Find the lengths of line segment 𝐶𝐵 and line segment 𝐴𝐷.
In the diagram, we can see that there are two triangles: triangle 𝐵𝐶𝐷 and triangle 𝐴𝐵𝐷. In the diagram, we can see that there are two angles that are denoted with one single arc. That’s the angle 𝐶𝐵𝐷 in triangle 𝐵𝐶𝐷 and the angle 𝐴𝐵𝐷 in triangle 𝐴𝐵𝐷. And since the angles are denoted in the same way, this means that the measure of these angles are equal to each other.
We have another pair of angles that are noted by a double arc. So we can say that the measure of angle 𝐵𝐷𝐶 is equal to the measure of 𝐵𝐷𝐴. Let’s have a look at the lines in our triangles. We can see that the line 𝐵𝐷 occurs in both triangles. And since this line is common to both triangles, we can say that the line 𝐵𝐷 in triangle 𝐵𝐶𝐷 is equal to the line 𝐵𝐷 in triangle 𝐴𝐵𝐷.
So now we know that our two triangles have two pairs of corresponding angles equal and a corresponding side equal. We can notice that the corresponding side that’s equal is in between the pairs of corresponding angles equal. And we can note this as angle-side-angle or ASA. And we can say that triangle 𝐵𝐶𝐷 and triangle 𝐴𝐵𝐷 are congruent using the angle-side-angle criterion.
Let’s now see how we can use the fact that the triangles are congruent to find the lengths of line segment 𝐶𝐵 and line segment 𝐴𝐷. So line segment 𝐶𝐵 in triangle 𝐵𝐶𝐷 must be equal to line segment 𝐴𝐵 in triangle 𝐴𝐵𝐷. We can see that line segment 𝐴𝐵 is labelled as 20 centimetres. Therefore, 𝐶𝐵 must also be equal to 20 centimetres. Next, to find our line segment 𝐴𝐷, we know that the corresponding side in triangle 𝐵𝐶𝐷 must be the line segment 𝐶𝐷. And since 𝐶𝐷 is labelled as 12 centimetres, then 𝐴𝐷 must also be 12 centimetres.
And so our final answer is line segment 𝐶𝐵 equals 20 centimetres. Line segment 𝐴𝐷 equals 12 centimetres.