Similar shapes:& Same shape
Congruent shapes:& Same shape and size
We haven't been given any information about the lengths of the triangles sides. This means we cannot say anything about the size of the triangles. However, we have been given two angles in both triangles. Therefore, we can only talk about the shape.
Are the triangles similar?
If the two triangles have at least two pairs of congruent angles, we know they have the same shape and are similar. Let's find the last angle by using the Triangle Angle Sum Theorem.
θ+18^(∘)+140^(∘)&=180^(∘) ⇔ θ=22^(∘)
β+18^(∘)+21^(∘)&=180^(∘) ⇔ β=141^(∘)
The triangles do not have the same angles and therefore, they cannot be similar.
b Like in Part A, we have to investigate the triangles' shape and size.
Are the triangles similar?
We already know one pair of congruent angles. From the diagram, we can identify two parallel lines that are cut by a transversal. The transversal creates a pair of alternate interior angles which also happens to be angles of the two triangles. Because the lines are parallel, these angles are congruent by the Alternate Interior Angles Theorem.
Since the angles have two pairs of congruent angles we can claim similarity by the AA Similarity Theorem.
Are the triangles congruent?
We have been given one side in each triangle. Since these sides are both between the same two corresponding angles, these sides must be corresponding.
Because two corresponding sides do not have the same length, the triangles cannot have the same size which means they are not congruent.
