Construct the following angles at the initial point of a given ray and justify the construction.

90^o

Construction of angle of 90°

Steps of construction:

Step 1: Draw a ray OA.

Step 2: With its initial point O as center and any radius, draw an arc, cutting OA at C.

Step 3: With center c and same radius (as in step 2) draw an arc cutting arc at D.

Step 4: With D as center and the same radius, draw an arc cutting the arc cutting at E.

Step 5: With D and E as centers and any convenient radius (more than DE). Draw to two arcs intersecting at P.

Step 6: Join OP. Then ∠AOP = 90°

Justification: -

By construction, OC = CD = OD

Therefore, ΔOCD is an equilateral triangle. So, ∠COD = 60°

Again OD = DE = OE

Therefore, ΔODE is also an equilateral triangle. So, ∠DOE = 60°

Since, OP bisects ∠DOE, so ∠POD = 30°.

Now,

∠AOP = ∠COD + ∠DOP

= 60° + 30°

= 90°