In triangle ABC, ∠ABC=30∘,∠ACB=60∘. D is a point in triangle ABC such that DB and DC bisect angles ABC and ACB respectively. What is the measure (in degrees) of ∠BDC?
Consider the sum of the angles in triangle BDC. The two acute angles of that triangle are 60/2 and 30/2.
�ÚBDC is the remaining obtuse angle.
�ÚBDC = 180 - (15 + 30) = 135 ∘
To find the measure of angle BDC, we can start by applying the angle bisector theorem.
The angle bisector theorem states that if a point D lies on the interior of an angle ABC, then it divides the angle into two smaller angles, with the measures of these angles inversely proportional to their opposite side lengths.
In triangle ABC, angle BDC is formed by the bisectors of angles ABC and ACB, which means that BD is the bisector of angle ABC and DC is the bisector of angle ACB.
Since angle ABC measures 30 degrees, and BD bisects this angle, the measure of angle ABD is half of 30 degrees, which is 15 degrees.
Similarly, since angle ACB measures 60 degrees, and DC bisects this angle, the measure of angle DCA is half of 60 degrees, which is 30 degrees.
Now, we can find the measure of angle BDC by adding the measures of angles ABD and DCA:
∠BDC = ∠ABD + ∠DCA
∠BDC = 15 degrees + 30 degrees
∠BDC = 45 degrees
Therefore, the measure of angle BDC is 45 degrees.
To find the measure of ∠BDC, we can use the angle bisector theorem.
The angle bisector theorem states that in a triangle, if a line bisects an angle, it divides the opposite side into two segments that are proportional to the adjacent sides.
In this case, we have angle bisectors BD and CD, which divide side AC into two segments. Let's denote the lengths of these two segments as x and y, respectively.
Based on the angle bisector theorem, we can set up the following proportion:
(x/y) = (AB/BC)
We know that ∠ABC = 30° and ∠ACB = 60°. Let's use these angles to find the lengths of AB and BC.
In triangle ABC, the internal angles must add up to 180°. So, we can find ∠CAB by subtracting the given angles from 180°:
∠CAB = 180° - ∠ABC - ∠ACB
= 180° - 30° - 60°
= 90°
Now, we have a right triangle. We can use trigonometric functions to find the lengths of AB and BC. Let's use sine:
sin(30°) = Opposite/Hypotenuse
sin(30°) = AB/AC
AB = AC * sin(30°)
AB = BC, as they are opposite sides of the same angle.
Therefore, BC = AC * sin(30°)
Now, we can substitute the lengths of AB and BC into the proportion we set up earlier:
(x/y) = (AB/BC)
(x/y) = (AC * sin(30°))/(AC * sin(30°))
The AC terms cancel out, and we are left with x/y = 1.
Since x and y represent the lengths of the segments BD and CD, respectively, we can conclude that BD = CD.
Therefore, ∠BDC is an isosceles triangle's angle, and its base angles are congruent.
Let's denote the measure of ∠BDC as α. Since it is an isosceles triangle, we know that each base angle is (180° - α)/2.
Therefore, ∠BDC = (180° - α)/2
Now, we can solve for α. Since we know that ∠ABC = 30° and ∠ACB = 60°, we can find the measure of ∠BAC:
∠BAC = 180° - ∠ABC - ∠ACB
= 180° - 30° - 60°
= 90°
In triangle BAC, we have an angle (∠BDC) and two adjacent angles (∠ABC and ∠ACB). Therefore, we can find α:
α = ∠BAC - ∠ABC - ∠ACB
= 90° - 30° - 60°
= 0°
Therefore, the measure of ∠BDC is 0°.