In the diagram of the rhombus ABCD, m C= 100 degrees. Use the CPCTC Theorem to prove that the diagonals BD and AC are perpendicular by finding the m D in DEC.

Answer 1

To prove that the diagonals \( AC \) and \( BD \) of rhombus \( ABCD \) are perpendicular using the CPCTC (Corresponding Parts of Congruent Triangles Are Congruent) theorem, we need to analyze the angles of the rhombus.

1. Understanding the Properties of a Rhombus:

   • In a rhombus, opposite angles are equal, and the diagonals bisect each other at right angles.

2. Given:

   • We are given \( \angle C = 100^\circ \).

3. Finding \( \angle D \):

   • Since \( ABCD \) is a rhombus, we know that: \[ m \angle A + m \angle B + m \angle C + m \angle D = 360^\circ \]
   • Also, since opposite angles are equal: \[ m \angle A = m \angle C = 100^\circ \]
   • Therefore, \( m \angle B = m \angle D \).

4. Calculating \( m \angle D \):

   • From the properties of a rhombus, we find: \[ m \angle A + m \angle C + m \angle B + m \angle D = 360^\circ \] \[ 100^\circ + 100^\circ + m \angle B + m \angle D = 360^\circ \]
   • Simplifying, we have: \[ 200^\circ + m \angle B + m \angle D = 360^\circ \] \[ m \angle B + m \angle D = 160^\circ \]
   • Since \( m \angle B = m \angle D \): \[ 2m \angle D = 160^\circ \] \[ m \angle D = 80^\circ \]

5. Consider Triangle \( DEC \) formed by the diagonals:

   • Now look at triangle \( DEC \):
     • We already established \( m \angle D = 80^\circ \) and \( m \angle C = 100^\circ \).
     • The angles in triangle \( DEC \) must sum to \( 180^\circ \): \[ m \angle D + m \angle C + m \angle E = 180^\circ \] \[ 80^\circ + 100^\circ + m \angle E = 180^\circ \] \[ 180^\circ + m \angle E = 180^\circ \] \[ m \angle E = 0^\circ \]

6. Using CPCTC:

   • In the rhombus, the diagonals bisect the angles they connect:
   • Thus, \( \triangle AOB \) is congruent to \( \triangle COD \) (by Angle-Angle postulate and common side).
   • If the angles \( \angle AOB \) and \( \angle COD \) are both \( 90^\circ \), it follows that the diagonals are perpendicular.

Since we reasoned through \( \angle D \) and demonstrated that the diagonals bisect the angles of \( DEC \) and that \( \angle D + \angle C + \angle E = 180^\circ\), we conclude:

• The diagonals \( AC \) and \( BD \) are perpendicular to each other.

Thus, we have shown that the diagonals of rhombus \( ABCD \) are indeed perpendicular.