Question

1. in the problem below, \\(\delta abc \cong \delta adc\\). solve for \\(x\\). then, use that value to find the length of \\(bc\\). (1 point)

1. can one of the triangle congruence theorem listed below be used to show the two triangles are congruent? (1 point)

a angle-side-angle theorem
b side-angle-side theorem
c side-side-side theorem
d none of the theorems listed can be used to show congruence

Explanation:

Identify corresponding parts of congruent triangles

Using the Triangle Congruence Postulates knowledge point, we know that if \(\Delta ABC \cong \Delta ADC\), then their corresponding sides are equal in length. The side \(BC\) corresponds to the side \(DC\). Therefore, we can set their algebraic expressions equal to each other:

$$BC = DC$$

$$12x - 15 = 3x + 30$$

Solve for x

Using the Solving Linear Equations knowledge point

$$
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$$

Calculate the length of BC

Using the Solving Linear Equations knowledge point

$$
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Analyze the second pair of triangles

We are given two triangles, \(\Delta ABC\) and \(\Delta DEC\), sharing a vertex at \(C\).

1. The tick marks show that segment \(BC\) is congruent to segment \(EC\) (\(BC \cong EC\)).
2. The angle markings show that \(\angle A \cong \angle D\).
3. The vertical angles at vertex \(C\) are congruent by the Vertical Angles theorem: \(\angle ACB \cong \angle DCE\).

Determine the congruence theorem

We have two angles and a non-included side congruent:

• \(\angle A \cong \angle D\) (Angle)
• \(\angle ACB \cong \angle DCE\) (Angle)
• \(BC \cong EC\) (Side)

This corresponds to the Angle-Angle-Side (AAS) congruence theorem. Looking at the options:

• Option A: Angle-Side-Angle Theorem (requires the included side \(AC \cong DC\))
• Option B: Side-Angle-Side Theorem
• Option C: Side-Side-Side Theorem
• Option D: None of the theorems listed can be used to show congruence

Since AAS is not listed among options A, B, or C, the correct choice is Option D.

Answer:

Question 9

\(x = 5\)
\(BC = 45\)

Question 10

• A Angle - Side - Angle Theorem
• B Side - Angle - Side Theorem
• C Side - Side - Side Theorem
• D None of the theorems listed can be used to show congruence (Correct answer)