Exercise 10 Page 357

We want to find m∠ 4 in the following triangle and determine what mistake our friend might have made.

Let's start by recalling that the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. Next we will recall the definitions of an exterior angle and its remote interior angles.

1. An exterior angle of a triangle is an angle formed by a side an extension of an adjacent side.
2. For each exterior angle of a triangle, the two nonadjacent interior angles are its remote interior angles.

   In this case, ∠ 4 is an exterior angle. Let's mark its remote interior angles on the diagram!

   
   We can see that remote interior angles of ∠ 4 are ∠ 1 and ∠ 2. The measure of ∠ 4 is equal to the sum of its remote interior angles. m∠ 4 = m∠ 1 + m∠ 2 In this case, m∠ 1 = (8x+7) ^(∘), m∠ 2 = (4x+14)^(∘), and m∠ 4 = (13x+12) ^(∘). Let's substitute these values into the formula! (13x+12) ^(∘) = (8x+7)^(∘) + (4x+14) ^(∘) Now we can solve the obtained equation for x.
   (13x+12)=(8x+7)+(4x+14)

   RemovePar

   Remove parentheses

   13x+12=8x+7+4x+14

   ▼

   Solve for x

   AddTerms

   Add terms

   13x+12=12x+21

   SubEqn

   LHS-12x=RHS-12x

   13x+12-12x=12x+21-12x

   SubTerms

   Subtract terms

   x+12=21

   SubEqn

   LHS-12=RHS-12

   x+12-12=21-12

   SubTerms

   Subtract terms

   x=9
   We obtained that x=9 is a solution to the equation. Finally, we can calculate m∠ 4.
   m∠ 4 = (13x+12) ^(∘)

   Substitute

   x= 9

   m∠ 4 = [13( 9)+12] ^(∘)

   Multiply

   Multiply

   m∠ 4 = [117+12] ^(∘)

   AddTerms

   Add terms

   m∠ 4 = 129 ^(∘)
   We got that m∠ 4 = 129 ^(∘). Our friend incorrectly says that m∠ 4 = 51 ^(∘). Let's find out what mistake they could make. To do so, we will find the measure of ∠ 3. Note that ∠ 3 and ∠ 4 form a straight angle and are supplementary. m∠ 3 + m∠ 4 = 180^(∘) [0.4em] ⇕ [0.4em] m∠ 3 = 180^(∘) - m∠ 4 Now we can find the measure of ∠ 3 using the correct measure of ∠ 4.
   m∠ 3 = 180^(∘) - m∠ 4

   Substitute

   m∠ 4= 129^(∘)

   m∠ 3 = 180^(∘) - 129^(∘)

   SubTerm

   Subtract term

   m∠ 3 = 51^(∘)
   We obtained that the measure of ∠ 3 is equal to the measure found by our friend. Therefore, they could mistake ∠ 4 for ∠ 3.