Question

Given that m∠AQB = 5x, m∠BQC = 8x - 24, and m∠AQC = 80°, find the value of 'x.' After finding the value of 'x,' determine whether QB→ is the angle bisector of ∠AQC. Please explain your reasoning in a short paragraph.

Answer 1

The value of 'x' is 8. After calculations, m∠AQB and m∠BQC are both equal to 40°, proving that QB→ is the angle bisector of ∠AQC.

Step-by-step explanation:

To find the value of 'x,' we use the given information about the angles:

• m∠AQB = 5x
• m∠BQC = 8x - 24
• m∠AQC = 80°

Since ∠AQC is made up of ∠AQB and ∠BQC, we can write an equation representing their relationship:

m∠AQC = m∠AQB + m∠BQC

This becomes:

80° = 5x + (8x - 24)

Combining like terms, we get:

80° = 13x - 24

Adding 24 to both sides yields:

104° = 13x

Dividing both sides by 13 gives us:

x = 8

Now, to determine whether QB→ is the angle bisector of ∠AQC, we plug x back into the original equations:

• m∠AQB = 5(8) = 40°
• m∠BQC = (8(8) - 24) = 40°

Since m∠AQB and m∠BQC are equal, QB→ is indeed the angle bisector of ∠AQC.