Geometry - How to find an angle of a line

Parellel

Examine the diagram. Which of these conditions does not prove that $m \parallel n$ ?

$\angle1 \cong \angle 5$

$l \parallel m$ and $l \parallel n$

$m \angle1 + m \angle 2 = 180$

$\angle3 \cong \angle 4$

Any of these statements can be used to prove that $m \parallel n$ .

Explanation

If $l \parallel m$ and $l \parallel n$ , then $m \parallel n$ , since two lines parallel to the same line are parallel to each other.

If $m \angle1 + m \angle 2 = 180$ , then $m \parallel n$ , since two same-side interior angles formed by transversal are supplementary.

If $\angle3 \cong \angle 4$ , then $m \parallel n$ , since two alternate interior angles formed by transversal are congruent.

However, $\angle1 \cong \angle 5$ regardless of whether and are parallel; they are vertical angles, and by the Vertical Angles Theorem, they must be congruent.

Lines A and B are parallel. Find the measurement of $\angle1$ .

$78^\circ$

$102^\circ$

$180^\circ$

The measurement of $\angle1$ cannot be determined.

Explanation

When parallel lines are cut by a transversal, corresponding angles are congruent. The given angle and $\angle1$ are corresponding angles.

$\angle1$ must also have the same mesaurement as the given angle.

$\angle1=78^\circ$

Varsity_question

AB and CD are two parrellel lines intersected by line EF. If the measure of angle 1 is , what is the measure of angle 2?

$48^{\circ}$

$132^{\circ}$

$64^{\circ}$

$180^{\circ}$

$24^{\circ}$

Explanation

The angles are equal. When two parallel lines are intersected by a transversal, the corresponding angles have the same measure.

If lines A and B are parallel, what is the measurement of $\angle2$ ?

$102^\circ$

$78^\circ$

$95^\circ$

$44^\circ$

Explanation

$\angle2$ and the given angle are supplementary, meaning that their angle measurements add up to .

$\angle2+78^\circ=180^\circ$

Subtracting 78 from each side to find the measurement of angle two.

$\angle2=102^\circ$

Mark is training for cross country and comes across a new hill to run on. After Mark runs meters, he's at a height of meters. What is the hill's angle of depression when he's at an altitude of meters?

$34^{\circ}$

Cannot be determined

The same as the angle of inclination

$37^{\circ}$

$35^{\circ}$

Explanation

Angle_of_a_line_hill

Upon reading the question, we're left with this spatial image of Mark in our heads. After adding in the given information, the image becomes more like

Angle_of_a_line_hill_resolution

The hill Mark is running on can be seen in terms of a right triangle. This problem quickly becomes one that is asking for a mystery angle given that the two legs of the triangle are given. In order to solve for the angle of depression, we have to call upon the principles of the tangent function. Tan, Sin, or Cos are normally used when there is an angle present and the goal is to calculate one of the sides of the triangle. In this case, the circumstances are reversed.

Remember back to "SOH CAH TOA." In this problem, no information is given about the hypotenuse and nor are we trying to calculate the hypotenuse. Therefore, we are left with "TOA." If we were to check, this would work out because the angle at Mark's feet has the information for the opposite side and adjacent side.

Because there's no angle given, we must use the principles behind the tan function while using a fraction composed of the given sides. This problem will be solved using arctan (sometimes denoted as $tan^{-1}$ ).

$tan^{-1}\left(\frac{17}{25}\right)= angle_{depression}$

$tan^{-1}\left(\frac{17}{25}\right)={\color{Blue} 34.2^{\circ}}$

If lines A and B are parallel, what is the measurement of angle 5?

$102^\circ$

$78^\circ$

$65^\circ$

$117^\circ$

Explanation

Notice that $\angle5$ and $\angle1$ are supplementary, meaning their angle measurements add up to .

$\angle5+\angle1=180^\circ$

Now, notice that $\angle1$ and the given angle are corresponding angles, meaning that their measurements are the same.

Thus,

$\angle5+78^\circ=180^\circ$

$\angle5=102^\circ$

If lines A and B are parallel, what is the measurement of $\angle6$ ?

$78^\circ$

$102^\circ$

$89^\circ$

$61^\circ$

$82^\circ$

Explanation

Notice that $\angle4$ and $\angle6$ are corresponding angles. This means that they possess the same angular measurements; thus, we can write the following:

$\angle4=\angle6$

Now, notice that $\angle4$ and the provided angle of $78^\circ$ angle are vertical angles. Vertical angles share the same angle measurements; therefore, we may write the following:

If $\angle4=\angle6$ and $\angle4=78^\circ$ , then $\angle6=78^\circ$

A student creates a challenge for his friend. He first draws a square, the adds the line for each of the 2 diagonals. Finally, he asks his friend to draw the circle that has the most intersections possible.

How many intersections will this circle have?

Explanation

The answer to this problem is 12. This can be drawn as shown below (intersections marked in red).

Square

We can also be sure that this is the maximal case because it is the largest answer selection. Were it not given as a multiple choice question, however, we could still be sure this was the largest. This is because no line can intersect a circle in more than 2 points. Keeping this in mind, we look at the construction of our initial shape. The square has 4 lines, and then each diagonal is an additional 2. We have thus drawn in 6 lines. The maximum number of intersections is therefore going to be twice this, or 12.

Angle $\angle ABC$ measures $20^{\circ}$

$\overrightarrow{BD}$ is the bisector of $\angle ABC$

$\overrightarrow{BE}$ is the bisector of $\angle CBD$

What is the measure of $\angle ABE$ ?

$5^{\circ}$

$10^{\circ}$

$15^{\circ}$

$30^{\circ}$

$40^{\circ}$

Explanation

Angle pic

Let's begin by observing the larger angle. $\angle ABC$ is cut into two 10-degree angles by $\overrightarrow{BD}$ . This means that angles $\angle ABD$ and $\angle CBD$ equal 10 degrees. Next, we are told that $\overrightarrow{BE}$ bisects $\angle CBD$ , which creates two 5-degree angles. $\angle ABE$ consists of $\angle ABD$ , which is 10 degrees, and $\angle DBE$ , which is 5 degrees. We need to add the two angles together to solve the problem.

$\angle ABE=\angle ABD+\angle DBE$

$\angle ABE=10^{\circ}+5^{\circ}$

$\angle ABE=15^{\circ}$

$\dpi{100} \small \overline{AB}$ is a straight line. $\dpi{100} \small \overline{CD}$ intersects $\dpi{100} \small \overline{AB}$ at point $\dpi{100} \small E$ . If $\dpi{100} \small \angle AEC$ measures 120 degrees, what must be the measure of $\dpi{100} \small \angle BEC$ ?

None of the other answers

$\dpi{100} \small 70$ degrees

$\dpi{100} \small 65$ degrees

$\dpi{100} \small 75$ degrees

$\dpi{100} \small 60$ degrees

Explanation

$\dpi{100} \small \angle AEC$ & $\dpi{100} \small \angle BEC$ must add up to 180 degrees. So, if $\dpi{100} \small \angle AEC$ is 120, $\dpi{100} \small \angle BEC$ (the supplementary angle) must equal 60, for a total of 180.

How to find an angle of a line

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