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In the figure given below, PQ || RS and ∠CTM = 110° and ∠BNU = 130° and I is incentre of ∆TVM and O is the orthocenter of ∆NVU, then what is the ratio of the measure of ∠TIM to the measure of ∠VOU?

In the figure given below, PQ || RS and ∠CTM = 110° and ∠BNU = 130° and I is incentre of ∆TVM and O is the orthocenter of ∆NVU, then what is the ratio of the measure of ∠TIM to the measure of ∠VOU? Correct Answer 12 : 13

Given:

PQ || RS and ∠CTM = 110° and ∠BNU = 130°

Calculation:

∠CTM = ∠VUS = 110°

⇒ ∠VUN = 180° – 110° = 70°

∠BNU = ∠VMQ = 130°

⇒ ∠VMT = 180° – 130° = 50°

In ∆TVM,

∠MTV = 180° – 110° = 70°

⇒ ∠TVM = 180° – 70° – 50° = 60°

∠TIM = 90° + (∠TVM/2)

⇒ ∠TIM = 90° + 30°

⇒ ∠TIM = 120°

In ∆NVU,

∠VNU = ∠VMT = 50°

∠VOU = 180° – ∠VNU

⇒ ∠VOU = 180° – 50°

⇒ ∠VOU = 130°

⇒ Required ratio = 120 : 130

∴ 12 : 13

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