Question

type the correct answer in each box. use numerals instead of words. if necessary, use / for the fraction bar(s)
abcd is a rectangle. the length of \\( \overline { b d } \\) is 8 units, and \\( m \angle a b d \\) is \\( 67 ^ { \circ } \\).
the length of \\( \overline { a c } \\) is units, and \\( m \angle c b d \\) is

Explanation:

Step 1: Recall the property of the diagonals of a rectangle

In a rectangle, the diagonals are equal in length. Given \(BD = 8\) units. Since \(AC\) and \(BD\) are diagonals of rectangle \(ABCD\), by the property of rectangle diagonals \(AC=BD\).

Step 2: Find the measure of \(\angle CBD\)

We know that \(\angle ABC = 90^{\circ}\) (angle of a rectangle). Given \(\angle ABD=67^{\circ}\).
Using the angle - addition property \(\angle ABC=\angle ABD+\angle CBD\).
Substitute \(\angle ABC = 90^{\circ}\) and \(\angle ABD = 67^{\circ}\) into the equation: \(90^{\circ}=67^{\circ}+\angle CBD\).
Solve for \(\angle CBD\): \(\angle CBD=90^{\circ}- 67^{\circ}\)

Answer:

The length of \(\overline{AC}\) is \(8\) units, and \(m\angle CBD\) is \(23^{\circ}\)