x = 5, m∠GPN = 40°, m∠SPH = 40°, m∠NPH = 140°, m∠GPS = 140°
Explanation:
Given data:
∠GPN = (5 + 7x)° and ∠SPH = (6x + 10)°
∠GPN and ∠SPH are vertically opposite angles.
Vertical angle theorem,
If two lines are intersecting, then vertically opposite angles are congruent.
∠GPN ≅ ∠SPH
(5 + 7x)° = (6x + 10)°
5° + 7x° = 6x° + 10°
Subtract 5° on both sides.
7x° = 6x° + 5°
Subtract 6x° on both sides.
x° = 5°
x = 5
m∠GPN = (5 + 7(5))°
m∠GPN = 40°
m∠SPH = (6(5) + 10)°
m∠SPH = 40°
Adjacent angles in a straight line is 180°.
m∠GPN + m∠NPH = 180°
40° + m∠NPH = 180°
m∠NPH = 180° – 40°
m∠NPH = 140°
∠GPS and ∠NPH are vertically opposite angles.
By vertical angle theorem, they are congruent.
∠GPS ≅ ∠NPH
m∠GPS = 140°
