m ∠1 + m ∠S + m ∠2 =180, m ∠4+ m ∠3 + m ∠7 = 180, m ∠5+ m ∠6 + m ∠V =180
2.
Triangle Angle-Sum Theorem
3.
m ∠1+ m ∠S+ m ∠2+ m ∠4+ m ∠3 + m ∠7+ m ∠5+ m ∠6 + m ∠V =540
3.
Addition Property
4.
m ∠VRS = m ∠1 + m ∠4+ m ∠5, m ∠TUV = m ∠7 + m ∠6, m ∠STU = m ∠2+ m ∠3
4.
Angle Addition Property
5.
m ∠S+ m ∠STU+ m ∠TUV+ m ∠V+ m ∠VRS =540
5.
Substitution Property of Equality
Practice makes perfect
Let's begin with reviewing the idea of a two-column proof. It lists each statement on the left, and the justification is on the right. Each statement must follow logically from the steps before it.
In this case, we are given that RSTUV is a pentagon. This how we will begin our proof!
Statement1)& RSTUV is a pentagon Reason1)& Given
From the graph, we can notice that our pentagon is divided into three triangles. Let's name the angles in each of them.
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!
Statements
Reasons
1.
RSTUV is a pentagon
1.
Given
2.
m ∠1 + m ∠S + m ∠2 =180, m ∠4+ m ∠3 + m ∠7 = 180, m ∠5+ m ∠6 + m ∠V =180
2.
Triangle Angle-Sum Theorem
3.
m ∠1+ m ∠S+ m ∠2+ m ∠4+ m ∠3 + m ∠7+ m ∠5+ m ∠6 + m ∠V =540
3.
Addition Property
4.
m ∠VRS = m ∠1 + m ∠4+ m ∠5, m ∠TUV = m ∠7 + m ∠6, m ∠STU = m ∠2+ m ∠3
