Question

mathematical connections in exercises 27 and 28, (a) write an equation that represents the sum of the angle measures of the triangle and (b) use your equation and the equation shown to find the values of x and y. 27. triangle with x°, right angle, y°, and equation x + 2 = 3y 28. triangle with (y − 18)°, y°, x°, and equation 3x − 5y = −22

Explanation:

Problem 27

Step 1: Sum of angles in triangle

A right triangle has one angle of \(90^\circ\), so the sum of angles is \(x + 90 + y = 180\), simplifying to \(x + y + 90 = 180\) or \(x + y = 90\).

Step 2: Solve the system of equations

We have the system:
\[

$$\begin{cases} x + y = 90 \\ x + 2 = 3y \end{cases}$$

\]
From the second equation, \(x = 3y - 2\). Substitute into the first equation:
\(3y - 2 + y = 90\)
\(4y - 2 = 90\)
\(4y = 92\)
\(y = 23\)
Then \(x = 3(23) - 2 = 69 - 2 = 67\)

Step 1: Sum of angles in triangle

The sum of angles in a triangle is \(180^\circ\), so \((y - 18) + y + x = 180\), simplifying to \(x + 2y - 18 = 180\) or \(x + 2y = 198\).

Step 2: Solve the system of equations

We have the system:
\[

$$\begin{cases} x + 2y = 198 \\ 3x - 5y = -22 \end{cases}$$

\]
From the first equation, \(x = 198 - 2y\). Substitute into the second equation:
\(3(198 - 2y) - 5y = -22\)
\(594 - 6y - 5y = -22\)
\(594 - 11y = -22\)
\(-11y = -616\)
\(y = 56\)
Then \(x = 198 - 2(56) = 198 - 112 = 86\)

Answer:

For part (a): \(x + 90 + y = 180\) (or \(x + y = 90\))
For part (b): \(x = 67\), \(y = 23\)

Problem 28