Answer:
∠DQE = 65°
Step-by-step explanation:
You want to know the measure of Angle DQE in the given figure.
Right triangle
Triangle ADE is inscribed in a semicircle, so is a right triangle. That makes acute angle AED complementary to acute angle EAD. The latter is given as 50°, so ...
∠AED = 90° -50° = 40°
Tangent
We know that tangent line QE is perpendicular to diameter AE, so angle AEQ is a right angle. Segment ED divides that right angle into complementary angles, so ...
∠DEA +∠DEQ = 90°
∠DEQ = 90° -∠DEA = 90° -40° = 50°
Isosceles triangle
A.pex angle DEQ in isosceles triangle DEQ being 50° means that the base angles QDE and DQE will have measure ...
∠DQE = (180° -50°)/2 = 130°/2
∠DQE = 65°
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Alternate solution
Let angles DQE and QDE be represented by x. Then the sum of angles in quadrilateral ADQE is 360°:
50° +(90+x)° +x° +90° = 360°
230° +2x = 360° . . . . . . simplify
x = 130°/2 = 65° . . . . . solve for x
