Next we will calculate the adjacent angles to 75^(∘) and 85^(∘). Since these angles form linear pairs, we can calculate their measures by equating their sum with 180^(∘) and solving for the adjacent angle.
m∠ α +75^(∘)=180^(∘)& ⇔ m∠ α = 105^(∘)
m∠ β +85^(∘)=180^(∘)& ⇔ m∠ β = 95^(∘)
Let's add these angles to the diagram.
b Like in Part A, we will start by finding the vertical angles to the given angles.
Next, we will calculate the adjacent angles to 110^(∘). Since these angles form a linear pair, we can calculate their measures by equating their sum with 180^(∘) and solving for the adjacent angle's measure.
m∠ θ +110^(∘)=180^(∘)& ⇔ m∠ θ = 70^(∘)
Let's add these angles to the diagram.
When we know two angles in the triangle, we can find the triangle's third angle by using the Triangle Angle Sum Theorem.
m∠ β+30^(∘)+70^(∘)= 180^(∘)
⇕
m∠ β= 80^(∘)
When we know the triangle's third angle, we can also determine it's vertical angle and adjacent angles with which it forms a linear pair.
Using the Alternate Interior Angles Theorem, we can identify two more angles. This also gives us enough information to identify the remaining two angles since they are vertical angles to the alternate interior angles we identify below.
