One important concept in geometry is the angles formed by intersecting lines. Some various characteristics and relationships can be identified between the angles formed by intersecting lines. This article will explore the specific types of angles and their properties. Through clear examples and step-by-step explanations, you will learn how to identify and analyze these angles and apply this knowledge to solve challenging problems.
The Linear Pair
When two lines intersect, the angles that are adjacent to each other and share a common vertex are known as linear pairs.
When there are several intersecting lines, we can identify several pairs of angles that are adjacent to each other, but not all of these pairs can be identified as linear pairs. Linear pairs are adjacent angles that form a straight line, i.e they share one common line and one other uncommon line.
This is described mathematically using the linear pair theorem. The linear pair theorem says that the measures of the two angles that make up a linear pair sum up to 180 degrees.
Example
Example 1. Identify the measures of all the angles formed by the intersecting lines
Answer:
Step 1: Identify Linear pairs
Since ∠BOD is known, it’s best to try and identify the linear pairs that consist of ∠BOD.
We an see that ∠BOC is adjacent to ∠BOD, hence it forms a linear pair.
Similarly ∠AOD is adjacent to ∠BOD, and forms a straight line, hence ∠AOD and ∠BOD form a linear pair as well.
Step 2: Use the linear pair theorem to build relationships between angles
The linear pair theorem states that the sum of measures of angles of a linear pair is 180 degrees. We can apply this theorem to the linear pairs identified in step 1.
∠BOC+∠BOD=180°
∠AOD+∠BOD=180°
Step 3: Solve the equations
By substituting for the measure of BOD (43 °) we can calculate both ∠BOC and ∠AOD as follows.
∠BOC=180°−∠BOD
∠BOC=137°
∠AOD=180°−∠BOD
∠AOD=137°
We’ve figured out 2 of the three unknown angles. Finally we can use the fact that the angles around a point is 360 to determine the size of AOC.
∠AOC+∠BOC+∠AOD+∠BOD=360°
∠AOC=360°−∠BOC−∠AOD−∠BOD
∠AOC=360°−137°−137°−43°
∠AOC=43°
The Linear Pair Perpendicular Theorem
The linear pair perpendicular theorem is an extension of the linear pair theorem, and it states that if the angles of a linear pair are of equal size, then they are both right angles. The converse of the theorem would show that the angles formed at the intersection of two perpendicular lines are linear pairs.
Vertical Angles
Vertical angles are a pair of non-adjacent angles formed when two lines intersect. They are opposite to each other, and their measures are always equal. Vertical angles can be seen in various shapes, polygons, geometrical objects, etc. and it is used to deduce other theorems such as the triangle sum theorem and exterior angle theorem.
Proof of the vertical angle theorem
To prove the vertical angle theorem, i.e the measure of vertical angles are equal, we can consider a pair of intersecting lines as shown below.
Using the definition of vertical angles we can first identify a pair of vertical angles.
Next, let's figure out how to answer the question “Are vertical angles congruent?”.
First consider angles ∠BOC and ∠BOD, which form a linear pair. Since they are a linear pair, the following statement is true.
∠BOC+∠BOD=180°
Similarly for angles BOD and AOD we can build a relationship:
∠AOD+∠BOD=180°
By combining statement 1 and statement 2, we can show that:
∠BOC−∠AOD=0°
∠BOC=∠AOD
∠BOC and ∠AOD are vertical angles, and the above proof shows they are congruent.
Similarly, by selecting a different linear pair you can also prove the vertical angle theorem for the vertical angles ∠AOC and ∠BOD as well.
Example
Example 2. Determine the measure of α
Solution:
Step 1. Identify the vertical angles
There are several vertical angle pairs formed by these intersecting lines. But to solve for alpha, we have to choose a pair that includes alpha and some known angles. According to that the vertical angles ∠AOE and ∠BOF are suitable.
Step 2. Apply the vertical angle theorem
From the vertical angle theorem we know that vertical angles are congruent. Hence, the relationship between the vertical angles chosen in step 1 can be given as:
∠AOE=∠BOF
Step 3: Substitute and solve for α
∠AOE=∠BOF
90°+α=119°
α=119°−90°
α=29°
Question
Quiz on angles formed by intersecting lines
Find the missing angles
Solution:
Step 1. Identify vertical angle pairs
∠EOB and ∠AOF are vertical angles, and so are ∠DOF and ∠COE. Therefore using the vertical angle theorem we can state that:
∠EOB=∠AOF
∠DOF=∠COE,
Hence ∠ COE and ∠ AOF are of size 50° and 68° respectively.
Step 2. Isolate the variable
∠BOC and ∠BOD form a linear pair, and so does ∠COF and ∠DOF. The linear pair theorem can be applied to these angles.
∠BOC+∠BOD=180°−−>(1)
Since ∠ BOC = ∠ BOE + ∠ COE, we can rewrite the equation as follows:
∠BOE+∠COE+∠BOD=180°−−>(1′)
Step 3. Substitute the angles found in Step 1 in the equations derived in Step 2
For equation 1:
68°+50°+∠BOD=180°
118°+∠BOD=180°
∠BOD=62°
As BOD and AOC are vertical angles, they are congruent. Hence, AOC is 62 degrees.
