Question

1. is there enough information to conclude the two triangles are congruent? if so, what is a correct congruence statement?

a yes, \\( \triangle c a b \cong \triangle d a c \\)
b yes, \\( \triangle a c b \cong \triangle a c d \\)
c no, the triangles cannot be proven congruent.
d yes, \\( \triangle a b c \cong \triangle a c d \\)

1. if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then

a the triangles are congruent by aas
b the triangles are congruent by hl
c the triangles are congruent by asa
d the triangles are congruent by sas

Explanation:

Question 4

Step 1: Analyze the given triangle

In the diagram, we have \( \triangle ACB \) and \( \triangle ACD \). \( AC \) is a common side, \( \angle ACB = \angle ACD = 90^\circ \) (right angles), and \( AB = AD \) (marked as equal). So by the Hypotenuse - Leg (HL) congruence criterion (for right triangles) or by SAS (since \( AC \) is common, \( \angle ACB=\angle ACD \), and \( AB = AD \) implies \( BC = CD \) as \( AC \) is the perpendicular bisector), the triangles \( \triangle ACB \) and \( \triangle ACD \) are congruent.

Step 2: Evaluate the options

• Option A: \( \triangle CAB\cong\triangle DAC \) is incorrect as the correspondence of vertices is wrong.
• Option B: \( \triangle ACB\cong\triangle ACD \) is correct as we have shown the triangles are congruent with the correct vertex correspondence.
• Option C: The triangles can be proven congruent, so this is incorrect.
• Option D: \( \triangle ABC\cong\triangle ACD \) has incorrect vertex correspondence.

The Hypotenuse - Leg (HL) congruence theorem states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent.

• Option A: AAS (Angle - Angle - Side) is not applicable here as we are dealing with hypotenuse and leg, not angles.
• Option B: HL is the correct criterion for this case.
• Option C: ASA (Angle - Side - Angle) is not applicable here.
• Option D: SAS (Side - Angle - Side) is not the criterion for hypotenuse - leg in right triangles.

Answer:

B. Yes, \( \triangle ACB\cong\triangle ACD \)

Question 5