Questions: Given the diagram, if Z is the incenter of triangle WXY, m angle WYX=86°, and m angle WXZ=33°, find the following measures: m angle WYZ= m angle WXY= m angle YWX= m angle ZWX=

Given the diagram, if Z is the incenter of triangle WXY, m angle WYX=86°, and m angle WXZ=33°, find the following measures:
m angle WYZ=
m angle WXY=
m angle YWX=
m angle ZWX=

Transcript text: Given the diagram, if $Z$ is the incenter of $\triangle W X Y, m \angle W Y X=86^{\circ}$, and $m \angle W X Z=33^{\circ}$, find the following measures: \[ m \angle W Y Z=\square \] \[ m \angle W X Y= \] $\square$ \[ m \angle Y W X= \] $\square$ \[ m \angle Z W X= \] $\square$

Solution

Solution Steps

Step 1: Find m∠WYZ

Since Z is the incenter, YZ bisects ∠WYX. Therefore, m∠WYZ = m∠WYX / 2. Given m∠WYX = 86°, m∠WYZ = 86° / 2 = 43°

Step 2: Find m∠WXY

The sum of the angles in a triangle is 180°. In ΔWXY, we have m∠WXY + m∠YWX + m∠WYX = 180° m∠YWX = m∠WXZ + m∠ZWX Given m∠WXZ = 33° We know m∠WYX = 86° Therefore, m∠WXY + (33° + m∠ZWX) + 86° = 180° m∠WXY + m∠ZWX = 180° - 86° - 33° m∠WXY + m∠ZWX = 61°

Since Z is the incenter, XZ bisects ∠YXW. Therefore, m∠WXZ= m∠YXZ=33°. Similarly, WZ bisects ∠YWX and YZ bisects ∠WYX. The measure of an angle formed by an angle bisector is equal to half the measure of the bisected angle. The measure of an angle formed by two angle bisectors is equal to 90° plus half the measure of the third angle. Therefore, m∠WXY = 1/2 (180° - 86°)= 1/2(94°) = 47° m∠YWX = 1/2 (180° - m∠X) = 1/2 (94°) = 47° + 16°= 63° So, m∠WXY = 61° - m∠ZWX= 47°

Step 3: Find m∠YWX

m∠YWX = 2 * m∠WXZ = 2 * 33° = 66°