Problem
of the Week
Problem
B and Solution
Angle
Adventures

Problem

In the diagram below, \(AC\), \(BD\), \(EJ\), \(HI\), and \(FG\) are line segments. Determine the
measure of each unknown angle \(w\),
\(x\), \(y\), and \(z\).

Horizontal line segment \(AC\) is
above horizontal line segment \(EJ\).

Point \(B\) is on \(AC\) and a ray starting at B divides
straight angle \(ABC\) into two angles:
one angle measures \(60\degree\) and
the other angled is labelled \(x\).

Line segment \(BD\) intersects \(EJ\) forming four quadrants around the
point of intersection. The angle forming the top-left quadrant measures
\(90\degree\) and the angle forming the
bottom-right quadrant (or opposite quadrant) is labelled \(w\).

Line segment \(HI\) intersects \(EJ\) at a different point, forming four
quadrants around this point of intersection. The angle forming the
top-left quadrant measures \(90\degree\). Line segment \(FG\) also passes through this point of
intersection and divides the top-right and bottom-left quadrants into
two parts. In the top-right quadrant, the angle between \(EJ\) and \(FG\) measures \(40\degree\). In the bottom-left quadrant,
the angle between \(EJ\) and \(FG\) (opposite the \(40\degree\) angle) is labelled \(y\), and the angle between \(FG\) and \(HI\) is labelled \(z\).

Solution

Solution 1

Since \(\angle w\) is opposite to
\(90\degree\), we know \(\angle w = 90\degree\).

Since \(\angle x\) supplementary to
\(60\degree\), we know that \(\angle x = 180\degree - 60\degree =
120\degree\).

Since \(\angle y\) is opposite to
\(40\degree\), we know \(\angle y = 40\degree\).

We know that \(90\degree + \angle y +
\angle z = 180\degree\), so we must have \(\angle y + \angle z = 90\degree\). Since
\(\angle y = 40\degree\), we have \(\angle z = 50\degree\).

Solution 2

Since the diagram is drawn to scale, you may use a protractor to find
the angles.