Exercise 45 Page 54

Angles ∠ VWX and ∠ UWV are adjacent and on a straight line. Therefore, these angles are supplementary which means their measures sum to 180^(∘). The same thing applies to ∠ TWU and ∠ TWX. lcm∠ VWX+m∠ UWV=180^(∘) & (I) m∠ TWU+m∠ TWX=180^(∘) & (II) Additionally, ∠ TWU and ∠ UWV are adjacent and complementary which means their measures sum to 90^(∘). m∠ TWU+m∠ UWV=90^(∘) (III) Since we are given m∠ VWX, we can solve for m∠ UWV by substituting the known angle into Equation (I).

m∠ VWX+m∠ UWV=180^(∘)

Substitute

m∠ VWX= 130^(∘)

130^(∘)+m∠ UWV=180^(∘)

SubEqn

LHS-130^(∘)=RHS-130^(∘)

m∠ UWV=50^(∘)

Having calculated m∠ UWV, we can solve for m∠ TWU in Equation (III).

m∠ TWU+m∠ UWV=90^(∘)

Substitute

m∠ UWV= 50^(∘)

m∠ TWU+ 50^(∘)=90^(∘)

SubEqn

LHS-50^(∘)=RHS-50^(∘)

m∠ TWU=40^(∘)

Finally, we have enough information to find m∠ TWX using Equation (II).

m∠ TWU+m∠ TWX=180^(∘)

Substitute

m∠ TWU= 40^(∘)

40^(∘)+m∠ TWX=180^(∘)

SubEqn

LHS-40^(∘)=RHS-40^(∘)

m∠ TWX=140^(∘)

Let's summarize! m∠ UWV&=50^(∘) m∠ TWU&=40^(∘) m∠ TWX&=140^(∘)