Question

homework 6.4
are you given enough information to determine whether the quadrilateral is a parallelogram? explain your answer.

1. diagram of a quadrilateral with two marked sides
2. diagram of a quadrilateral with diagonals marked
3. diagram of a quadrilateral with four marked sides
4. diagram of a quadrilateral with two marked sides and an arrow
5. diagram of a quadrilateral with a diagonal and marked angles
6. diagram of a quadrilateral with angle measures 120°, 60°, 60°

Explanation:

1.

Step1: Use the property of parallelogram (one pair of opposite sides are equal)

In a parallelogram, one of the properties is that if one pair of opposite sides are equal, then it is a parallelogram. Here, we are given that one pair of opposite sides are equal.

2.

Step1: Use the property of parallelogram (diagonals bisect each other)

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. The marks on the diagonals show that they bisect each other.

3.

Step1: Use the property of parallelogram (both pairs of opposite sides are equal)

A quadrilateral with both pairs of opposite sides equal is a parallelogram. The marks on the sides show that both pairs of opposite sides are equal.

4.

Step1: Analyze the given information

Only one pair of sides is equal and there is no information about the other pair of sides or angles or diagonals. So, not enough information.

5.

Step1: Use the property of parallelogram (alternate - interior angles and congruent triangles)

The equal angles (alternate - interior angles) formed by the transversal and the sides can be used to show that the triangles formed by the diagonal are congruent. Then, we can get that the opposite sides are parallel (by corresponding angles) and equal (by congruent triangles).

6.

Step1: Use the property of parallelogram (consecutive angles are supplementary)

In a parallelogram, consecutive angles are supplementary. Here, \(120^{\circ}+60^{\circ} = 180^{\circ}\), and also, we can use the property that if both pairs of opposite angles are equal (two angles of \(60^{\circ}\) and two angles of \(120^{\circ}\)), then it is a parallelogram.

Answer:

1. Yes, because one pair of opposite sides are equal.
2. Yes, because the diagonals bisect each other.
3. Yes, because both pairs of opposite sides are equal.
4. No, only one pair of sides is equal and no other information.
5. Yes, by using the properties of alternate - interior angles and congruent triangles to show opposite sides are parallel and equal.
6. Yes, because consecutive angles are supplementary and both pairs of opposite angles are equal.