(1)Are two triangles congruent if two sides and an angle of one triangle are equal to two sides and an angle of the other?If not then under what conditions will they be congruent?

(2)How to construct Triangle(ABC) and Triangle(PQR)such that they are equal in area but not congruent.

It has to be the contained angle

This is a right triangle.

This is right triangle

(1) Two triangles are indeed congruent if two sides and an angle of one triangle are equal to two sides and an angle of the other triangle. This is known as the Side-Angle-Side (SAS) congruence criterion. However, it is important to note that this is not the only condition for triangles to be congruent.

In addition to SAS, there are other congruence criteria such as Side-Side-Side (SSS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), and Hypotenuse-Leg (HL). Each of these criteria has specific requirements for congruence, and all of them must be met in order for the triangles to be declared congruent.

For example, if we have two triangles with sides AB = XY, BC = YZ, and ∠B = ∠Y, the triangles are indeed congruent by the SAS criterion. However, if we only had two sides and an angle that don't satisfy any of the other congruence criteria, such as AB = XY, BC = YZ, and ∠A = ∠X, we cannot conclude that the triangles are congruent.

(2) To construct two triangles (ABC) and (PQR) that have equal areas but are not congruent, you can follow these steps:

1. Start by drawing a line segment AB.
2. From point A, draw an arc with any radius to intersect line segment AB at point C.
3. From point B, draw an arc with the same radius to intersect line segment AB at point D (on the opposite side of AB from C).
4. Now, you have line segments AB and CD that intersect at point B.
5. From point C, draw a line segment CE to any desired length.
6. From point D, draw a line segment DF of the same length as CE.
7. From point B, draw a line segment BG to intersect line segment DF.
8. From point B, draw a line segment BH to intersect line segment CE.
9. The intersection point of line segments BG and BH will be point P.
10. Connect points A and P to form triangle ABC.
11. Connect points B and C to form triangle PQR.

Now, triangle ABC and triangle PQR have the same area because they have the same base (AB) and the same height (distance between AB and PR). However, they are not congruent because their corresponding angles and sides are not equal.