Question

1. auto racing some facts about the maximum banking angles of daytona international speedway at corners 1, 2, 3, and 4 are at the right. find m∠3. explain your steps. (banked corners are described on page 98.) m∠1 + m∠3 + m∠4 = 93° m∠2 + m∠4 = 62° m∠2 = m∠3 m∠1 = m∠2

Explanation:

Step1: Substitute equal - angle relationships

Since \(m\angle1 = m\angle2\) and \(m\angle2=m\angle3\), then \(m\angle1 = m\angle3\). The first equation \(m\angle1 + m\angle3 + m\angle4=93^{\circ}\) can be rewritten as \(m\angle3 + m\angle3 + m\angle4 = 93^{\circ}\), or \(2m\angle3 + m\angle4=93^{\circ}\). The second equation is \(m\angle2 + m\angle4 = 62^{\circ}\), and since \(m\angle2 = m\angle3\), it can be rewritten as \(m\angle3 + m\angle4 = 62^{\circ}\).

Step2: Solve the system of equations

Let \(x = m\angle3\) and \(y = m\angle4\). We have the system of equations \(

$$\begin{cases}2x + y=93^{\circ}\\x + y=62^{\circ}\end{cases}$$

\). Subtract the second - equation from the first equation: \((2x + y)-(x + y)=93^{\circ}-62^{\circ}\).
\[

$$\begin{align*} 2x + y - x - y&=93^{\circ}-62^{\circ}\\ x&=31^{\circ} \end{align*}$$

\]
Since \(x = m\angle3\), then \(m\angle3 = 31^{\circ}\).

Answer:

\(m\angle3 = 31^{\circ}\)