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
Which type of angles are shown in the figure below?
Examples with solutions for Vertically Opposite Angles
Step-by-step solutions included
Exercise #1
Does the diagram show an adjacent angle?
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
Video Solution
Exercise #2
Is it possible to have two adjacent angles, one of which is obtuse and the other right?
Step-by-Step Solution
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.
Answer:
No
Video Solution
Exercise #3
Does the diagram show an adjacent angle?
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
Video Solution
Exercise #4
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
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.