Question

In the given figure, EAD perpendicular to BCD. Ray FAC cuts ray EAD at a point A such that angle EAF = 30°. Also, in ∆BAC, angle BAC = x° and angle ABC = (x+10)°. Then, the value of x is
(a) 20 (b) 25
(c) 30 (d) 35

(JUSTIFY YOUR ANSWER)

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Answer 1

Answer:

In the given triangle the value of x is 25°

Step-by-step explanation:

Given: In a triangle EAD perpendicular to BCD.

∠EAF  = 30° ∠BAC = x° ∠ABC = (x+10)°

To find: The value of x°

Solution

In the given figure, ∠CAD = ∠EAF

∠CAD Vertically opposite ∠EAF

Given that ∠EAF  = 30°

Therefore ∠EAF  = ∠CAD = 30°

∠CAD = 30°

Angle sum property:

• According to the triangle's "Angle sum property," the sum of its internal angles is equal to  180°.
• The sum of the angles will always be 180° whether a triangle is an acute, obtuse, or a right triangle.
• The triangle ABC can be represented as, ∠A + ∠B + ∠C = 180°.

in the given triangle ∆ABD,

∠ABD + ∠BAD + ∠ADB = 180°

∠BAC = x°  

∠BAD =  (x° + 30°)

∠ABC = (x+10)°

∠ABD = (x+10)°

∠ADB = 90 (D is the perpendicular to C)

⇒ (x° + 10)° + (x° + 30°) + 90° = 180°

⇒x° + 10° +  x° + 30° + 90° = 180°

⇒ 2x° + 130° = 180°

⇒ 2x° = 180° − 130°

⇒ 2x° = 50°

⇒ x  = 25

Thus, the value of x  =  25°

Final answer:

In the given triangle the value of x is 25°

Answer 2

Answer: Option b is the correct answer, i.e., x= 25°.

Step-by-step explanation:

Concept:

• Vertically Opposite Angles: The opposite (X) angles of two intersecting lines are equal. The two green angles and the two yellow angles in the accompanying diagram are both equal. Because they are vertically opposing at a vertex, these X angles are known as vertically opposite angles.

• According to Theorem 1: Angle Sum Property of Triangle, the sum of a triangle's interior angles is 180°, parallel to the triangle's side BC. As a result, a triangle's interior angles add up to 180°.
• An angle that is equivalent to two right angles and whose sides extend in the same direction from its vertex.

Here we are given that,  EAD perpendicular to BCD. Ray FAC cuts ray EAD at a point A such that angle EAF = 30°. Also, in ∆BAC, angle BAC = x° and angle ABC = (x+10)°.

Now, we perform the following steps to find out the value of x.

Step 1: First we consider the ΔACD.

Here, we are said that the ray FAC cuts ray EAD at a point A. So the two rays can be considered as two transversals. Hence we can apply the concept of vertically oppsite angle and hence we obtain,

∠EAF = ∠CAD = 30° (Vertically opposite angles)

Step 2: Now by using the sum angle property of the ΔACD, we can write,

90° + 30°+ ∠ACD = 180°

⇒ ∠ACD = 60°

Step 3: By using the straight line angle property we get,

∠ACD+∠ACB = 180°

⇒60°+∠ACB = 180°

⇒∠ACB = 120°

Step 4: Now we consider the ΔABC. Here we use again the sum angle property of the triangle as follows:

$x+ (x+10)+120 = 180$

⇒ $2x+130=180$

⇒$x=$ 25°

Final ans: Value of x is 25° .

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