Math Problem Statement

Quadrilateral ABCD is shown on the coordinate plane.

A figure shows a projector is placed 10 meters from a screen ABC with an angle of A 36 degrees projector and ABC with an angle of A36 degrees. What needs to be proven to conclude that quadrilateral ABCD is a parallelogram?

A. side length of AB = side length of CD and slope of BC × slope of CD = -1 B. side length of BC = side length of DA and slope of DA × slope of AB = -1 C. slope of AB = slope of CD and slope of BC = slope of DA D. slope of AB = slope of BC and slope of CD = slope of DA

Solution

To prove that quadrilateral ABCD is a parallelogram, we need to establish that both pairs of opposite sides are parallel. This is one of the defining properties of parallelograms.

  • Two lines are parallel if they have the same slope.
  • Thus, for quadrilateral ABCD to be a parallelogram, we need to show that the opposite sides, AB and CD, and BC and DA, are parallel.

This means:

  • The slope of AB must be equal to the slope of CD.
  • The slope of BC must be equal to the slope of DA.

Therefore, the correct answer is:

C.
slope of AB = slope of CD and slope of BC = slope of DA

This condition proves that both pairs of opposite sides are parallel, which is sufficient to conclude that ABCD is a parallelogram.

Would you like more details or have any questions?

Here are 5 related questions to explore this further:

  1. What other properties can define a parallelogram?
  2. How can the slopes of diagonals help in proving a quadrilateral is a parallelogram?
  3. What if only one pair of opposite sides has equal slopes? What type of quadrilateral would that be?
  4. How can the distance formula be used to prove sides are equal in a parallelogram?
  5. What are the conditions to prove a quadrilateral is a rectangle or rhombus?

Tip: To prove lines are parallel, always check their slopes—if they are equal, the lines are parallel.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Properties of Parallelograms
Slopes of Lines

Formulas

Slope formula: (y2 - y1) / (x2 - x1)
Properties of parallelograms: Opposite sides are parallel and equal in length

Theorems

Slope Theorem
Parallel Line Theorem for Parallelograms

Suitable Grade Level

Grades 9-10