Question

1. which congruence rule can be used to show that the two triangles are congruent?

a. ssa
b. sas
c. asa
d. not enough information

1. the transformation shown is:

a. a reflection over the x - axis followed by a reflection over the y - axis
b. a translation 2 units right and a reflection over the x - axis
c. a 90° counterclockwise rotation

Explanation:

14. Check congruence rules

• SSA:
• SSA (Side - Side - Angle) is not a valid congruence rule in general (except for the right - angled triangle case (HL which is a special form of SSA)).
• SAS:
• SAS (Side - Angle - Side) requires two sides and the included angle to be equal. From the figure, we can assume that the two sides (marked with the same tick - marks) and the included angle (the non - marked angle between the two sides) are equal.
• ASA:
• ASA (Angle - Side - Angle) requires two angles and the included side to be equal. There is no information about two angles and an included side from the given figure description.

15. Analyze the transformation

• Option A:
• A reflection over the \(x\) - axis changes the \(y\) - coordinate sign (\((x,y)\to(x, - y)\)), and then a reflection over the \(y\) - axis changes the \(x\) - coordinate sign (\((x, - y)\to(-x, - y)\)).
• Let's assume a point \(A(x,y)\) in \(\triangle ABC\). After reflection over the \(x\) - axis, it becomes \(A_1(x, - y)\), and then after reflection over the \(y\) - axis, it becomes \(A'( - x, - y)\).
• Option B:
• A translation 2 units right changes the \(x\) - coordinate (\((x,y)\to(x + 2,y)\)), and a reflection over the \(x\) - axis changes the \(y\) - coordinate sign (\((x+2,y)\to(x + 2, - y)\)). This does not match the transformation from \(\triangle ABC\) to \(\triangle A'B'C'\).
• Option C:
• A \(90^{\circ}\) counter - clockwise rotation about the origin has the transformation rule \((x,y)\to(-y,x)\). This does not match the transformation from \(\triangle ABC\) to \(\triangle A'B'C'\).

Answer:

1. B. SAS
2. A. A reflection over the \(x\) - axis followed by a reflection over the \(y\) - axis