Name: Illustration Based on the SAS Criterion
Uploaded: 2019-07-19
Duration: 4 min 36 s
Description: One of the easiest ways to learn a concept is through a visual illustration. To learn how the SAS congruence criterion is applied in mathematical problems, w...

Question

In the adjoining figure, $A D$ bisects $∠ A$ . Arrange $A B, B D$ and $D C$ in ascending order.

Accepted Answer

Given, $A D$ bisects $∠ A$ , $∠ A B C = 75 °$ and $∠ A C B = 35 °$ .

Angle sum property of triangle states that the sum of interior angles of a triangle is $180 °$ .

$∴ ∠ A B C + ∠ A C B + ∠ B A C = 180 °$

$\Rightarrow 75 ° + 35 ° + ∠ B A C = 180 °$

$\Rightarrow ∠ B A C = 180 ° - 110 °$

$\Rightarrow ∠ B A C = 70 °$

Given, $A D$ bisects $∠ B A C$ .

$∴ ∠ B A D = ∠ D A C = \frac{∠ B A C}{2}$

$\Rightarrow ∠ B A D = ∠ D A C = \frac{70 °}{2}$

$\Rightarrow ∠ B A D = ∠ D A C = 35 °$

Now, $∠ B A C > ∠ A C B$

Sides opposite greater angles is greater.

$\Rightarrow B C > A B$ _________ $(1)$

In $△ A D C, ∠ A C D = ∠ D A C = 35 °$

Sides opposite to equal angles are equal.

$\Rightarrow A D = D C$

Similarly, in $△ A B D, ∠ A B D + ∠ A D B + ∠ B A D = 180 °$

$\Rightarrow 75 ° + ∠ A D B + 35 ° = 180 °$

$\Rightarrow ∠ A D B = 180 ° - 110 °$

$\Rightarrow ∠ A D B = 70 °$

In $△ A B D, ∠ A B D = 75 °$ and $∠ A D B = 70 °$

$\Rightarrow ∠ A B D > ∠ A D B$

$\Rightarrow A D > A B$

$\Rightarrow D C > A B$ $[∵ A D = D C]$ _____ $(2)$

In $△ A B D, ∠ A D B = 70 °$ and $∠ B A D = 35 °$

$∴ ∠ A D B > ∠ B A D$

$\Rightarrow A B > B D$ ______________ $(3)$

From $(2)$ and $(3)$ , we get,

$D C > A B > B D$

Hence, the required answer is $B D < A B < D C$ .

M L Aggarwal Solutions for Chapter: Triangles, Exercise 7: Higher Order Thinking Skills (HOTS)

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Triangles, Exercise 7: Higher Order Thinking Skills (HOTS)

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