Question

find the values of x, y, and z. the diagram is not to scale.
46°
13°
58°
x° z° y°
o x = 63, y = 104, z = 76
o x = 76, y = 104, z = 63
o x = 76, y = 63, z = 104
o x = 63, y = 76, z = 104

Explanation:

Step1: Find the third - angle of the large triangle

The sum of angles in a triangle is 180°. For the large triangle with angles 46° and 58°, the third - angle is $180-(46 + 58)=76^{\circ}$. So $x = 76$.

Step2: Find angle $z$

For the smaller triangle with angles 13° and the angle we just found (76°), using the angle - sum property of a triangle, the third - angle $z$ is $180-(13 + 76)=91^{\circ}$ (This is wrong. Let's correct it. The outer - angle of a triangle is equal to the sum of the two non - adjacent interior angles). The outer - angle of the smaller triangle where one non - adjacent interior angle is 13° and the other is part of the large - triangle's angle. The outer - angle $x$ (76°) is the sum of 13° and $z$. So $z=76 - 13=63$.

Step3: Find angle $y$

Since $x + y=180$ (linear pair of angles), and $x = 76$, then $y = 180 - 76=104$.

Answer:

$x = 76, y = 104, z = 63$