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Answer 1

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1. Given, ∠SPR = 135° and ∠PQT = 110°
To find ∠PRQ.

∠SPR = 135°
∠PQT = 110°
∠P = 180° - 135° = 45° ( linear pair )

∠Q = 180° - 110° = 70° ( linear pair )

∠RPQ + ∠PQR + ∠PRQ = 180° ( Angle Sum Property )
45° + 70° + ∠PRQ = 180°
115° + ∠PRQ = 180°
∠PRQ = 180° - 115° = 65°

2. Given, ∠X = 62° and ∠XYZ = 54°
To find ∠OZY and ∠YOZ.

∠X + ∠Y + ∠Z = 180° ( Angle Sum Property )
62° + 54° + ∠Z = 180°
∠Z + 116° = 180°
∠Z = 180° - 116° = 64°

∠OYZ = 54/2 = 27° ( YO is the bisector of ∠XYZ )

∠OZY = 64/2 = 32° ( OZ is the bisector of ∠XYZ )

∠OYZ + ∠OZY + ∠YOZ = 180° ( Angle Sum Property )
27° + 32° + ∠YOZ = 180°
59° + ∠YOZ = 180°
∠YOZ = 180° - 59° = 121°

3. Given, ∠BAC = 35° and ∠CDE = 53°
To find ∠DCE.

∠E = 35° ( Alternate Angle )

∠D + ∠E + ∠C = 180° ( Angle Sum Property )
53° + 35° + ∠C = 180°
88° + ∠C = 180°
∠C = 180° - 88° = 92°

Hope this helps.

Have a nice day =)

Answer 2

Hi there !

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Your answer is in the attachment. Kindly refer to it.

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Let's see the properties used here :

☛ ᏞᏆNᎬᎪᎡ ᏢᎪᏆᎡ ᎪXᏆᎾᎷ - If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and vice versa. This property is called as the linear pair axiom.

☛ ᏢᎡᎾᏢᎬᎡᎢY - If two lines intersect each other, then the vertically opposite angles are equal.

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Thanks for the question !

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