Question

In triangle ABC of the figure , BD and CD are internal bisectors of angle B and angle C , respectively.Prove that 180° + y = 2x

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Answer 1

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Answer 2

Answer: To prove: 2x=180+y

Step-by-step explanation:

Given that:

A triangle ABC is given.

BD and CD are internal bisectors of angle B and C respectively.

To prove: 2x=180+y

For ΔABC;

We know that:

• sum of all the angles of a triangle is 180.

• So, $A+B+C=180^{0}$.

By adjusting;

$B+C=180^{0} -A$

Multiplying both the sides by 1/2, we get;

$\frac{1}{2}B+\frac{1}{2}C=90^{0} -\frac{1}{2}A$   (equation 1)

Now, for ΔBDC;

$x+\frac{1}{2} B+\frac{1}{2} C=180^{0}$   (sum of all angles of a triangle)

Substituting from the equation 1 in this equation, we get;

$x+90^{0}-\frac{1}{2}A=180^{0}$

$x-\frac{1}{2}A=90^{0}$

$x=90^{0}+\frac{1}{2}y$    (A=y from the figure in the question)

By solving above equation, we get;

$2x=180^{0}+y$

Hence proved.