Question

How to draw a triangle?

Answer 1

How to construct a triangle given by its side and the two adjacent interior angles using a compass and a ruler
You are given a segment a and two angles $\angle B$ and $\angle C$ in a plane (Figure 1).
You need to construct a triangle which has one side congruent to the segment a and two angles at the endpoints of this side congruent to the angles $\angle B$ and $\angle C$ using a compass and a ruler.

Construction

Make the following steps (Figure 2):
1) Draw an arbitrary straight line in the plane using the ruler.
Figure 1. The segment a and the angles $\angle B$ and $\angle C$
2) Then construct the segment BC on this line congruent to the given segment a using the compass.
It is described in the lesson How to draw a congruent segment and a congruent angle using a compass and a ruler under the current topic how to do it.
3) Then construct the \angle CBA1 with the vertex at the point B adjacent to the straight line BC and congruent to the given angle \angle B using a compass and a ruler. The side of this angle which lies on the straight line BC should be directed consistently with the direction of the ray BC.
The description on how to do it is in the lesson How to draw a congruent segment and a congruent angle using a compass and a ruler.
4) Then construct the angle $\angle BCA2$ with the vertex at the point C adjacent to the straight line BC and congruent to the given angle $\angle C$ using a compass and a ruler. The side of this angle which lies on the straight line BC should be directed consistently with the direction of the ray CB. The description on how to do it is in the same lesson.
5) Extend the straight lines BA1 and CA2 till the intersection at the point A if necessary.
The triangle ABC is what has to be constructed.
Indeed, the side BC of this triangle is congruent to the given segment a, and the adjacent angles $\angle CBA$ and $\angle BCA$ are congruent to the given angles $\angle B$ and $\angle C$ respectively.
Figure 2. Constructing the triangle given by its side and the two adjacent interior angles
How to construct a triangle given by its two sides and the interior angle included between them using a compass and a ruler You are given rwo segments a and c and the angle $\angle B$ in a plane (Figure 3).
You need to construct a triangle which has two sides congruent to the segments a and c and the angle between these sides congruent to the angle $\angle B$
using a compass and a ruler.


Figure 3. The segments a and c and the angle $\angle B$
Construction
Make the following steps (Figure 4):
1) Draw an arbitrary straight line in the plane using the ruler.
Figure 3. The segments a and c and the angle $\angle B$
2) Then construct the segment BC on this line congruent to the given segment a using the compass.
It is described in the lesson How to draw a congruent segment and a congruent angle using a compass and a ruler under the current topic how to do it.
3) Then construct the angle $\angle CBA1$ with the vertex at the point B adjacent to the straight line BC and congruent to the given angle $\angle B$ using a compass and a ruler. The side of this angle which lies on the straight line BC should be directed consistently with the direction of the ray BC.
The description on how to do it is in the lesson How to draw a congruent segment and a congruent angle using a compass and a ruler.
4) Then construct the segment BA on the straight line BA1 congruent to the given segment c using a compass.
It gives you the position of the point A on the straight line BA1.
5) Connect the points A and C by the segment AC using a ruler.
The triangle ABC is what has to be constructed.
Indeed, the sides BC and BA of this triangle are congruent to the given segments a and c respectively, and the included angle $\angle CBA$ is congruent to the given angles $\angle B$.


Figure 4. Constructing the triangle given by its two sides and the interior angle included between them
How to construct a triangle given by its three sides using a compass and a ruler
You are given three segments a, b and c in a plane (Figure 5).
You need to construct a triangle which has three sides congruent to the segments
a, b and c using a compass and a ruler.

Figure 5. The segments a, b and c
Construction
Make the following steps (Figure 6):
1) Draw an arbitrary straight line in the plane using the ruler.
Figure 5. The segments a, b and c
2) Then construct the segment BC on this line congruent to the given segment a using the compass.
It is described in the lesson How to draw a congruent segment and a congruent angle using a compass and a ruler under the current topic how to do it.
3) Make the compass opening equal to the length of the segment c and draw an arc of this radius with the center at the point B (Figure 6).
4) Make the compass opening equal to the length of the segment b and draw an arc of this radius with the center at the point C (Figure 6).
5) Connect the intersection point A of these two arcs with the points B and C by the segments AB and AC using a ruler.
The triangle ABC is what has to be constructed.
Indeed, the sides BC, AC and AB of this triangle are congruent to the given segments a, b and c respectively.
Figure 6. Constructing the triangle given by its three sides