If in two , two angles of one triangle are respectively equal to the two angles of the other triangle, then the two are similar (AA similarity criterion). Fig. 6.40 Fig. 6.41 98
Step-by-Step Solution
Key Concept: AA similarity criterion – if two angles of one triangle are respectively equal to two angles of another triangle, the third angles are also equal (since the sum of angles in a triangle is $180^{\circ}$). Hence the triangles are similar and their corresponding sides are in proportion.
1. Given: Two triangles $\triangle ABC$ and $\triangle DEF$ such that $\angle A = \angle D$ and $\angle B = \angle E$.
2. Find the third angle:
$$\angle C = 180^{\circ} - (\angle A + \angle B)$$
$$\angle F = 180^{\circ} - (\angle D + \angle E)$$
Since $\angle A = \angle D$ and $\angle B = \angle E$, we have $\angle C = \angle F$.
3. All three angles are equal: $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$.
4. Apply AA similarity criterion: Because two (hence all three) corresponding angles are equal, $\triangle ABC \sim \triangle DEF$.
5. Resulting proportion of sides (optional for full marks):
$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}.$$
Correct Answer: The two triangles are similar (AA similarity criterion).