Questions: m angle ABC=40° and overrightarrowBD is the angle bisector of angle ABC y=[?]

m angle ABC=40° and overrightarrowBD is the angle bisector of angle ABC y=[?]
Transcript text: $m \angle A B C=40^{\circ}$ and $\overrightarrow{B D}$ is the angle bisector of $\angle A B C$ $y=[?]$
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Solution

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Solution Steps

Step 1: Set up the equation

Since BD is the angle bisector of ∠ABC, we know that m∠ABD = m∠DBC. We are given that m∠ABC = 40°. Therefore, m∠ABD = m∠DBC = 20°. We are given the following expressions for the angles: m∠ABD = 3x - 1 m∠DBC = 34 - 2x

Since both angles equal 20°, we set their expressions equal to each other: 3x - 1 = 34 - 2x

Step 2: Solve for x

Add 2x to both sides of the equation: 5x - 1 = 34

Add 1 to both sides: 5x = 35

Divide by 5: x = 7

Step 3: Substitute the value of x into the expression for angle BCD

Since ray BD bisects angle ABC, Ray CD must bisect angle ACB by the Angle Bisector Theorem. Therefore, Angle BCD equals 2 times angle ACD. We are given that Angle ACD = 5y-18 and Angle BCD= 3y+6. Using the relation from the angle bisector theorem, we get 2(5y-18)=3y+6

10y-36=3y+6

7y=42

y=6

Final Answer: The value of y is 6.

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