Lesson 4 Parallelogram Conjectures and Proof Solidify Understanding

Ready

Use a compass and a straightedge to bisect the following line segments.

1.

2.

3.

Construct the midpoint, $B$ , of $\overset{―}{M S}$ . Then connect point $B$ to point $H$ .

4.

Construct the angle bisector of $∠ X Y Z$ .

5.

Construct a square inscribed in the circle.

6.

Construct a regular hexagon inscribed in the circle.

Set

7.

Quadrilateral $B C D E$ below was formed by $2$ sets of intersecting parallel lines. Figure 2 is the image of Figure 1. It has been rotated $180 °$ . Find the center of rotation for Figure 1. Make a list of everything that has been preserved in the rotation. Then make a list of anything that has changed.

Is quadrilateral $B C D E$ a parallelogram? How do you know?

8.

Given: $\overset{―}{B M} ≅ \overset{―}{D M}$ , $\overset{―}{A M} ≅ \overset{―}{C M}$

Prove: Quadrilateral $A B C D$ is a parallelogram.

9.

Given: $\overset{―}{A B} ≅ \overset{―}{C D}$ and $\overset{―}{B C} ≅ \overset{―}{A D}$

Prove: Quadrilateral $A B C D$ is a parallelogram.

10.

Given: Quadrilateral $M A R N$ is a parallelogram.

Prove: $∠ M A R ≅ ∠ R N M$ and $∠ A R N ≅ ∠ N M A$

11.

Given: $M A R N$ is a parallelogram.

$A B D M$ is an image of $M A R N$ after a $180 °$ rotation about point $F$

$M A C E$ is an image of $M A R N$ after being reflected over $\overset{―}{M A}$

Prove: $△ A B C ≅ △ M D E$

Go

State whether each statement is true or false. If it is false, explain why or rewrite the statement to make it true.

12.

If a triangle is equilateral, then the median and the altitude are the same segments.

13.

The perpendicular bisectors of the sides of a triangle also bisect the angles.

14.

Some of the angles in a triangle equal $180 °$ .

15.

An altitude of a triangle may fall on the exterior of the triangle.

16.

The third angle in a triangle is always the supplement to the sum of the other $2$ angles.

17.

In a right triangle, the $2$ acute angles are always complementary.

18.

All squares are also rectangles.

19.

A rhombus is always a square.

20.

If a figure is a trapezoid, then it is also a parallelogram.

21.

The diagonals of a rectangle bisect the angles.

22.

A parallelogram can have $3$ obtuse angles.

23.

The figure made by two pairs of intersecting parallel lines is always a parallelogram.

24.

All of the angles in a parallelogram can be congruent.

25.

A diagonal always divides a quadrilateral into $2$ congruent triangles.

26.

If a quadrilateral goes through a translation, the sides of the pre-image and image will remain parallel.