Ask a New Question
Menu

1. Draw acute Δ ABC and acute Δ DEF below to where they look congruent.

2. Write a congruent statement for the two triangles. 3. Label ∠D as 5x - 2, ∠E as 9x + 3, ∠F as 11x-21, and ∠B as 5y + 35 4. Solve for x and solve for y.

1 answer

1. Drawing Acute Triangles:

Unfortunately, I cannot draw or provide visual content directly. However, I can guide you on how to create two congruent acute triangles, ΔABC and ΔDEF.

To draw two congruent acute triangles:

  • Draw triangle ΔABC with the following vertices:

    • A(1, 3)
    • B(3, 7)
    • C(5, 3)

  • For triangle ΔDEF, draw it in the same shape and position:

    • D(1, 3)
    • E(3, 7)
    • F(5, 3)

Make sure that both triangles have the same corresponding angles and side lengths, ensuring they look congruent.

2. Write a congruent statement for the two triangles:

The congruence statement can be written as: \[ \Delta ABC \cong \Delta DEF \]

3. Label the angles:

  • Label the angles as follows:
    • ∠B = \(5y + 35\)
    • ∠D = \(5x - 2\)
    • ∠E = \(9x + 3\)
    • ∠F = \(11x - 21\)

Since triangles ΔABC and ΔDEF are congruent, their corresponding angles are equal.

This gives us the following equations based on the corresponding angles:

  1. \( \angle D = \angle B \) \[ 5x - 2 = 5y + 35 \]

  2. \( \angle E = \angle F \) \[ 9x + 3 = 11x - 21 \]

4. Solving for x and y:

Solving for \(x\):

From the second equation: \[ 9x + 3 = 11x - 21 \] Subtract \(9x\) from both sides: \[ 3 = 2x - 21 \] Add 21 to both sides: \[ 24 = 2x \] Divide by 2: \[ x = 12 \]

Solving for \(y\):

Now substituting \(x = 12\) into the first equation: \[ 5(12) - 2 = 5y + 35 \] \[ 60 - 2 = 5y + 35 \] \[ 58 = 5y + 35 \] Subtract 35 from both sides: \[ 23 = 5y \] Divide by 5: \[ y = \frac{23}{5} = 4.6 \]

Final values:

  • \(x = 12\)
  • \(y = 4.6\)

Feel free to check these solutions in the original angle equations to ensure accuracy!