A triangle ABC is right angled at A. L is a point on BC such that AL ⊥ BC. Prove that ∠ BAL = ∠ ACB
Solution:
Given, ABC is a triangle right angled at A.
The point L lies on BC such that AL ⊥ BC
We have to prove that ∠BAL = ∠ACB.
Considering triangle ABC,
∠A = 90°
Considering triangle ALC,
Since AL ⊥ BC, ∠L = 90°
In triangle ABC and ALC,
∠BAC = ∠ALC = 90° ------------------- (1)
Considering triangles ABC and ABL,
∠B is common to both the triangles.
So, ∠ABC = ∠ABL --------------------- (2)
On adding (1) and (2),
∠BAC + ∠ABC = ∠ALC + ∠ABL ----------------- (3)
In triangle ABC,
We know that the sum of all three interior angles of a triangle is always equal to 180 degrees.
∠ABC + ∠BAC + ∠ACB = 180°
∠ABC + ∠BAC = 180° - ∠ACB -------------------------- (4)
Similarly, in triangle ABL,
∠ABL + ∠ALB + ∠LAB = 180°
We know L = 90°
So, ∠ALB = ∠ALC
Now, ∠ABL + ∠ALC + ∠LAB = 180°
∠ALC + ∠ABL = 180° - ∠LAB -------------------------------- (5)
Substituting (4) and (5) in (3),
180° - ∠ACB = 180° - ∠LAB
On rearranging,
180° - ∠ACB - 180° = - ∠LAB
-∠ACB = -∠LAB
Therefore, ∠ACB = ∠BAL
✦ Try This: Find the ∠DCE ?
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 6
NCERT Exemplar Class 9 Maths Exercise 6.3 Problem 9
A triangle ABC is right angled at A. L is a point on BC such that AL ⊥ BC. Prove that ∠ BAL = ∠ ACB
Summary:
A right triangle or right-angled triangle or orthogonal triangle is a triangle in which one angle is equal to 90 degrees. A triangle ABC is right angled at A. L is a point on BC such that AL ⊥ BC. It is proven that ∠BAL = ∠ACB
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