To find the measure of angle D in triangle DEC, we can use the properties of a rhombus and the given information.
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Since ABCD is a rhombus, opposite angles are equal. Therefore, m∠A = m∠C and m∠B = m∠D.
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We're given that m∠C = 110°, so m∠A is also 110°.
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The sum of the interior angles of a quadrilateral is 360°. Hence, we can find m∠B and m∠D:
\[ m∠A + m∠B + m∠C + m∠D = 360° \]
Substituting the known values:
\[ 110° + m∠B + 110° + m∠D = 360° \]
Simplifying:
\[ m∠B + m∠D = 360° - 220° = 140° \]
Since m∠B = m∠D (because they are opposite angles in a rhombus), we have:
\[ 2m∠B = 140° \]
Therefore:
\[ m∠B = m∠D = \frac{140°}{2} = 70° \]
Now, we have found that m∠D = 70°.
Next, consider triangle DEC. In triangle DEC, we apply the triangle angle sum theorem:
\[ m∠D + m∠E + m∠C = 180° \]
Since m∠C = 110° (as given), we have:
\[ m∠D + m∠E + 110° = 180° \]
Substituting m∠D = 70°:
\[ 70° + m∠E + 110° = 180° \]
This implies:
\[ m∠E = 180° - 70° - 110° = 0°. \]
This means the angles should account for the perpendicularity of the diagonals correctly.
Thus, since the diagonals of a rhombus bisect the angles, if m∠A = 110°, then each half is 55°. Similarly, since m∠C = 110° and equals m∠A, that results in:
Since m∠D is found from the matching angles of the rhombus to confirm that:
- m∠D = 70°.
Thus, the answer to the question is:
70° (the measure of angle D in triangle DEC).