Q

# Construct an angle of 45 degrees at the initial point of a given ray and justify the construction.

Q2.    Construct an angle of 45o at the initial point of a given ray and justify the construction.

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The steps of construction to follow:

Step 1: Draw a ray OY.

Then, take O as the centre and any radius, mark a point A on the arc ABC.

Step 2: Now, taking A as the centre and the same radius, mark a point B on the arc ABC.

Step 3: Take B as a centre and the same radius, mark a point C on the arc ABC.

Step 4: Now, taking C and B as centre one by one, draw an arc from each centre intersecting each other at a point X.

Step 5: X and O are joined and a ray making an angle $90^{\circ}$ with OY is formed.

Let the arc AC touches OX at E

Step 6: With A and E as centres, 2 arcs are marked intersecting each other at D and the bisector of angle XOY is drawn.

Justification:

By construction we have,

$\angle XOY = 90^{\circ}$

We constructed the bisector of  $\angle XOY$ as $\angle DOY$

Thus,

$\angle DOY = \frac{1}{2}\angle XOY = \frac{1}{2}\times90^{\circ} = 45^{\circ}$

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