A triangle ABC is right angled at A. L is a point on BC such that AL ⊥ BC. Prove

that BAL = ACB.

Two lines are respectively perpendicular to two parallel lines. Show that they are

parallel to each other.

A triangle ABC is right angled at A. L is a point on BC such that AL  ⊥  BC. Prove that BAL = ACB.

Solution:  

In ΔCAB, 

∠B + ∠C + ∠CAB = 180°.............(i)

In ΔBLA, 

∠B + ∠BLA + ∠BAL = 180°.............(ii)

now, from (i) and (ii) , we get,

∠B+∠C+ ∠CAB = ∠B + ∠BLA + ∠BAL

∠C +∠CAB = ∠BLA + ∠BAL
∠ACB  = ∠BAL  [∠BLA = ∠BAC = 90°]

Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other.

Solution: 

Given: Two lines are respectively perpendicular to two parallel lines.

now, let the slope of parallel line be k.

as it is given that, two lines are perpendicular to two parallel lines
therefore, 

slope of first line which is perpendicular to the parallel line = -1/k....(i)

and 

slope of second line which is perpendicular to the parallel line = -1/k....(ii)

from (i) and (ii) , 

its clear that , slope of the perpendicular  lines is equal.

hence, they are parallel to each other.