Questions: Find the values of (x, y), and (z). The diagram is not to scale. (x=76, y=104, z=63) (x=76, y=63, z=104) (x=63, y=76, z=104) (x=63, y=104, z=76)

Solution Steps

The angles in a triangle add up to 180°. In the larger triangle, we have angles of 46°, 58°, and x + 13°. So, 46° + 58° + x + 13° = 180°. Simplifying, we have 117° + x = 180°, so x = 63°.

z and 13° are adjacent angles that form a straight line with the other side of the triangle, so they are supplementary angles. This means z + 13° = 180°, so z = 180° - 13° = 76° = 167°. However, since x + z + y is along a straight line, they should be supplementary angles. We can see that x = 63 and z must be less than x, thus making the first calculation for z incorrect. Instead, we know that x and the angle made up of the sum of z and 13 are supplementary angles. This means, x + z + 13 = 180 or 63 + z + 13 = 180 which gives us z = 76.

Since x, z, and y lie on a straight line, they must add up to 180°. We found x = 63° and z = 76°, so 63° + 76° + y = 180°. This simplifies to 139° + y = 180°, so y = 41°. Alternatively, since we know y and the sum of the two other angles form a straight line, we have y + 46 + 58 = 180 or y + 104 = 180 or y = 76. Since y must be greater than x or z, this makes the latter equation for calculating y more suitable. Another way to calculate y is to know that y + x + z = 180 so y + 63 + 13 = 180 or y = 104. Since y must be greater than x or z, this makes the last equation more plausible.

Final Answer: