Exercise 43 Page 342

Now, we can use the Triangle Angle-Sum Theorem and write that the sum of the measures of the angles in each triangle is equal to 180. Statement2) & m ∠ 1 + m ∠ S + m ∠ 2 = 180 & m ∠ 4+ m ∠ 3 + m ∠ 7 =180 & m ∠ 5 + m ∠ 6 + m ∠ V =180 Reason2)& Triangle Angle-Sum Theorem Next, we can use the Addition Property and add the left-hand sides and the right-hand sides together. m ∠ 1 + m ∠ S + m ∠ 2 + m ∠ 4+ m ∠ 3 + m ∠ 7 + m ∠ 5 + m ∠ 6 + m ∠ V = 180 +180+180 ⇕ m ∠ 1 + m ∠ S + m ∠ 2 + m ∠ 4+ m ∠ 3 + m ∠ 7 + m ∠ 5 + m ∠ 6 + m ∠ V = 540 As we can see, the measures of all the angles add up to 540. Statement3)& m ∠ 1 + m ∠ S + m ∠ 2 + & m ∠ 4+ m ∠ 3 + m ∠ 7 + & m ∠ 5 + m ∠ 6 + m ∠ V =540 Reason3)& Addition Property Notice that by the Angle Addition Postulate we can conclude that m ∠ VRS is the sum of the measures of ∠ 1, ∠ 4, and ∠ 5. Similarly, m ∠ TUV is the sum of the measures of ∠ 7 and ∠ 6, and m ∠ STU is the sum of the measures of ∠ 2 and ∠ 3. Statement4)& m ∠ VRS = m ∠ 1 + m ∠ 4+ m ∠ 5 & m ∠ TUV = m ∠ 7 + m ∠ 6 & m ∠ STU = m ∠ 2 + m ∠ 3 Reason4)& Angle Addition Property Now, let's rearrange our equation so that we can identify these sums. m ∠ 1 + m ∠ S + m ∠ 2 + m ∠ 4+ m ∠ 3 + m ∠ 7 + m ∠ 5 + m ∠ 6 + m ∠ V = 540 ⇕ m ∠ S + m ∠ 2 + m ∠ 3 + m ∠ 7 + m ∠ 6 + m ∠ V + m ∠ 1 + m ∠ 4+ m ∠ 5 =540 Next, we can substitute m ∠ STU, m ∠ TUV, and m ∠ VRS, and for m ∠ 2+ m ∠ 3, m ∠ 7 + m ∠ 6, and m ∠ 1 + m ∠ 4 + m ∠ 5 respectively. This gives us what we wanted to prove! Statement5) & m ∠ S + m ∠ STU + m ∠ TUV+ & m ∠ V + m ∠ VRS =540 Reason5)& Substitution Prop. of Equality Finally, we can complete our two-column table!