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
Identify the angles shown in the diagram below?
Examples with solutions for Vertically Opposite Angles
Step-by-step solutions included
Exercise #1
is parallel to
Determine which of the statements is correct.
Step-by-Step Solution
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.
Answer:
Colaterales Adjacent
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
In which of the diagrams are the angles vertically opposite?
Step-by-Step Solution
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.
Answer:
Video Solution
Exercise #4
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 #5
Does the diagram show an adjacent angle?
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
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.