To find the measure of angle C in triangle BCD using the Triangle Angle Sum Theorem, we can follow these steps:
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Understand the Triangle Angle Sum Theorem: This theorem states that the sum of the angles in a triangle is always 180 degrees. So for triangle BCD, we have: \[ \angle B + \angle C + \angle D = 180^\circ \]
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Identify the angles:
- We are given \( \angle D = 22^\circ \).
- We also know that angle A is the exterior angle at vertex B (i.e., \( \angle ABC \)), which is 103 degrees.
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Use the Exterior Angle Theorem: The Exterior Angle Theorem states that an exterior angle is equal to the sum of the two opposite interior angles. Therefore: \[ \angle ABC = \angle B + \angle C \]
Since \( \angle ABC = 103^\circ \): \[ \angle B + \angle C = 103^\circ \]
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Now set up the equations: We have the two equations:
- \( \angle B + \angle C = 103^\circ \) (1)
- \( \angle B + \angle C + \angle D = 180^\circ \) (2)
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Substituting the value of \( \angle D \): We know \( \angle D = 22^\circ \), so substituting that into equation (2): \[ \angle B + \angle C + 22^\circ = 180^\circ \] Simplifying gives: \[ \angle B + \angle C = 180^\circ - 22^\circ \] \[ \angle B + \angle C = 158^\circ (3) \]
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Now we can solve the system of equations (1) and (3): From equation (1): \[ \angle B + \angle C = 103^\circ \] From equation (3): \[ \angle B + \angle C = 158^\circ \]
Setting both equal: \[ 103^\circ + 22^\circ = 180^\circ \]
Since \( \angle D \) remains constant, to find \( C \): Substituting \( \angle D = 22^\circ \) in any equation derived: From (1): \[ \angle C = 103^\circ - \angle B \]
From (3): \(\angle C = 158 - 22 = 136^\circ\)
Finally: By placing all these together to find the final output: \[ \angle C = 180^\circ - \angle B - 22^\circ \]
Since there is nothing left, summing correctly will yield \( C = 103 \), verbally verifying that: \[ \angle C = 158^\circ - 22^\circ = 136^\circ \]
Thus confirming, The measure of angle C is: \[ \boxed{136} \] degrees.