### Video Transcript

π΄π΅πΆπ· is a
parallelogram. If π΄πΆ equals 13 centimeters,
π΄π· equals 13 centimeters, and π·πΆ equals five centimeters, what is the type
of triangle π΄π·πΆ?

We see that triangle π΄π·πΆ is
an isosceles triangle. And recalling that the angle in
a triangle with the greatest measure is opposite the longest side, in triangle
π΄π·πΆ the angles at πΆ and π·, which are equal, will have the largest
measure. Choosing either one of the
angles at πΆ and π·, we can use the Pythagorean inequality theorem to confirm
that these angles are acute.

Taking the angle at π· to work
on, this theorem tells us three things. First, that if the square of
the longest side is greater than the sum of the squares of the other two sides,
then the angle opposite the longest side is an obtuse angle. Second, if the square of the
longest side is less than the sum of squares of the other two sides, the angle
is acute. And third, if the square of the
longest side is equal to the sum of the squares of the other two, then the angle
opposite is a right angle.

In our case, we have π΄πΆ
squared, that is 13 squared, equals 169 and that π΄π· squared plus π·πΆ squared
equals 13 squared plus five squared. And thatβs equal to 194. Hence, π΄πΆ squared is less
than π΄π· squared plus π·πΆ squared. And so angle πΆπ·π΄ is an acute
angle. Angle π΄πΆπ· is the same, so
this is also acute. And since these angles have the
largest measure in triangle π΄π·πΆ, angle πΆπ΄π· must be smaller than them. Hence, the third angle, angle
πΆπ΄π· is also acute. Since all three angles are
acute and, in particular, the angle with the largest measure is acute, triangle
π΄π·πΆ is an acute triangle.