Question

In triangle ABC, line segment CX is an altitude. Angle ACB is a right angle. Which statements are true? Select two options.

1) ΔABC is congruent to ΔBXC
2) ΔAXC is congruent to ΔCXB
3) ΔBCX is congruent to ΔACX
4) ΔACB is congruent to ΔAXC
5) ΔCXA is congruent to ΔCBA

Answer 1

In triangle ABC with CX as an altitude and angle ACB as a right angle, the only true statement is that ΔAXC is congruent to ΔCXB by the Hypotenuse-Leg theorem. All other statements are false as they do not accurately describe the relationships between the triangles in the given figure.

Step-by-step explanation:

In the context of triangle ABC, where CX is an altitude and angle ACB is a right angle, we need to determine which of the given statements is true

Analysis of Statements

1. ΔABC is not congruent to ΔBXC since ΔABC includes the hypotenuse of the right triangle ABC while ΔBXC does not.
2. ΔAXC is congruent to ΔCXB by the Hypotenuse-Leg (HL) theorem, as both triangles share the hypotenuse CX of the right triangle and have a right angle.
3. ΔBCX is not congruent to ΔACX as they do not share the same angles and sides. ΔBCX includes the right angle, while ΔACX does not.
4. ΔACB being the entire triangle, cannot be congruent to ΔAXC which is a smaller right triangle within ΔACB.
5. ΔCXA cannot be congruent to ΔCBA as they again do not share identical angles and sides and are different in size.

Therefore, the true statements are that ΔAXC is congruent to ΔCXB and none of the other statements provided are true.