Can some one please check these for me.Thanks

Angle DEF is similar to Angle HJK and the scale factor of angle DEF to anle HJK IS 5/2. If EF =15 FIND JK
Here is what I came up with Am I correct
5/2 = EF/JK
5/2 =15/JK
5(JK)=30
JK=30/5
JK=6

Given: ABCD is a parallelogram; <1 is congruent to <2

To Prove: ABCD is a Rhombus

Plan: Show <2 is congruent to < CAB.
hence CB is congruent to AB,
making ABCD a parallelogram
with consecutive sides congruent

PROOF

Statements Reasons
1. ABCD is a parallelogram 1.Given
<1 is congruent to <2
2. Draw segment from A to C 2. Two points determine exactly one line
3.< 1 & <3 are congruent 3. Alternate angles are congruent
4. AC bisects angle 1&2 4. Def. of Angle bisector
5 . m<DAB=m<BCD 5. Opposite angles in a parallelogram are equal
6. <2 & <3 are congruent 6. Transitive prop 7 . AB=BC 7. Sides oposite are congruent angles
8.<1 & <2 8. Cor.parts are eual
9. AB=BC=CD=DA 9. ABCD is a rhombus

Your solution for the first problem is correct. If EF = 15 and the scale factor is 5/2, then JK = 6.

For the second problem, the proof is mostly correct, but there are a few minor things that need to be fixed or clarified:

1) Instead of stating "<1 is congruent to <2" twice in the Statements column, you should state "1.Given" and "8. <1 is congruent to <2" in the Reasons column.

2) In statement 5, it should be "m<DAB = m<BCA" instead of "m<BCD"

3) In statement 6, it should be "If <1 is congruent to <2 and <1 is congruent to <3, then <2 is congruent to <3" in the Reasons column.

4) In statement 8, it should be "<CAB is congruent to <CBD" or "<DAB is congruent to <BCA" instead of "<1 and <2"

Here is the modified proof:

PROOF

Statements Reasons
1. ABCD is a parallelogram 1. Given
2. <1 is congruent to <2 1. Given
3. Draw segment from A to C 3. Two points determine exactly one line
4. < 1 & <3 are congruent 4. Alternate angles are congruent
5. AC bisects angle 1 & 2 5. Def. of Angle bisector
6. m<DAB = m<BCA 6. Opposite angles in a parallelogram are equal
7. <2 is congruent to <3 7. If <1 is congruent to <2 and <1 is congruent to <3, then <2 is congruent to <3
8. <CAB is congruent to <CBD 8. Cor.parts are equal
9. AB=BC=CD=DA 9. Sides opposite congruent angles are congruent
10. ABCD is a rhombus 10. A parallelogram with consecutive sides congruent is a rhombus

For the first problem:

You are correct in setting up the proportion EF/JK = 5/2.

To solve for JK, you can cross multiply and then divide:

(2)(EF) = (5)(JK)
2(15) = 5(JK)
30 = 5(JK)
JK = 30/5
JK = 6

So your answer for JK is correct, it is 6.

For the second problem:

The proof provided is not complete. You have listed some statements and reasons, but there are missing steps and justifications.

Here is a corrected proof:

Statements Reasons
1. ABCD is a parallelogram Given
2. <1 is congruent to <2 Given
3. Draw segment AC Two points determine exactly one line
4. <1 & <3 are congruent Alternate angles are congruent
5. AC bisects angle 1 & 2 Definition of Angle bisector
6. m<DAB = m<BCD Opposite angles in a parallelogram are equal
7. <2 & <3 are congruent Transitive property
8. Opposite sides AB and CD are congruent Opposite sides of a parallelogram are congruent
9. ABCD is a rhombus Definition of a rhombus

To prove that ABCD is a rhombus, you need to show that all sides are congruent, not just the consecutive sides AB and BC. You can do this by using the fact that opposite sides of a parallelogram are congruent, as shown in step 8.

For the first question:

To find JK, we know that the scale factor of angle DEF to angle HJK is 5/2. This means that the corresponding sides of these angles are in the same ratio.

From the given information, we have EF = 15 and we want to find JK. We can set up a proportion using the scale factor:

EF/JK = 5/2

Substituting the values we have:

15/JK = 5/2

Now, cross-multiply to solve for JK:

2 * 15 = JK * 5
30 = 5 * JK

To isolate JK, divide both sides by 5:

JK = 30/5
JK = 6

Therefore, JK is equal to 6.

For the second question:

To prove that ABCD is a rhombus, we need to show that the opposite sides are congruent. Let's go through the given proof step by step:

1. ABCD is a parallelogram (Given)
<1 is congruent to <2

2. Draw segment from A to C (Two points determine exactly one line)

3. <1 and <3 are congruent (Alternate angles are congruent)

4. AC bisects angle 1 and 2 (Definition of angle bisector)

5. m<DAB = m<BCD (Opposite angles in a parallelogram are equal)

6. <2 and <3 are congruent (Transitive property)

7. AB = BC (Since <1 and <2 are congruent, their adjacent sides AB and BC are congruent)

8. <1 and <2 (Corresponding parts of congruent triangles are equal)

9. AB = BC = CD = DA (AB and BC are congruent, and by symmetry, CD and DA are also congruent)

Therefore, based on the proof, we can conclude that ABCD is a rhombus since all four sides are congruent.