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29 In the figure, ABCD is a parallelogram and CDE is a triangle. /_EDA=52^(@) and /_BCE=19^(@). Find /_DEC.

2929 In the figure, ABCD A B C D is a parallelogram and CDE C D E is a triangle. ∠EDA=52∘ \angle E D A=52^{\circ} and ∠BCE=19∘ \angle B C E=19^{\circ} . Find ∠DEC \angle D E C .

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Q. 2929 In the figure, ABCD A B C D is a parallelogram and CDE C D E is a triangle. ∠EDA=52∘ \angle E D A=52^{\circ} and ∠BCE=19∘ \angle B C E=19^{\circ} . Find ∠DEC \angle D E C .
  1. Identify Relationship: Identify the relationship between angles in a parallelogram; opposite angles are equal.
  2. Angle Equality in Parallelogram: Since ABCDABCD is a parallelogram, angle ABCABC equals angle CDACDA.
  3. Calculate Angle CDA: Angle CDA is supplementary to angle EDA because they are consecutive angles in a parallelogram, so angle CDA =180∘−52∘=128∘= 180^\circ - 52^\circ = 128^\circ.
  4. Calculate Angle DEC: In triangle CDE, angles CDE and EDA are supplementary to angle DEC, so angle DEC = 180∘−angle CDE−angle EDA.180^\circ - \text{angle CDE} - \text{angle EDA}.
  5. Find Angle CDECDE: But we need to find angle CDECDE first, which is equal to angle BCEBCE since they are alternate interior angles created by a transversal cutting parallel lines BCBC and DEDE.
  6. Substitute and Calculate DEC: So, angle CDE=CDE = angle BCE=19∘BCE = 19^\circ.
  7. Substitute and Calculate DEC: So, angle CDE=BCE=19°CDE = BCE = 19°.Now we can find angle DECDEC by substituting the values we have: angle DEC=180°−128°−19°=33°DEC = 180° - 128° - 19° = 33°.

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