Question 13

We copy an angle by transferring the arc length subtended by the angle's arms using a compass.

Step 1 — Copying the first angle Let the first angle be $\angle ABC$. Its vertex is B. We want to make a copy of this angle. First, draw a ray $\text{B'A'}$. This will be one arm of the new angle. Place the compass point at B. Draw an arc of any convenient radius. This arc cuts arm BA at point M. This arc cuts arm BC at point N. Now, without changing the compass opening, place the compass point at B'. Draw another arc of the same radius. This arc cuts the ray $\text{B'A'}$ at point M'. Next, open the compass. Measure the distance between M and N. Place the compass point at M. Adjust the compass so its pencil end is at N. Now, do not change this compass opening. Place the compass point at M'. Draw an arc that cuts the previous arc. This arc was drawn from B'. Let this intersection point be N'. Finally, draw a ray from B' through N'. Let this ray be $\text{B'C'}$. The angle $\angle A'B'C'$ is a copy of $\angle ABC$.

Step 2 — Copying the second angle Let the second angle be $\angle DEF$. Its vertex is E. We want to make a copy of this angle. First, draw a ray $\text{E'D'}$. This will be one arm of the new angle. Place the compass point at E. Draw an arc of any convenient radius. This arc cuts arm ED at point O. This arc cuts arm EF at point P. Now, without changing the compass opening, place the compass point at E'. Draw another arc of the same radius. This arc cuts the ray $\text{E'D'}$ at point O'. Next, open the compass. Measure the distance between O and P. Place the compass point at O. Adjust the compass so its pencil end is at P. Now, do not change this compass opening. Place the compass point at O'. Draw an arc that cuts the previous arc. This arc was drawn from E'. Let this intersection point be P'. Finally, draw a ray from E' through P'. Let this ray be $\text{E'F'}$. The angle $\angle D'E'F'$ is a copy of $\angle DEF$.

Step 3 — Copying the third angle Let the third angle be $\angle GHI$. Its vertex is H. We want to make a copy of this angle. First, draw a ray $\text{H'G'}$. This will be one arm of the new angle. Place the compass point at H. Draw an arc of any convenient radius. This arc cuts arm HG at point Q. This arc cuts arm HI at point R. Now, without changing the compass opening, place the compass point at H'. Draw another arc of the same radius. This arc cuts the ray $\text{H'G'}$ at point Q'. Next, open the compass. Measure the distance between Q and R. Place the compass point at Q. Adjust the compass so its pencil end is at R. Now, do not change this compass opening. Place the compass point at Q'. Draw an arc that cuts the previous arc. This arc was drawn from H'. Let this intersection point be R'. Finally, draw a ray from H' through R'. Let this ray be $\text{H'I'}$. The angle $\angle G'H'I'$ is a copy of $\angle GHI$.

Step 4 — Copying the fourth angle Let the fourth angle be $\angle JKL$. Its vertex is K. We want to make a copy of this angle. First, draw a ray $\text{K'J'}$. This will be one arm of the new angle. Place the compass point at K. Draw an arc of any convenient radius. This arc cuts arm KJ at point S. This arc cuts arm KL at point T. Now, without changing the compass opening, place the compass point at K'. Draw another arc of the same radius. This arc cuts the ray $\text{K'J'}$ at point S'. Next, open the compass. Measure the distance between S and T. Place the compass point at S. Adjust the compass so its pencil end is at T. Now, do not change this compass opening. Place the compass point at S'. Draw an arc that cuts the previous arc. This arc was drawn from K'. Let this intersection point be T'. Finally, draw a ray from K' through T'. Let this ray be $\text{K'L'}$. The angle $\angle J'K'L'$ is a copy of $\angle JKL$.

Answer

(i) The angle $\angle A'B'C'$ is a copy of the first given angle. (ii) The angle $\angle D'E'F'$ is a copy of the second given angle. (iii) The angle $\angle G'H'I'$ is a copy of the third given angle. (iv) The angle $\angle J'K'L'$ is a copy of the fourth given angle.