Answer:
See explanation
Step-by-step explanation:
If [tex]PX\cong PY,[/tex] then triangle PXY is isosceles triangle. Angles adjacent to the base XY of an isosceles triangle PXY are congruent, so
[tex]\angle 1\cong \angle 2[/tex]
and
[tex]m\angle 1=m\angle 2[/tex]
Angles 1 and 3 are supplementary, so
[tex]m\angle 3=180^{\circ}-m\angle 1[/tex]
Angles 2 and 4 are supplementary, so
[tex]m\angle 4=180^{\circ}-m\angle 2[/tex]
By substitution property,
[tex]m\angle 4=180^{\circ}-m\angle 2=180^{\circ}-m\angle 1=m\angle 3[/tex]
Hence,
[tex]\angle 3\cong \angle 4[/tex]
Consider triangles APX and BPY. In these triangles:
- [tex]PX\cong PY[/tex] - given;
- [tex]\angle 5\cong \angle 6[/tex] - given;
- [tex]\angle 3\cong \angle 4[/tex] - proven,
so [tex]\triangle APX\cong \triangle BPY[/tex] by ASA postulate.
Congruent triangles have congruent corresponding sides, then
[tex]AP\cong BP[/tex]
Therefore, triangle APB is isosceles triangle (by definition).
