Chapter 9, Problem 19CR

To determine

To show: Sides AC and DBare congruent.

Explanation of Solution

Given information:

  $\overline{AB }$ lies in m , $\overline{CD }$ lies in nand m is parallel to n .

  $\overline{AC }$ intersects $\overline{BD }$ at P .

  $\overline{AD }$ isperpendicular to n .

  $\overline{BC }$ isperpendicular to n .

Formula used:The below properties are used: Two intersecting lines determine a plane.

If a plane intersects two parallel planes, the lines of intersection are parallel.

If two lines are perpendicular to third, the lines are parallel.

If both pairs of opposite sides of a quadrilateral are parallel, the figure is a parallelogram.

A line perpendicular to a plane is perpendicular to every line in the plane passing through its foot.

Perpendicular lines intersect to form right angles.

If a parallelogram contains a right angle, then it is a rectangle.

In a rectangle, diagonals are congruent.

Proof:

  

It is given that,

  $\overline{AB }$ lies in m , $\overline{CD }$ lies in n and m is parallel to n .

  $\overline{AC }$ intersects $\overline{BD }$ at P .

  $\overline{AD }$ isperpendicular to n .

  $\overline{BC }$ is perpendicular to n .

Two intersecting lines determine a plane.

ABCDis a plane.

If a plane intersects two parallel planes, the lines of intersection are parallel.

  $\overline{AB }\parallel \overline{DC}$

If two lines are perpendicular to third, the lines are parallel.

  $\overline{AD }\parallel \overline{BC}$

If both pairs of opposite sides of a quadrilateral are parallel, the figure is a parallelogram.

ABCD is a parallelogram.

A line perpendicular to a plane is perpendicular to every line in the plane passing through its foot.

  $\overline{AD }\perp \overline{DC}$

Perpendicular lines intersect to form right angles.

  $\angle ADC$ is a right angle.

If a parallelogram contains a right angle, then it is a rectangle.

ABCDis a rectangle.

In a rectangle, diagonals are congruent.

  $AC\cong DB$

  $\overline{AC}$ is congruent to $\overline{DB}$ .