Question

In the adjacent figure, DO and CO are the bisectors of angle D and angle C, respectively. Find angle DOC and angle A​

Answer 1

$\blue\bigstar$Answer:

$\angle$DOC = 135° and $\angle$A = 150°

$\blue\bigstar$ Given:

• DO is the bisector of $\angle$D
• CO is the bisector of $\angle$C
• $\angle$ADO = 20°
• $\angle$DCO = 25°
• $\angle$ABC = 120°

$\blue\bigstar$To find:

• $\angle$DOC
• $\angle$A

$\blue\bigstar$ Solution:

$\because$ DO is the bisector of $\angle$ADC,

$\therefore\boxed{\angle ADO \:= \:\angle ODC \: = 20°\:}$

In $\triangle$ODC,

$\because$$\angle$ODC + $\angle$DCO + $\angle$DOC = 180°

(Angle Sum Property Of A Triangle)

$\implies$ 20° + 25° + $\angle$DOC = 180°

(By substituting their values)

$\implies$ 45° + $\angle$DOC = 180°

$\implies$ $\angle$DOC = 180° - 45°

$\implies$ $\boxed{\angle DOC\: = \:135°\:}$

$\therefore$ $\angle$DOC = 135°

So, $\angle$D = $\angle$ADO + $\angle$ODC

$\implies$ $\angle$D = 20° + 20°

( $\because$ $\angle$D is bisected, therefore the bisected angles are equal)

$\implies$ $\boxed{\angle D\: = 40°\:}$

$\therefore$ $\angle$D = 40°

$\angle$C = $\angle$ OCD + $\angle$ OCB

$\implies$C = 25° + 25°

( $\because$ $\angle$C is bisected, therefore the bisected angles are equal)

$\implies$ $\boxed{\angle C \:= 50°\:}$

$\therefore$ $\angle$C = 50°

Now in Quadrilateral ABCD,

$\because$ $\angle$ A + $\angle$ B + $\angle$C + $\angle$D = 360°

( Angle Sum Property Of A Quadrilateral)

$\implies$ $\angle$A + 120° + 50° + 40° = 360°

$\implies$ $\angle$A + 210° = 360°

$\implies$ $\angle$A = 360° - 210°

$\implies$ $\boxed{\angle A\:= \:150°\:}$

$\therefore$ $\angle$DOC = 135° and $\angle$A = 150°

$\pink\bigstar$ Concepts Used:

• Angle Sum Property Of A Triangle
• Angle Sum Property Of A Quadrilateral
• When an angle is bisected, then both the angles will be equal.

Answer 2

Step-by-step explanation:

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