Since abcd is a parallelogram, the two pairs of sides (ab¯ and cd¯, as well as ad¯ and bc¯) are congruent. then, since ∠9 and ∠11 are vertical angles, it can be concluded that ∠9≅∠11. since abcd is a parallelogram, ab¯∥cd¯. since ∠2 and ∠5 are alternate interior angles along these parallel lines, the alternate interior angles theorem allows that ∠2≅∠5. since two angles of △aeb are congruent to two angles of △ced, the third angles theorem supports that ∠8≅∠3. therefore, using the it can be stated that △aeb≅△ced. then, applying the definition of congruent triangles, it can be stated that ae¯≅ce¯, which makes e the midpoint of ac¯. use a similar argument to prove that △aed≅△ceb; then it can be concluded that e is also the midpoint of bd¯. since the midpoint of both line segments is the same point, the segments bisect each other by definition. match each number (1 and 2) with the word or phrase that correctly fills in the corresponding blank in the proof.

1. Opposite

2. angle-side-angle criterion

Step-by-step explanation:

Since ABCD is a parallelogram, the two pairs of (opposite) sides (AB¯ and CD¯, as well as AD¯ and BC¯) are congruent. Then, since ∠9 and ∠11 are vertical angles, it can be concluded that ∠9≅∠11. Since ABCD is a parallelogram, AB¯∥CD¯. Since ∠2 and ∠5 are alternate interior angles along these parallel lines, the Alternate Interior Angles Theorem allows that ∠2≅∠5. Since two angles of △AEB are congruent to two angles of △CED, the Third Angles Theorem supports that ∠8≅∠3. Therefore, using the (angle-side-angle criterion), it can be stated that △AEB≅△CED. Then, applying the definition of congruent triangles, it can be stated that AE¯≅CE¯, which makes E the midpoint of AC¯. Use a similar argument to prove that △AED≅△CEB; then it can be concluded that E is also the midpoint of BD¯. Since the midpoint of both line segments is the same point, the segments bisect each other by definition. Match each number (1 and 2) with the word or phrase that correctly fills in the corresponding blank in the proof.

A parallelogram posses the following features:

   1. The opposite sides are parallel.

   2. The opposite sides are congruent.

   3. It has supplementary consecutive angles.

   4. The diagonals bisect each other.

step-by-step explanation:

we are given a graph for system of inequalities.

in order to write the system of inequalities for the given graph, we need to find the inequality for each of the line separately.

let us find the slope of dotted line first.

taking two coordinates on the graph.

let us say (0,0) and (3,1).

let us find the slope of the line now.

slope of the dotted line would be =.

y-intercept of dotted line is y=0 only.

plugging slope m= and b=0 in slope-intercept form y=mx+b, we get

.

the line is shaded up of that dotted line.

therefore, for shading up we would take greater than inequality symbol > .

and because it's a dotted line we would take only > symbol in place of = sign in above equation of line.

therefore, inequality for dotted line would be

let us take two points (3,1) and (0,4) on the solid line.

slope of the solid line would be =.

for the solid line we have y-intercept b= 4.

plugging slope m= -1 and b=4 in slope-intercept form y=mx+b, we get

y = -1x + 4.

the line is shaded down of that solid line.

therefore, for shading down we would take less than inequality symbol < .

and because it's a solid line we would take ≤ symbol in place of = sign in above equation of line.

therefore, inequality for solid line would be