Question

what are the values of y and u? diagram: a triangle with angle ( y - 12^circ ), and a lower triangle with angle ( y + 32^circ ) and ( y ), with ( u ) between the two triangles on a vertical line.

Explanation:

Step1: Use triangle angle sum property

In a triangle, the sum of interior angles is \(180^\circ\). For the larger triangle, angles are \(y - 12^\circ\), \(y\), and the right angle? Wait, no, looking at the diagram, the straight line implies \(u+(y + 32^\circ)=180^\circ\) (linear pair), and in the triangle, \((y - 12^\circ)+y+(y + 32^\circ)=180^\circ\)? Wait, no, let's re - examine. The triangle has angles \(y-12^\circ\), \(y\), and the angle adjacent to \(u\) and \(y + 32^\circ\). Wait, actually, the two angles \(y-12^\circ\), \(y\) and the angle that is supplementary to \(y + 32^\circ\) (since \(u\) and \(y + 32^\circ\) are linear pair, so the angle in the triangle is \(180^\circ-(y + 32^\circ)=u\)? Wait, no, let's use the exterior angle theorem or angle sum.

Wait, the sum of angles in a triangle: Let's consider the triangle with angles \(y-12^\circ\), \(y\), and the angle at the vertex where \(u\) is formed. Wait, actually, the angle at that vertex is equal to \(180^\circ - u\), and also, since \(u\) and \(y + 32^\circ\) are supplementary (linear pair), \(u=180^\circ-(y + 32^\circ)\). Also, in the triangle, the sum of angles is \(180^\circ\), so \((y-12^\circ)+y+(180^\circ-(y + 32^\circ))=180^\circ\).

Simplify the left - hand side:
\(y-12 + y+180 - y - 32=180\)
Combine like terms:
\((y + y - y)+(180-12 - 32)=180\)
\(y + 136 = 180\)

Step2: Solve for y

Subtract 136 from both sides:
\(y=180 - 136\)
\(y = 44\)

Step3: Solve for u

Since \(u\) and \(y + 32^\circ\) are supplementary (linear pair), \(u + (y + 32^\circ)=180^\circ\). Substitute \(y = 44\):
\(u+(44 + 32)=180\)
\(u+76 = 180\)
Subtract 76 from both sides:
\(u=180 - 76\)
\(u = 104\)

Answer:

\(y = 44\), \(u=104\)