Question

Using the image below, identify the angles that are vertical angles, list the angles that are linear pairs, and then solve for , , and .

Answer 1

x and z are vertical angles

y and 76 are vertical angles

x and 76 are linear pairs

x and y are linear pairs

z and 76 are linear pairs

y and z are linear pairs

y = 76°

x = 104°

z = 104°

How to find the vertical angles?

Vertical angles are a pair of non-adjacent angles formed by the intersection of two lines. They are always equal to each other, or in other words, congruent.

Thus:

x and z are vertical angles

y and 76 are vertical angles

A linear pair is a pair of adjacent angles that are supplementary, meaning they add up to 180 degrees.

Thus:

x and 76 are linear pairs

x and y are linear pairs

z and 76 are linear pairs

y and z are linear pairs

Thus:

y = 76°

x = 180 - 76 = 104°

z = 104°

Answer 2

Vertically opposite angles are
`x,z;y,76^(\circ)`

Linear pair of angles are
`x,y;y,z;z,76^(\circ);x,76^(\circ)`

`y=76^(\circ)`,
`z=x=104^(\circ)`

Explanation:

Given: image

To find: vertical angles, the angles that form linear pairs, value of x, y and z

Solution:

If two lines intersect each other then the vertically angles formed are equal.

Two adjacent angles are said to be linear if their sum is
`180^(\circ)`.

From the given image,

vertically opposite angles are
`x,z;y,76^(\circ)`

Linear pair of angles are
`x,y;y,z;z,76^(\circ);x,76^(\circ)`

As vertically opposite angles are equal,
`y=76^(\circ)`

As sum of angles that form a linear pair is
`180^(\circ)`,

`x+76^(\circ)=180^(\circ)\\x=180^(\circ)-76^(\circ)\\=104^(\circ)`

Also, as x and z are vertically opposite angles,

`z=x=104^(\circ)`