Use the image to answer the question.

Question

A rhombus labeled upper A upper B upper C upper D has 2 interior, diagonal lines.

In the diagram of the rhombus, m∠EDC=42°
and DE¯¯¯¯¯¯¯¯
is perpendicular to AC¯¯¯¯¯¯¯¯
. Using what you know about perpendicular bisectors of a rhombus, what is the m∠ECD
?

(1 point)
m∠ECD
is
°.

Answer 1

In a rhombus, the diagonals bisect each other at right angles (90 degrees). Given that DE is perpendicular to AC, we know that angle EDC forms part of two right triangles created by the diagonals.

Given \( m∠EDC = 42° \), and knowing that the diagonals bisect the angles in the rhombus, we can calculate \( m∠ECD \) as follows:

Since triangle EDC is a right triangle (because DE is perpendicular to AC), we can express the angles in triangle EDC:

• \( m∠EDC + m∠ECD + m∠DCE = 90° \) (since DE is perpendicular to AC).

Also, because of the properties of the rhombus:

• \( m∠DCE = m∠EDC = 42° \).

Thus, we can write:

\[ m∠ECD + 42° + 42° = 90° \]

This simplifies to:

\[ m∠ECD + 84° = 90° \]

Subtracting 84° from both sides gives:

\[ m∠ECD = 90° - 84° = 6° \]

Therefore, \( m∠ECD \) is \( 6° \).