Math

QuestionFind the measures of angles mABDm \angle A B D, mCBDm \angle C B D, and mABCm \angle A B C given that BDundefined\overrightarrow{B D} bisects ABC\angle A B C.

Studdy Solution

STEP 1

Assumptions1. BDundefined\overrightarrow{B D} bisects ABC\angle A B C. This means that ABD\angle A B D and CBD\angle C B D are equal. . The sum of the measures of the angles in a triangle is180 degrees.

STEP 2

Let's denote the measure of ABD\angle A B D and CBD\angle C B D as xx (since they are equal). Then, we can write the following equationABD=CBD=x\angle A B D = \angle C B D = x

STEP 3

Since BDundefined\overrightarrow{B D} bisects ABC\angle A B C, the measure of ABC\angle A B C is the sum of the measures of ABD\angle A B D and CBD\angle C B D. We can write this asABC=ABD+CBD\angle A B C = \angle A B D + \angle C B D

STEP 4

Substitute the values of ABD\angle A B D and CBD\angle C B D from2 into the equation from3ABC=x+x\angle A B C = x + x

STEP 5

implify the equation from4 to find the measure of ABC\angle A B CABC=2x\angle A B C =2x

STEP 6

We don't know the exact values of xx and ABC\angle A B C yet, but we know their relationship ABC\angle A B C is twice as large as ABD\angle A B D and CBD\angle C B D.
To find the exact values, we would need more information about the triangle, such as the measure of another angle or the length of a side.
Without additional information, we can only say thatmABD=mCBD=xm \angle A B D = m \angle C B D = xmABC=2xm \angle A B C =2x

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