The value is m∠XQW=24° , WZ=18, and m∠XRZ=48°.
The angle bisectors of a triangle are concurrent, meaning that they all meet at a single point. This point is called the incenter of the triangle.
In the diagram, PQR is a triangle with angle bisectors PZ, QZ, and RZ. These angle bisectors meet at the incenter, Z.
We are given that YZ=17, QZ=18, m∠WPY=92° , and m∠XQZ=24° . We are asked to find m∠XQW, WZ, and m∠XRZ.
To find m∠XQW:
Since QZ is the angle bisector of ∠PQZ, we know that ∠PQZ=∠ZQW. We are also given that m∠WPY=92° , so m∠PQZ=92° −24° = 68° . Therefore, m∠ZQW = 68° .
Since XZ is the angle bisector of ∠PZX, we know that ∠PZX=∠XQW. We are also given that m∠XQZ=24° , so m∠PZX=24°. Therefore, m∠XQW=24°.
To find WZ:
We can use the Angle Bisector Theorem to find the ratio of the side lengths QZ and WZ. The Angle Bisector Theorem states that the ratio of the side lengths of a triangle is equal to the ratio of the segments of the opposite side created by the angle bisector.
In this case, the Angle Bisector Theorem tells us that
QZ / WZ = PQ / PZ
. We are given that QZ=18 and PQ=17, so we can substitute these values into the equation to get
18 / WZ = 17 / 17 . Solving for WZ, we get 18 / WZ= 17 / 17
To find m∠XRZ:
Since PZ is the angle bisector of ∠RPZ, we know that ∠RPZ=∠ZPR. We are also given that m∠XQZ=24° , so m∠RPZ=24°.
Therefore, m∠XRZ=24° + 24° = 48°.