Angle with bisector

Question

PQS 2x + 1. RQS 4x - 15

Answer 1

I'm assuming that you have a triangle PQR with angle bisector QS bisecting angle PQR.

Angle PQS=Angle RQS

Therefore 2x+1=4x-15

2x+1=4x-15

1=4x-2x-15

1=2x-15

1+15=2x

16=2x

2x=16

x=8

2x+1=16+1=17 degrees

4x-15=32-15=17 degrees

Answer 2

You don't specify whether line sement PQ is the angle bisector or if it is RQ.

Suppose the angle bisector is PQ. Then the measure of angle PQS is one-half of the measure of angle RQS:

2x+1 = (1/2) * (4x-15) = 2x -7.5

Subtracting 2x from both sides of the equation results in:

1 = -7.5 and that is a false statement. So PQ cannot be the angle bisector. That means RQ must be the angle bisector.

That leads to (1/2) * (2x+1) = 4x - 15, or

x + 0.5 = 4x - 15

Subtracting x from and adding 15 to both sides of the equation leads to:

3x = 15.5

Dividing both sides by the coefficient of x results in:

x = 31/6

We can check this value using a calculator and decimal fractions.

The mixed fraction value for this number is 5 1/6.

The measure of angle PQS is 34/3 (or 11 1/3) degrees.

The measure of angle RQS is 17/3 (or 5 2/3) degrees.

You can quickly verify that these are the correct answers by using a calculator and comparing the decimal fractions.