Question

1. use the given information to prove that a || b.

given: ∠4≅∠8
prove: a || b

1. use the given information to prove that x || y.

given: ∠3≅∠6
prove: x || y

1. for each of the following quadrilaterals, select all the properties that must be true.

	two pairs of parallel lines	only one pair of parallel sides	all sides congruent	four right angles
rhombus
rectangle
square

1. in the coordinate plane the point a(-2,1) is translated to the point a(2, - 1). under the same translation the points b(-5,3) and c(0,4) are translated to b and c, respectively. what are the coordinates of b and c?
2. in the coordinate plane the point a(-4,4) is translated to the point a(-3,1). under the same translation the points b(-2,0) and c(-6,6) are translated to b and c, respectively. what are the coordinates of b and c?

Explanation:

5. Proving \(a\parallel b\)

Step1: Recall corresponding - angles postulate

If corresponding angles are congruent, then the lines are parallel. \(\angle4\) and \(\angle8\) are corresponding angles.

Step2: State the given and conclusion

Given \(\angle4\cong\angle8\). By the corresponding - angles postulate, \(a\parallel b\).

Statement	Reason
\(a\parallel b\)	Corresponding - angles postulate

6. Proving \(x\parallel y\)

Step1: Recall alternate - interior angles theorem

If alternate - interior angles are congruent, then the lines are parallel. \(\angle3\) and \(\angle6\) are alternate - interior angles.

Step2: State the given and conclusion

Given \(\angle3\cong\angle6\). By the alternate - interior angles theorem, \(x\parallel y\).

Statement	Reason
\(x\parallel y\)	Alternate - interior angles theorem

7. Properties of quadrilaterals

Quadrilateral	Two pairs of parallel lines	Only one pair of parallel sides	All sides congruent	Four right angles
Rhombus	Yes	No	Yes	No
Rectangle	Yes	No	No	Yes
Square	Yes	No	Yes	Yes

8.

Step1: Find the translation rule

To get from \(A(-2,1)\) to \(A'(2, - 1)\), we have \(x\) - coordinate change: \(2-(-2)=4\) and \(y\) - coordinate change: \(-1 - 1=-2\). The translation rule is \((x,y)\to(x + 4,y-2)\).

Step2: Translate point \(B\)

For \(B(-5,3)\), \(x=-5,y = 3\). New \(x=-5 + 4=-1\), new \(y=3-2 = 1\). So \(B'(-1,1)\).

Step3: Translate point \(C\)

For \(C(0,4)\), \(x = 0,y = 4\). New \(x=0 + 4=4\), new \(y=4-2 = 2\). So \(C'(4,2)\).

9.

Answer:

Step1: Find the translation rule

To get from \(A(-4,4)\) to \(A'(-3,1)\), \(x\) - coordinate change: \(-3-(-4)=1\) and \(y\) - coordinate change: \(1 - 4=-3\). The translation rule is \((x,y)\to(x + 1,y-3)\).

Step2: Translate point \(B\)

For \(B(-2,0)\), \(x=-2,y = 0\). New \(x=-2 + 1=-1\), new \(y=0-3=-3\). So \(B'(-1,-3)\).

Step3: Translate point \(C\)

For \(C(-6,6)\), \(x=-6,y = 6\). New \(x=-6 + 1=-5\), new \(y=6-3 = 3\). So \(C'(-5,3)\).