+912 
The measure of ∠BCD is 120°. The measure of ∠ABC is 85°.
What is measure of ∠BAC?
+536 The measure of ∠ACB is 60°.
The measure of ∠ABC is 85°.
...So... ∠BAC is 35°
+2446 Another option is to utilize something called the Exterior Angle Theorem. This theorem states that the measure of the exterior angle of a triangle is equal to sum of the two nonadjacent interior (also known as remote) angles. The diagram below does all the speaking for me.
Now that this theorem is established, we can save one step. In the given diagram, m∠CAB+m∠ABC=m∠BCD
| m∠CAB+m∠ABC=m∠BCD | Exterior Angle Theorem |
| m∠CAB+85=120 | Substitution Property of Equality |
| m∠CAB=35∘ | Subtraction Property of Equality |
Why is this the case? Well, I'm happy to show you why!
In this triangle, one is given that this is △ABC with an exterior angle ∠BAD. Our goal here is to prove that m∠B+m∠C=m∠BAD. I will utilize a two-column proof.
Let's assume that m∠BAC=x∘.
| m∠BAC+m∠B+m∠C=180∘ | Triangle Sum Theorem |
| x+m∠B+m∠C=180∘ | Substitution Property of Equality |
| m∠B+m∠C=(180−x)∘ | Subtraction Property of Equality |
| ∠BAC and ∠BAD form a linear pair | Definition of a linear pair |
| ∠BAC and ∠BAD are supplementary | Linear Pair Theorem |
| m∠BAC+m∠BAD=180∘ | Definition of supplementary angles |
| x∘+m∠BAD=180∘ | Substitution Property of Equality |
| m∠BAD=(180−x)∘ | Subtraction Property of Equality |
| m∠B+m∠C=m∠BAD | Transitive Property of Equality |
