Question

. ∠ABC is adjacent to ∠CBD. If the m∠ABC=4x+23, m∠CBD=6x+7, and m∠ABD=130°, what is the measure of angle ABC?

Answer 1

By applying the vertical angles theorem, the measure of angle ABC is equal to 55°.

In Mathematics and Euclidean Geometry, the vertical angles theorem states that two opposite (adjacent) vertical angles that are formed whenever two lines intersect each other are always congruent, which means being equal to each other.

By applying the vertical angles theorem to the lines, we have the following congruent angles:

m∠ABC ≅ m∠CBD

4x + 23 = 6x + 7

6x - 4x = 23 - 7

2x = 16

x = 8

Now, we can find the measure of angle ABC;

m∠ABC = 4(8) + 23

m∠ABC = 32 + 23

m∠ABC = 55°

Answer 2

The measure of angle ABC is

`63 \degree`

Explanation:

As illustrated in the diagram,we can conclude that m<ABC+m<CBD=m<ABD

This implies that

`4x + 23 + 6x + 7 = 130 \degree`

`4x+6x+23+7 =130 \degree`

`10x + 30=130 \degree`

`10x=130 \degree - 30 \degree`

`10x=100 \degree`

Dividing through by 10, we obtain

`x=10 \degree`

This implies

m<ABC=4x+27

`= 4(10)+23`

`=40+23`

`=63 \degree`

Therefore the measure of angle ABC is

`63 \degree`