Alternate interior angles

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Alternate interior angles

Alternate interior angles are alternate angles located in the internal area between parallel lines. They are not on the same side of the transversal nor are they on the same level (floor) relative to the line.

Diagram showing corresponding interior angles in geometry with marked arcs in blue and connecting lines, featuring a quadrilateral structure and labeled by Tutorela.

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Does the diagram show an adjacent angle?

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Alternate interior angles

In this article, we will learn about alternate interior angles, how to identify them, as well as their characteristics.
First, we need to remember what alternate angles are in general:
Alternate angles
Alternate angles between parallel lines are equal.
They are called alternate angles because they:
• Are not on the same side of the transversal line
• Are not on the same "level" relative to the line

Here are alternate angles for example:

Diagram showing corresponding interior angles in geometry with marked arcs in blue and connecting lines, featuring a quadrilateral structure and labeled by Tutorela.

The two marked angles are not on the same level and not on the same side, therefore they are alternate angles.
In order to confirm the presence of an alternate interior angle, you must observe that:
There is an exterior part - outside the two parallel lines
As well as an interior part - between the two parallel lines.
Let's examine the illustration:

Diagram explaining corresponding exterior and interior angles with labeled sections: the exterior part in orange, the interior part in blue, and highlighted angles for geometry demonstration, designed for teaching angle relationships.

In the illustration, we can see that the two alternate angles located between the two parallel lines in the inner part are alternate interior angles. Let's examine another example of a pair of alternate interior angles:

Diagram showing corresponding interior angles in geometry with marked arcs in blue and connecting lines, featuring a quadrilateral structure and labeled by Tutorela.


Note that in this illustration as well, you can observe that the two alternate angles are located in the internal part between the two parallel lines, and therefore they are alternate interior angles.

Bonus tip!
Alternate angles located in the external part outside the two parallel lines are called exterior alternate angles.

And now let's practice!
Here are two parallel lines and a line intersecting them.
a. Determine whether the angles shown are alternate angles.
b. Determine whether they are also alternate interior angles.

Diagram showing corresponding interior angles in geometry with marked arcs in blue and connecting lines, featuring a quadrilateral structure and labeled by Tutorela.

Solution:
a. Yes, the angles in the figure are alternate angles. They are not on the same side of the transversal and not on the same level relative to the line.
b. Yes, the alternate angles in the figure are interior since they are located in the inner part between the two parallel lines.

Another exercise:
Two parallel lines and a line intersecting them are shown.
a. Determine whether the angles shown are alternate angles
b. Determine whether they are alternate interior angles.

Diagram explaining corresponding exterior and interior angles with labeled sections: the exterior part in orange, the interior part in blue, and highlighted angles for geometry demonstration, designed for teaching angle relationships.

Solution:
a. Yes, the angles in the figure are alternate angles. They are not on the same level relative to the line and not on the same side of the transversal.
b. No. The angles are located in the external part outside the two parallel lines, therefore they are alternate angles but not interior ones.

Another exercise:
Here are two parallel lines and a line that intersects them.
Find the size of angle AA
and determine whether angle AA and angle BB are alternate interior angles.
Given that: B=100B=100

Diagram illustrating the relationship between angles A and B formed by parallel lines and a transversal, with a mathematical equation overlay for calculating angles in polygons, presented by Tutorela.

Solution:
According to the given information as well as the provided figue , we can determine that angle AA and angle BB are alternate angles. They are located between two parallel lines, each on a different side of the transversal and not on the same level relative to the line.
Alternate angles are equal to each other, therefore if B=100B=100 we can conclude that angle A=100A=100

Additionally, we can also determine that the two angles are alternate interior angles because they are both located in the interior part between the two parallel lines.

Additional Exercise:

In all drawings, the two lines are parallel to each other.
a. Determine whether there are alternate interior angles in both drawings.
b. If in drawing 11 the marked angle QQ equals 130130, what is angle WW?
c. Determine true or false - only alternate exterior angles are equal to each other.

1.

Diagram demonstrating corresponding exterior angles in geometry, marked as 'w' and 'q' in blue, within a quadrilateral structure for educational purposes.

2.

Diagram showing corresponding interior angles in geometry with marked arcs in blue and connecting lines, featuring a quadrilateral structure and labeled by Tutorela.

Solution:
a. No, only in the second drawing the two angles are alternate interior angles given that they are located in the inner part of the lines.
In the first drawing, the two angles are alternate exterior angles since they are located in the outer part of the lines.

b. The two angles marked in the drawing 11 are alternate angles and therefore they are equal.
From this we can conclude that angle WW is also equal to 130130.

