Math

QuestionFind 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 and mABC=(25x+34)m \angle A B C=(25 x+34)^{\circ}.

Studdy Solution

STEP 1

Assumptions1. The vector BDundefined\overrightarrow{B D} bisects ABC\angle A B C. . The measure of ABC\angle A B C is given by (25x+34)(25 x+34)^{\circ}.

STEP 2

Since BDundefined\overrightarrow{B D} bisects ABC\angle A B C, it means that ABD\angle A B D and CBD\angle C B D are equal. We can express this mathematically asmABD=mCBDm \angle A B D = m \angle C B D

STEP 3

We also know that the sum of ABD\angle A B D and CBD\angle C B D is equal to the measure of ABC\angle A B C. We can express this mathematically asmABD+mCBD=mABCm \angle A B D + m \angle C B D = m \angle A B C

STEP 4

Substitute the given measure of ABC\angle A B C into the equationmABD+mCBD=(25x+34)m \angle A B D + m \angle C B D = (25 x+34)^{\circ}

STEP 5

Since mABD=mCBDm \angle A B D = m \angle C B D, we can substitute mABDm \angle A B D for mCBDm \angle C B D in the equation2mABD=(25x+34)2m \angle A B D = (25 x+34)^{\circ}

STEP 6

To find the measure of ABD\angle A B D, we divide both sides of the equation by2mABD=(25x+34)2m \angle A B D = \frac{(25 x+34)^{\circ}}{2}

STEP 7

Since mABD=mCBDm \angle A B D = m \angle C B D, the measure of CBD\angle C B D is also (25x+34)2\frac{(25 x+34)^{\circ}}{2}.

STEP 8

The measure of ABC\angle A B C is given by (25x+34)(25 x+34)^{\circ}.
So, the measures of the angles aremABD=(25x+34)2m \angle A B D = \frac{(25 x+34)^{\circ}}{2}mCBD=(25x+34)2m \angle C B D = \frac{(25 x+34)^{\circ}}{2}mABC=(25x+34)m \angle A B C = (25 x+34)^{\circ}

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