In each diagram, BD bisects &lt;ABC. Find m&lt;ABC. (2x+20) (4x)

Question

In each diagram, BD bisects &amp;lt;ABC. Find m&amp;lt;ABC. (2x+20) (4x)

fudzisako · Accepted Answer

To solve the problem, we need to find the measure of angle ABC, given that line BD bisects it. Let's denote the measure of angle ABC as

m ∠ A B C

.
According to the angle bisector theorem, when a line bisects an angle, it divides the angle into two congruent angles. In this case, since line BD bisects angle ABC, we have:

m ∠ A B D = m ∠ D B C

We are given the measures of angles ABD and DBC in terms of x:

m ∠ A B D = 2 x + 20

m ∠ D B C = 4 x

Since these two angles are congruent, we can set them equal to each other:

2 x + 20 = 4 x

To find the value of x, we can solve this equation:

2 x - 4 x = - 20

- 2 x = - 20

Dividing both sides by -2:

x = 10

Now that we have the value of x, we can substitute it back into one of the expressions for the angles to find the measure of angle ABC. Let's use the expression for angle DBC:

m ∠ A B C = m ∠ D B C = 4 x

Substituting x = 10:

m ∠ A B C = 4 \times 10

m ∠ A B C = 40

Therefore, the measure of angle ABC is 40 degrees.

In each diagram, BD bisects <ABC. Find m<ABC.

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