Question

please solve this problem​

Answer 1

$\large \red{\bf \: ⟼ Given :- } ✍$

AD = BD = AC

$\large \red{\bf \: ⟼ To \: Find :- } ✍$

∠C or ∠4

$\large \red{\bf \: ⟼ Solution :- } ✍$

❥︎ In triangle ABD

$\bf \: ⟼ AD = BD$

⟼ As, angles opposite to equal sides are equal.

$\bf \: ⟼ ∠1 = ∠2$

Also, Using exterior angle property in triangle ABD, we get

$\bf \: ⟼ ∠3 = ∠1 + ∠2$

⟼On Substituting the value of ∠1, we get

$\bf \: ⟼ ∠3 = ∠2 + ∠2$

$\bf \: ⟼ ∠3 = 2∠2 \: ⟼ \: (1)$

❥︎ Now, In triangle ADC

$\bf \: ⟼ AD = AC$

As, angle opposite to equal sides are always equal.

$\bf \: ⟼ ∠3 = ∠4 \: ⟼ \: (2)$

⟼Also, using angle sum property, we get

$\large \red{\bf \: ⟼ ∠3 + ∠4 + ∠DAC = 180}$

$\bf \: ⟼ 2∠2 + 2∠2 + ∠DAC = 180$

$\bf \: ⟼ ∠DAC = 180 - 4∠2 \: ⟼ \: (3)$

⟼ Now, using straight line angle, we get

$\large \red{\bf \: ⟼∠2 + ∠DAC +71 = 180 }$

On substituting the value from equation (3), we get

$\bf \: ⟼ ∠2 + 180 - 4∠2 + 71 = 180$

$\bf \: ⟼ 71 - 3∠2 = 0$

$\bf \: ⟼ 3∠2 = 71$

$\bf \: ⟼ ∠2 = \dfrac{71}{3}$

$\begin{gathered}\bf\red{So,} \end{gathered}$

$\large \red{\bf \: ⟼ ∠4 = 2∠2 = 2 \times \dfrac{71}{3} = \dfrac{142}{3} } ✍$

$\large{\boxed{\boxed{\bf{Option \: (b) \: is \: correct}}}}$

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$\large \red{\bf \: ⟼ Explore \: more } ✍$

❥︎Properties of a triangle

• A triangle has three sides, three angles, and three vertices.

• The sum of all internal angles of a triangle is always equal to 180°. This is called the angle sum property of a triangle.

• The sum of the length of any two sides of a triangle is greater than the length of the third side.

• The side opposite to the largest angle of a triangle is the largest side.

• Any exterior angle of the triangle is equal to the sum of its interior opposite angles. This is called the exterior angle property of a triangle.

❥︎Based on the angle measurement, there are three types of triangles:

• Acute Angled Triangle : A triangle that has all three angles less than 90° is an acute angle triangle.

• Right-Angled Triangle : A triangle that has one angle that measures exactly 90° is a right-angle triangle.

• Obtuse Angled Triangle : triangle that has one angle that measures more than 90° is an obtuse angle triangle.

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