Video Transcript
In the figure, segments 𝐸𝐷 and
𝐵𝐺 meet at 𝐴. If the measure of angle 𝐹 equals
90 degrees, the measure of angle 𝐺 equals 97 degrees, the measure of angle 𝐸
equals 97 degrees, the measure of angle 𝐷 equals 137 degrees, and the measure of
angle 𝐶 equals 95 degrees, find the measure of angle 𝐵.
Looking at the figure, it consists
of two connected quadrilaterals, 𝐴𝐵𝐶𝐷 and 𝐴𝐸𝐹𝐺. We’ve been given some information
about the measures of some of the interior angles in these quadrilaterals. So let’s begin by adding this
information to the figure. The measure of angle 𝐹 is 90
degrees. The measures of angles 𝐺 and 𝐸
are each 97 degrees. The measure of angle 𝐷 is 137
degrees. And finally the measure of angle 𝐶
is 95 degrees. Using all this information, we’re
then asked to find the measure of angle 𝐵.
Now, looking at the figure, we can
identify that in addition to angle 𝐵, there are actually only two other angles
whose measures are unknown: angle 𝐷𝐴𝐵 and angle 𝐺𝐴𝐸. Furthermore, as these two angles
are formed by the intersection of two straight lines, they are vertically opposite
angles. And so their measures must be
equal. In quadrilateral 𝐴𝐸𝐹𝐺, angle
𝐺𝐴𝐸 is the only unknown angle. So we can calculate its measure by
recalling the formula for the sum of the interior angle measures in a polygon.
In general, the sum of the measures
of the interior angles in an 𝑛-sided polygon, which we denote as 𝑆 sub 𝑛, is
equal to 𝑛 minus two multiplied by 180 degrees. A quadrilateral has four sides. So substituting 𝑛 equals four
gives 𝑆 sub four equals 360 degrees. This should be a familiar fact. The sum of the interior angle
measures in a quadrilateral is 360 degrees.
We can now form an equation using
the interior angle measures in polygon 𝐴𝐸𝐹𝐺. The measure of angle 𝐺𝐴𝐸 plus 97
degrees plus 90 degrees plus 97 degrees equals 360 degrees. Simplifying on the left-hand side
gives the measure of angle 𝐺𝐴𝐸 plus 284 degrees equals 360 degrees. The measure of angle 𝐺𝐴𝐸 is
therefore equal to 360 degrees minus 284 degrees, which is 76 degrees. Recalling from earlier that angles
𝐷𝐴𝐵 and 𝐺𝐴𝐸 are vertically opposite angles and hence are of equal measure, it
follows that the measure of angle 𝐷𝐴𝐵 is also 76 degrees.
We now know the measures of three
of the interior angles in polygon 𝐴𝐵𝐶𝐷. And so we can use the sum of the
interior angle measures in a quadrilateral again to calculate the measure of angle
𝐵. We have that the measure of angle
𝐵 plus 95 degrees plus 137 degrees plus 76 degrees is equal to 360 degrees. This simplifies to the measure of
angle 𝐵 plus 308 degrees equals 360 degrees. And hence the measure of angle 𝐵
is equal to 360 degrees minus 308 degrees, which is 52 degrees.
So, we’ve completed the
problem. Using the fact that vertically
opposite angles are of equal measure and the sum of the interior angle measures in a
quadrilateral twice, we’ve found that the measure of angle 𝐵 is 52 degrees.
