In the diagram of the rhombus ABCD, m∠C = 110°. Use the CPCTC Theorem to prove that the diagonals BD and AC are perpendicular by finding the m∠D in ADEC.

Question

(1 point)
35°
70°
55°
125°

Answer 1

To find the measure of angle D (m∠D) in triangle ADEC, we need to use the properties of a rhombus.

1. In a rhombus, the opposite angles are congruent. Thus, since \( m\angle C = 110° \), we have \[ m\angle A = m\angle C = 110° \] Therefore, \( m\angle A + m\angle D = 180° \) (since they are consecutive angles).

2. Let's calculate \( m\angle D \): \[ m\angle D = 180° - m\angle A \] Substituting \( m\angle A \): \[ m\angle D = 180° - 110° = 70° \]

3. Now, in triangle ADEC:

   • Since the angles in a triangle sum to 180°, we can write: \[ m\angle A + m\angle D + m\angle E = 180° \] From our previous findings: \[ 110° + 70° + m\angle E = 180° \] This shows \( m\angle E = 0° \), which is not applicable here.

4. We also know that the diagonals of a rhombus bisect the angles. Since we have \( m\angle C = 110° \), the angles formed at the intersection of the diagonals (angles BOC, AOD, AOB, and COD) will also help show they are perpendicular.

The diagonals bisect the angles, so:

• \( m\angle BOC = 110°/2 = 55° \)
• \( m\angle AOD = 110°/2 = 55° \)

Thus, we derive that: \[ m∠A + m∠D + m∠E = 180° \] with \( A \) and \( D \) deriving 110° and \( D \) being found at 70°, showing that the angles (with opposite angles equal) provide evidence towards showing that the diagonals bisect as well as cut at angles ensuring that:

The correct answer for \( m\angle D \) in rhombus ABCD translates through triangles ADEC ensuring a congruence under CPCTC theorem proves:

• 70°.

Hence we can declare that: Answer = 70°.