Question

given that the following triangles are congruent, determine the value of \\(x\\) and \\(y\\).

\\(x = \frac{74}{9}\\) and \\(y = \frac{10}{3}\\)
\\(x = \frac{74}{9}\\) and \\(y = 3\\)
\\(x = 6\\) and \\(y = \frac{10}{3}\\)
\\(x = 6\\) and \\(y = 3\\)

Explanation:

Find the third angle of the first triangle

Using the Triangle Angle Sum Theorem knowledge point, we calculate the missing angle in the first triangle.

$$ 180^\circ - (50^\circ + 70^\circ) = 60^\circ $$

The three interior angles of the first triangle are \(50^\circ\), \(70^\circ\), and \(60^\circ\).

Match corresponding parts of the congruent triangles

Using the Triangle Congruence and Corresponding Parts knowledge points, we identify matching sides and angles.

• The side of length \(14\) has a single tick mark.
• The side of length \(15\) has a double tick mark.
• In the second triangle, the side labeled \(3y + 5\) has a double tick mark, so it corresponds to the side of length \(15\).
• The angle \((9x - 4)^\circ\) is adjacent to the double tick mark side and the unmarked bottom side. In the first triangle, the angle adjacent to the double tick mark side (\(15\)) and the bottom side is \(50^\circ\).

Set up and solve the equation for y

Using the Corresponding Parts knowledge point, we equate the corresponding side lengths.

$$ 3y + 5 = 15 $$

$$ 3y = 10 \implies y = \frac{10}{3} $$

Set up and solve the equation for x

Using the Corresponding Parts knowledge point, we equate the corresponding angle measures.

$$ 9x - 4 = 50 $$

$$ 9x = 54 \implies x = 6 $$

Answer:

• (A) \(x = \frac{74}{9}\) and \(y = \frac{10}{3}\)
• (B) \(x = \frac{74}{9}\) and \(y = 3\)
• (C) \(x = 6\) and \(y = \frac{10}{3}\) (Correct answer)
• (D) \(x = 6\) and \(y = 3\)