Question

21 write and solve an equation based on the diagram. then find the measure of each angle. notes: vertical angles are equal/congruent. angles that are a linear pair are supplementary. *supplementary is two angles that add up to 180 degrees. a ((x + 16)^circ + (4x - 5)^circ = 90); the angles are (31.8^circ) and (58.2^circ) . b ((x + 16)^circ + (4x - 5)^circ = 180); the angles are (49.8^circ) and (130.2^circ) c ((x + 16)^circ = (4x - 5)^circ); the angles are each (23^circ) d ((x + 16)^circ = (4x - 5)^circ); the angles are each (14^circ) show hint

Explanation:

Step1: Analyze vertical angles

Vertical angles are equal, so set \((x + 16)^\circ=(4x - 5)^\circ\).

Step2: Solve the equation

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Wait, this is wrong. Wait, maybe the angles are vertical angles? Wait, no, maybe I misread. Wait, let's check option C: \((x + 16)=(4x - 5)\), solving: \(x+16 = 4x - 5\), \(16 + 5=4x - x\), \(21 = 3x\), \(x = 7\)? No, option C says angles are each \(23^\circ\). Wait, maybe I made a mistake. Wait, let's check option C: if \(x+16 = 4x - 5\), then \(16 + 5=4x - x\), \(21 = 3x\), \(x = 7\), then \(x + 16=23\), \(4x - 5=28 - 5 = 23\). Oh, right! I miscalculated \(4x\) when \(x = 7\)? No, \(4\times7=28\), \(28 - 5 = 23\). So \(x = 7\), then \(x + 16=23\), \(4x - 5=23\). So the equation is \((x + 16)^\circ=(4x - 5)^\circ\), and angles are each \(23^\circ\), which is option C.

Answer:

C. \((x + 16)^\circ=(4x - 5)^\circ\); the angles are each \(23^\circ\)