a Acute angles have measures that are less than 90∘.
B
b Obtuse angles have measures that are greater than 90∘ and less than 180∘.
C
c Where is the intersection located with respect to the sides?
A
aDrawing:
What appears to be true?: The angle bisectors intersect inside of the triangle and is equidistant from the sides of the triangle.
B
bDrawing:
What appears to be true?: The angle bisectors intersect inside of the triangle and is equidistant from the sides of the triangle.
C
cConjecture: Angle bisectors of any triangle will intersect in the triangle and be equidistant from the sides of the triangle.
Practice makes perfect
a A triangle with three acute angles is unsurprisingly called an acute triangle. This means all of the angles are less than 90∘. Let's draw an arbitrary acute triangle.
Next, we will show how to create an angle bisector for one of the angles, ∠A. To do this, we will have to use a compass. Put the compass point on ∠A and draw an arc that intersects the sides that make up ∠A.
Next, we will put the compass point on D and E and draw two arcs using the same compass settings. The line from A and through the intersection point of these arcs, is the angle bisector to ∠A.
If we repeat this procedure for the remaining angles, we can draw all of the angle bisectors.
We notice one thing and one thing only. The point where the angle bisectors intersect, appears to be equidistant from the triangle's three sides.
b A triangle where one of the angles is obtuse is called an obtuse triangle. This means one of the angles are greater than 90∘. Note that this necessarily makes the remaining two angles less than 90∘. Let's draw an arbitrary acute triangle.
Next, we will show how to create an angle bisector for one of the angles, ∠A. To do this, we will have to use a compass. Put the compass point on ∠A and draw an arc that intersects the sides of ∠A.
Next, we will put the compass point on D and E and draw two arcs using the same compass settings. The line from A and through the intersection point of these arcs, is the angle bisector to ∠A.
If we repeat this procedure for the remaining angles, we can draw all of the angle bisectors.
We notice one thing and one thing only. The point where the angle bisectors intersect, appears to be equidistant from the triangle's three sides.
c With each of the triangles in the previous parts of the exercise, the angle bisectors all intersect inside the triangle. The point of intersection also appears to be the same distance from all of the sides of the triangle. The measures of the angles of the triangle do not seem to affect the location of the point of intersection.
Mathleaks uses cookies for an enhanced user experience. By using our website, you agree to the usage of cookies as described in our policy for cookies.
