The SSS Congruence Theorem Practice

Question

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Question
Which of the following triangles are congruent by the SSS Congruence Theorem? Enter the lesser number first.

Triangle #1: a triangle with sides measuring 2.5 cm, 8 cm, and 6.75 cm in length

Triangle #2: a triangle with an angle measuring 60°
between sides measuring 2.5 cm and 8 cm in length

Triangle #3: a triangle with an angle measuring 111.17°
between sides measuring 2.5 cm and 6.75 cm in length, and a third side with a length of 8 cm

(1 point)
Triangle #
and Triangle #
are congruent.

Answer 1

To determine which triangles are congruent using the SSS (Side-Side-Side) Congruence Theorem, we will compare the side lengths of each triangle.

1. Triangle #1 (sides: 2.5 cm, 6.75 cm, 8 cm)

2. Triangle #2 (angle: 60° between sides: 2.5 cm and 8 cm)

   • To apply the SSS theorem, we need to find the length of the third side of Triangle #2 using the Law of Cosines: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \] Here, \( a = 2.5 \), \( b = 8 \), and \( C = 60° \). \[ c^2 = (2.5)^2 + (8)^2 - 2 \times 2.5 \times 8 \times \cos(60°) \] \[ c^2 = 6.25 + 64 - (2.5 \times 8) \] \[ c^2 = 6.25 + 64 - 10 \] \[ c^2 = 60.25 \] \[ c = \sqrt{60.25} \approx 7.76 \text{ cm} \]

3. Triangle #3 (angle: 111.17° between sides: 2.5 cm and 6.75 cm, third side: 8 cm)

   • We already have one side \( c = 8 \), and the two other sides \( a = 2.5 \) cm and \( b = 6.75 \) cm.

Now we compare the side lengths:

• For Triangle #1: 2.5 cm, 6.75 cm, and 8 cm.
• For Triangle #3: The sides are 2.5 cm, 6.75 cm, and 8 cm.

Since Triangle #3 has the same three side lengths as Triangle #1, they are congruent by SSS.

Conclusion:

Triangle #1 and Triangle #3 are congruent.

So, the answer is: Triangle #1 and Triangle #3 are congruent.