C. Incorrect – exterior alternate angles are also equal to each other.

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Examples with solutions for Angles in Parallel Lines

Exercise #1

Does the diagram show an adjacent angle?

Video Solution

Step-by-Step Solution

To determine if the diagram shows adjacent angles, we need to analyze the geometric arrangement shown:

  • Step 1: Identify the common vertex.

    In the diagram, both the vertical line and the diagonal line intersect at a point. This intersection point serves as the common vertex for the angles in question, as they radiate outward from this shared point.

  • Step 2: Identify the common side.

    Adjacent angles must share a common side or arm. In the diagram, the vertical line acts as one common side for both angles, with one angle extending upwards and the other horizontally from the vertex.

  • Step 3: Ensure no overlap of interiors.

    It is equally essential to ensure that these two angles do not overlap. Each angle branches from the vertex in a different direction, maintaining distinct interiors.

By confirming the presence of a common vertex and a common side without overlap of the angle interiors, the angles satisfy the definition of being adjacent.

Therefore, the diagram does indeed show adjacent angles.

Consequently, the correct answer is Yes.

Answer

Yes

Exercise #2

Does the diagram show an adjacent angle?

Video Solution

Step-by-Step Solution

To determine whether the diagram shows adjacent angles, we need to confirm the presence of two properties:
1. Two angles must share a common vertex.
2. These angles must have a common arm and should not overlap.

Based on the given representation, the provided diagram consists solely of a single line. There are no visible intersecting lines or vertices from which angles can originate. Without intersection, there cannot be distinct angles, and thereby no adjacent angles can be identified.

Therefore, the diagram lacks the necessary properties to demonstrate adjacent angles. Hence, the correct choice is No.

Answer

No

Exercise #3

Does the diagram show an adjacent angle?

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Inspect the given diagram for angles.
  • Step 2: Determine if any angles share a common vertex and a common side.
  • Step 3: Verify that the angles do not overlap.

Now, let's work through each step:

Step 1: Inspecting the diagram, we notice several intersecting lines.

Step 2: To check for adjacent angles, we look for pairs of angles that share both a common vertex and a common side. An adjacent angle must be formed by such pairs, ensuring they do not overlap.

Step 3: Based on our definition, after closely examining the diagram, no pair of angles in the diagram seems to satisfy the definition of adjacent angles. The intersecting lines form angles that don't share a common arm with any other angle at the same vertex in the manner required for adjacency.

Therefore, the solution to the problem is No, the diagram does not show an adjacent angle.

Answer

No

Exercise #4

If two adjacent angles are not right angles, then one of them is obtuse and the other is acute.

Video Solution

Step-by-Step Solution

To solve the problem, let’s consider the nature of adjacent angles:

  • Step 1: Adjacent angles are two angles that share a common side and vertex. If two adjacent angles form a straight line, their measures sum up to 180180^\circ.
  • Step 2: According to the problem, neither angle is a right angle, meaning neither is 9090^\circ.
  • Step 3: Given this constraint, analyze the possibilities:
    • If one angle is acute (less than 9090^\circ), then the other must be more than 9090^\circ to make the total 180180^\circ. Therefore, the other angle is obtuse.
    • If one angle is obtuse (greater than 9090^\circ), then the other must be less than 9090^\circ to make the total 180180^\circ. Thus, the other angle is acute.

Since both scenarios involve one angle being acute and the other obtuse, we verify that the statement is correct.

Therefore, the statement is true.

Answer

True

Exercise #5

It is possible for two adjacent angles to be right angles.

Video Solution

Step-by-Step Solution

To determine if it is possible for two adjacent angles to be right angles, we start by considering the definition of adjacent angles. Adjacent angles share a common side and a common vertex. We must think about this scenario in terms of the angles lying on a straight line or a flat plane.

A right angle is exactly 9090^\circ. Hence, if we have two right angles that are adjacent, their measures would be:

  • First angle: 9090^\circ
  • Second angle: 9090^\circ

When these two angles are adjacent, as defined in the problem, their sum is:

90+90=180 90^\circ + 90^\circ = 180^\circ

Angles that are adjacent along a straight line add up exactly to 180180^\circ. Therefore, it is indeed possible for two adjacent angles to be both 9090^\circ. This configuration simply means that these two angles lie along a straight line, dividing it into two right angles.

Hence, the statement is True.

Answer

True

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