Angles in Parallel Lines - Examples, Exercises and Solutions

Remember that adjacent angles are angles that are formed when two lines intersect one another.

These angles are created at the point of intersection, one adjacent to the other, and that's where their name comes from.

Adjacent angles always complement one another to one hundred and eighty degrees, meaning their sum is 180 degrees. 

Let's remember that vertical angles are angles that are formed when two lines intersect. They are are created at the point of intersection and are opposite each other.

Alternate angles are a pair of angles that can be found on the opposite side of a line that cuts two parallel lines.

Furthermore, these angles are located on the opposite level of the corresponding line that they belong to.

First let's remember that corresponding angles can be defined as a pair of angles that can be found on the same side of a transversal line that intersects two parallel lines.

Additionally, these angles are positioned at the same level relative to the parallel line to which they belong.

Let's review the definition of adjacent angles:

Adjacent angles are angles formed where there are two straight lines that intersect. These angles are formed at the point where the intersection occurs, one next to the other, and hence their name.

Now let's review the definition of collateral angles:

Two angles formed when two or more parallel lines are intersected by a third line. The collateral angles are on the same side of the intersecting line and even are at different heights in relation to the parallel line to which they are adjacent.

Therefore, answer C is correct for this definition.

To solve this problem, consider the following explanation:

When dealing with the concept of corresponding angles, we are typically considering two parallel lines cut by a transversal. The property of corresponding angles states that if two lines are parallel, then any pair of corresponding angles created where a transversal crosses these lines are equal.

Given the problem: If one of the corresponding angles is a right angle, we need to explore if this necessitates that the other corresponding angle is also a right angle.

Let’s proceed with the steps to solve the problem:

• Step 1: Recognize that we are discussing corresponding angles formed by a transversal cutting through two parallel lines.
• Step 2: Apply the property that corresponding angles are equal when lines are parallel. This means if one angle in such a pair is a right angle, then the other must be equal to it.
• Step 3: Since a right angle measures $90^\circ$, the other corresponding angle must also measure $90^\circ$ since they are equal by the property of corresponding angles.

Therefore, based on the equality of corresponding angles when lines are parallel, if one corresponding angle is a right angle, the other angle will also be a right angle.

The final conclusion for the problem is that the statement is True.

To solve this problem, we need to understand the properties of vertically opposite angles:

• Vertically opposite angles are the angles that are opposite each other when two lines intersect.
• One key property of vertically opposite angles is that they are always equal in measure.
• An acute angle is defined as an angle that is less than $90^\circ$.
• An obtuse angle is defined as an angle that is greater than $90^\circ$.

Given that vertically opposite angles are equal, if one angle is acute, the opposite angle must also be acute. This contradicts the statement in the problem that if one is acute, the other will be obtuse.

Therefore, the correct analysis of the problem reveals that the statement is incorrect.

Thus, the solution to the problem is False.

Remember the definition of angles opposite by the vertex:

Angles opposite by the vertex are angles whose formation is possible when two lines cross, and they are formed at the point of intersection, one facing the other. The acute angles are equal in size.

The drawing in answer A corresponds to this definition.

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 $180^\circ$.
• Step 2: According to the problem, neither angle is a right angle, meaning neither is $90^\circ$.
• Step 3: Given this constraint, analyze the possibilities:
• If one angle is acute (less than $90^\circ$), then the other must be more than $90^\circ$ to make the total $180^\circ$. Therefore, the other angle is obtuse.
• If one angle is obtuse (greater than $90^\circ$), then the other must be less than $90^\circ$ to make the total $180^\circ$. 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.

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 $90^\circ$. Hence, if we have two right angles that are adjacent, their measures would be:

• First angle: $90^\circ$
• Second angle: $90^\circ$

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

Angles that are adjacent along a straight line add up exactly to $180^\circ$. Therefore, it is indeed possible for two adjacent angles to be both $90^\circ$. This configuration simply means that these two angles lie along a straight line, dividing it into two right angles.

Hence, the statement is True.

To determine if two adjacent angles can both be obtuse, we first need to recall the definition of an obtuse angle and what it means for angles to be adjacent.

• An obtuse angle measures more than $90^\circ$ but less than $180^\circ$.
• Adjacent angles are two angles that share a common side and vertex.
• When we consider adjacent angles that form a linear pair, their sum must equal $180^\circ$.

For two angles to both be obtuse, each must measure more than $90^\circ$. Let's consider two angles, $a$ and $b$, that are adjacent and both obtuse:

• $a > 90^\circ$
• $b > 90^\circ$

Adding both inequalities gives:

$a + b > 180^\circ$

This sum $a + b$ would contradict the requirement that adjacent angles forming a linear pair sum to exactly $180^\circ$.

Therefore, two adjacent angles cannot both be obtuse, as their sums would exceed the allowable amount for a linear pair.

Thus, it is not possible for two adjacent angles to be obtuse. The correct answer is False.

Remember the definition of adjacent angles:

Adjacent angles always complement each other up to one hundred eighty degrees, that is, their sum is 180 degrees.

This situation is impossible since a right angle equals 90 degrees, an obtuse angle is greater than 90 degrees.

Therefore, together their sum will be greater than 180 degrees.

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.

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.

To solve this problem, let's first understand the concept of adjacent angles. Adjacent angles share a common vertex and a common side. When two adjacent angles are formed by two intersecting lines, they often form what is known as a linear pair.

According to the Linear Pair Postulate, if two angles form a linear pair, then the sum of these adjacent angles is $180$ degrees. This is because these angles lie on a straight line, effectively forming a straight angle, which measures $180$ degrees.

Let's apply this knowledge to the statement in the problem:
The statement says, "The sum of adjacent angles is 180 degrees." In the context of the Linear Pair Postulate, this is indeed correct as adjacent angles that create a linear pair sum to $180$ degrees.

Therefore, when the statement refers specifically to linear pairs, it is true.

Thus, the solution to the problem is True.