Find the measures of the 4 angles.

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°

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