Question

The interior angles of a quadrilateral are in an arithmetic sequence. If the largest angle is 96°, find the other angles.

Answer 1

Answer:

smallest angle = 84°

2nd angle = 88°

third angle = 92°

Step-by-step explanation:

Given:

The interior angles of a quadrilateral form an arithmetic sequence

The largest angle = 96°

To find:

the other angles

The sum of the interior angles of a quadrilateral = 360°

For an arithmetic sequence, there is a common difference

let the common difference = d

let the sequence:

a, a + d, a + 2d, a + 3d

a = first angle

a + 3d = last angle = 96°

if a + 3d = 96

a + 2d = 96 - common difference

a + 2d = 96 - d

a + d = 96 - 2d

a = 96 - 3d

The sequence:

96-3d, 96-2d, 96-d, 96

Sum of the angles in the interior:

`\begin{gathered} 96-3d\text{ + 96-2d + 96 - d + 96 = 360\degree} \\ \\ 96\text{ + 96 + 96 + 96 + -3d - 2d - d = 360} \\ \\ 384\text{ - 6d = 360} \end{gathered}`
`\begin{gathered} add\text{ 6 to both sides:} \\ 384\text{ - 6d + 6d = 360 + 6d} \\ \\ 384\text{ - 360 = 6d} \\ \\ 24\text{ = 6d} \\ \\ divide\text{ both sides by 6:} \\ d\text{ = 24/6 = 4} \end{gathered}`

substitute for d, to get the angle measure

`\begin{gathered} 1st\text{ angle = smallest angle = 96 - 3d} \\ smallest\text{ angle = 96-3\lparen4\rparen} \\ smallest\text{ angle = 96-3\lparen4\rparen= 84\degree} \\ \\ Next\text{ angle = 96 - 2d } \\ Next\text{ angle = 96 - 2\lparen4\rparen} \\ Next\text{ angle = 88\degree} \\ \\ Third\text{ angle = 96 - d} \\ Third\text{ angle = 96 - 4} \\ Third\text{ angle = 92\degree} \end{gathered}`