone A, which covers the stores, and Zone B, which covers the hallways.

If Zone A measures $64^\circ 35' 50''$, calculate the exact size (in degrees, minutes, and seconds) that Zone B must have to cover the rest of the field of view.

A communications satellite orbits the Earth and makes three angular rotation measurements to correct its trajectory. The first rotation is $42^\circ \ 35' \ 48''$. The second rotation is double the first. If at the end of the day the satellite must have rotated a total angle of a full straight angle to be aligned with the ground base, what must the third rotation measure? Express the result in complex form (degrees, minutes and seconds).

We are analyzing the structure of a modern shelving unit. The horizontal shelves are parallel to each other and are crossed by a slanted support.

Knowing that the angle $\hat{1}$ measures $124^\circ$, use the properties of angles formed by parallel lines intersected by a transversal to calculate, without using a protractor, the size of angles $\hat{2}, \hat{3}$ and $\hat{4}$. Justify each answer (e.g., because they are vertical angles, corresponding, alternate interior, etc.).

Look at the following figure showing several related angles. We know that lines $m$ and $n$ are parallel. If angle $\hat{A} = 48^\circ$, calculate with reasoning the value of angles $\hat{B}$ and $\hat{C}$.

Hint: Use the concept of supplementary angles and corresponding angles between parallel lines.

Draw a line $r$ and a point $P$ that does not belong to it in your notebook. Using set squares, draw a line $s$ that passes through $P$ and is parallel to $r$. How many lines parallel to $r$ can pass through that point $P$?

On the map of a playground, they want to place a lamppost that is exactly the same distance from the slide ($T$) and the swings ($C$).

1. a) What is the technical name for the imaginary line where all the points that meet this condition should be?
2. b) If the distance between the slide and the swings is $12$ feet, what will be the minimum distance of the lamppost from both if it is placed at the midpoint?
3. c) Draw a diagram in your notebook of the line segment $TC$ and its perpendicular bisector.

An architect is designing a park in the shape of a scalene triangle whose vertices are three fountains located at points $A$, $B$ and $C$. They want to place a streetlight at a point $P$ that meets two conditions:

1. That it is at the same distance from fountain $A$ as from fountain $B$.

2. That it is at the same distance from the street connecting $A$ and $C$ as from the street connecting $B$ and $C$.

What geometric constructions (perpendicular bisector, angle bisector, etc.) must the architect carry out to find the exact point for the streetlight? Explain the property of the points on these lines that justifies your answer.

Two main avenues in a city cross forming an angle of $72^\circ 15'$. They want to build a pedestrian street that divides that angle exactly in half.

1. a) What is the technical name for the line that represents this pedestrian street?
2. b) Calculate the value of the angle that the new street will form with each of the avenues.
3. c) What type of angle is the result (acute, right, or obtuse)?

In the design of a metal structure for a roof, there are two parallel main beams, $r$ and $s$. A third transversal beam $t$ intersects them forming an obtuse angle of $135^\circ \ 20'$. It is decided to install a reinforcement that coincides exactly with the angle bisector of the acute angle formed by the intersection of beams $s$ and $t$. What angle will this reinforcement form with beam $r$ on the inside of the structure? Reason whether the angles will be alternate interior or corresponding.

Consider two adjacent angles, $\widehat{AOB}$ and $\widehat{BOC}$. The angle bisectors $OX$ of angle $\widehat{AOB}$ and $OY$ of angle $\widehat{BOC}$ are drawn.

Prove, by calculating step by step, the size of the angle $\widehat{XOY}$ formed by both angle bisectors. Does this result depend on the individual size of the initial angles? Justify your answer.

In a strategy video game, a laser beam bounces off a metal surface. It is known that the angle of incidence $\alpha$ and the angle of reflection $\beta$ are equal.

If the total angle formed by the incoming and outgoing beams is $115^\circ 20'$, calculate the size of angle $\alpha$. Then, indicate whether the supplementary angle of $\alpha$ is acute, right, or obtuse.

Angle $\alpha$ is such that its supplementary angle measures three times its complementary angle.

1. a) Set up a logical equation using the concepts of complementary angle ($90^\circ - \alpha$) and supplementary angle ($180^\circ - \alpha$).
2. b) Calculate the value of angle $\alpha$.
3. c) How would you classify angle $\alpha$ according to its size?

A wall clock shows 3:40 in the afternoon.

1. a) Calculate the angle formed by the hour hand and the minute hand. Keep in mind that the hour hand moves slightly as the minutes advance.
2. b) How would you classify this angle according to its size (acute, right, obtuse, or straight)?

Calculate the exact angle formed by the hands of an analog clock when it shows 03:40.

Hint: Bear in mind that the hour hand does not stay fixed on the 3, but advances proportionally as the minute hand moves.

1. a) How many degrees does the hour hand advance for each minute that passes?
2. b) Calculate the position in degrees of both hands with respect to 12:00 and find the difference.

Four intersecting lines have been drawn on a plane in such a way that each one intersects all the others, but three lines never pass through the same point.

1. a) What is the maximum number of intersection points that can be formed?
2. b) Into how many regions is the plane divided?

Draw a diagram to represent this and justify your answer by counting the regions.

Given a line $r$ and an external point $P$, we know that there is only one line parallel to $r$ that passes through $P$.

Imagine now that we have three lines parallel to each other ($r, s, t$) and two transversal lines ($m, n$) that are not parallel to each other.

1. a) What is the total number of angles formed at the intersection points?
2. b) If all the acute angles formed by the line $m$ with the parallel lines measure $70^\circ$, and the acute angles formed by the line $n$ with the parallel lines measure $55^\circ$, calculate the angle formed between the lines $m$ and $n$.