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# In the Given Figure, Rs is a Diameter of the Circle. Nm is Parallel to Rs and ∠Mrs = 29°. Calculate: (I) ∠Rnm, (Ii) ∠Nrm - ICSE Class 10 - Mathematics

ConceptArc and Chord Properties - Angle in a Semi-circle is a Right Angle

#### Question

In the given figure, RS is a diameter of the circle. NM is parallel to RS and ∠MRS = 29°.
Calculate:
(i) ∠RNM,
(ii) ∠NRM

#### Solution

(i) Join RN and MS.

∴ ∠RMS = 90°
(Angle in a semicircle is a right angle)
∴ ∠RSM = 90° - 29° = 61°
(By angle sum property of triangle RMS)
∴ ∠RNM =180° ∠RSM =180° - 61° = 119°
(pair of opposite angles in a cyclic quadrilateral are supplementary)

(ii) Also, RS || NM
∴ ∠NMR = ∠MRS = 29°      (Alternate angles)
∴ ∠NMS = 90° + 29° = 119°
Also, ∠NRS + ∠MS = 180°
(pair of opposite angles in a cyclic quadrilateral are supplementary)
⇒ ∠NMR + 29° +119° = 180°
⇒ ∠NRM = 180° -148°
∴ ∠NRM = 32°

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Solution In the Given Figure, Rs is a Diameter of the Circle. Nm is Parallel to Rs and ∠Mrs = 29°. Calculate: (I) ∠Rnm, (Ii) ∠Nrm Concept: Arc and Chord Properties - Angle in a Semi-circle is a Right Angle.
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