Before going deeper into opposite angles, we will pause a moment to visualize the types of scenarios where this type of angle can be found. To make it easier to understand, we will draw two parallel straight lines cut by a secant or transversal, as shown in the following illustration:
Angles in Parallel Lines Practice Problems - Opposite & Corresponding
Master angles in parallel lines with practice problems on opposite angles, corresponding angles, alternate angles, and collateral angles. Step-by-step solutions included.
📚Master Angles in Parallel Lines with Guided Practice
- Identify opposite angles formed by intersecting lines and apply the equal angle property
- Calculate corresponding angles when a transversal cuts through parallel lines
- Solve for unknown angles using alternate angle relationships in parallel line systems
- Apply collateral angle properties to find supplementary angle measures
- Work with complex geometric figures involving parallelograms and trapezoids
- Use angle relationships to solve multi-step problems with algebraic expressions
Understanding Vertically Opposite Angles
Complete explanation with examples
What are opposite angles?
What do we see here? The transversal intersects with each one of the straight lines and (in our case and are parallel, although this is not required in order to get opposite angles).
With this example in mind, we are ready to move on to the formal definition of opposite angles, which will help us to identify them more easily:
Opposite angles are a pair of angles that arise when two straight lines intersect. These angles are formed at the point of intersection (which we will call the vertex), one in front of the other. Opposite angles are equal.
In the following illustration, we can see two examples of opposite angles, the first pair is marked in red and the second pair in blue.
Practice Vertically Opposite Angles
Test your knowledge with 48 quizzes
Does the diagram show an adjacent angle?
Examples with solutions for Vertically Opposite Angles
Step-by-step solutions included
Exercise #1
If one of two corresponding angles is a right angle, then the other angle will also be a right angle.
Step-by-Step Solution
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 , the other corresponding angle must also measure 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.
Answer:
True
Video Solution
Exercise #2
It is possible for two adjacent angles to be right angles.
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 . Hence, if we have two right angles that are adjacent, their measures would be:
- First angle:
- Second angle:
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 . Therefore, it is indeed possible for two adjacent angles to be both . 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
Video Solution
Exercise #3
The sum of adjacent angles is 180 degrees.
Step-by-Step Solution
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 degrees. This is because these angles lie on a straight line, effectively forming a straight angle, which measures 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 degrees.
Therefore, when the statement refers specifically to linear pairs, it is true.
Thus, the solution to the problem is True.
Answer:
True
Video Solution
Exercise #4
If one vertically opposite angle is acute, then the other will be obtuse.
Step-by-Step Solution
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 .
- An obtuse angle is defined as an angle that is greater than .
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.
Answer:
False
Video Solution
Exercise #5
Does the drawing show an adjacent angle?
Step-by-Step Solution
Adjacent angles are angles whose sum together is 180 degrees.
In the attached drawing, it is evident that there is no angle of 180 degrees, and no pair of angles can create such a situation.
Therefore, in the drawing there are no adjacent angles.
Answer:
Not true
Video Solution
Frequently Asked Questions
What are opposite angles and how do I identify them?
+Opposite angles are formed when two straight lines intersect, creating angles that face each other at the intersection point. They are always equal in measure and can be identified by their position directly across from each other at the vertex.
How do I solve problems with angles in parallel lines?
+When working with parallel lines cut by a transversal, use these key relationships: 1) Corresponding angles are equal, 2) Alternate angles are equal, 3) Collateral angles sum to 180°, 4) Opposite angles at intersections are equal.
What's the difference between corresponding and alternate angles?
+Corresponding angles are on the same side of the transversal and same relative position on each parallel line. Alternate angles are on opposite sides of the transversal but between the parallel lines (alternate interior) or outside them (alternate exterior).
How do I calculate unknown angles in parallel line problems?
+Set up equations using angle relationships: equal angles for corresponding/alternate/opposite angles, or supplementary relationships (sum = 180°) for collateral angles. Substitute known values and solve algebraically.
What are collateral angles in parallel lines?
+Collateral angles (also called co-interior or same-side interior angles) are on the same side of a transversal cutting parallel lines, but on opposite sides of the parallel lines themselves. They are supplementary, meaning they always add up to 180°.
How do opposite angles help solve geometry problems?
+Opposite angles are always equal, so if you know one angle at an intersection, you automatically know its opposite angle. This property is especially useful in problems involving intersecting diagonals in parallelograms, rhombuses, and other quadrilaterals.
What angle relationships exist when parallel lines are cut by a transversal?
+Four key relationships exist: 1) Corresponding angles are equal, 2) Alternate interior angles are equal, 3) Alternate exterior angles are equal, 4) Co-interior (collateral) angles are supplementary (sum to 180°).
How do I apply angle properties in parallelogram diagonal problems?
+When diagonals intersect in a parallelogram, they create opposite angles that are equal. Use this property along with triangle angle sum (180°) and given measurements to find unknown angles systematically.