Question

Parallelogram Proofs Quick Check
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Question
Kite ABCD
has diagonal lines AC¯¯¯¯¯¯¯¯
and BD¯¯¯¯¯¯¯¯
that intersect at E
, with BD¯¯¯¯¯¯¯¯
being the shortest diagonal line. Given that BE¯¯¯¯¯¯¯¯=x+3
and DE¯¯¯¯¯¯¯¯=4x
, investigate the pattern and make a conjecture about line BD¯¯¯¯¯¯¯¯
. What is the total length of BD¯¯¯¯¯¯¯¯
?(1 point)
Responses

4
4

12
12

8
8

1
1
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Parallelogram Proofs Quick Check
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Question
Use the image to answer the question.

A quadrilateral is labeled upper A upper B upper C upper D. A segment connects upper B and upper D. Another segment connects upper A and upper C. The point where the segments intersect is upper E.

In the rectangle BCDA
, AE¯¯¯¯¯¯¯¯
can be expressed as 3x
and ED¯¯¯¯¯¯¯¯
as −4x+21
. Identify a pattern to find the total length of BD¯¯¯¯¯¯¯¯
.

(1 point)
Responses

3
3

18
18

27
27

9
9
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Parallelogram Proofs Quick Check
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Question
Use the image to answer the question.

Two parallel horizontal lines a and b, intersecting with two parallel slanted lines e and f. A parallelogram is formed in the center of the intersecting lines.

In the diagram, a∥b
and e∥f
. Sylvia writes a proof to prove that opposite angles, ∠6
and ∠11
, are congruent in the parallelogram. Drag and drop the statements and reasons into their correct locations in the two-column proof.

(2 points)
Put responses in the correct response input area to answer the question. Select a response, navigate to the desired input area and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
\
Complete the Table to Prove Congruency Between Angles
Statement Reason
1. a∥b
and e∥f
given
2. Press enter key to drop response here.statement 2 that is proven by the Corresponding angles theorem Corresponding Angles Theorem
3. m∠5+m∠6=180°
and m∠9+m∠11=180°
definition of linear pair
4. m∠13+m∠6=180°
and m∠13+m∠11=180°
Press enter key to drop response here.reason that proves statement 4
5. Press enter key to drop response here.statement that is proven by the Transitive Property of Equality Transitive Property of Equality
6. m∠6=m∠11
Press enter key to drop response here.reason that proves statement 6

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Parallelogram Proofs Quick Check
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Question
Use the image to answer the question.

Parallelogram upper X upper Y upper Z upper W is shown. Vertex upper X is on the top left. Vertex upper W is on the bottom left.

In parallelogram WXYZ
, the interior angle at point Y
measures 30 degrees. What is the measure of the interior angle at point X
?

(1 point)
Responses

330 degrees
330 degrees

150 degrees
150 degrees

30 degrees
30 degrees

60 degrees
60 degrees
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Parallelogram Proofs Quick Check
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Question
Use the image to answer the question.

Parallelogram upper D upper E upper F upper G is shown. Diagonals are drawn from upper D to upper F and upper E to upper G to meet at upper H.

In parallelogram DEFG
, which segment bisects EG¯¯¯¯¯¯¯¯
?

(1 point)
Responses

EH¯¯¯¯¯¯¯¯
upper E upper H

EF¯¯¯¯¯¯¯¯
upper E upper F

FG¯¯¯¯¯¯¯¯
upper F upper G

DF¯¯¯¯¯¯¯¯
upper D upper F
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Answer 1

Here are the answers:

1) BD = 8

(Reason: In a kite the longer diagonal bisects the shorter, so BE = DE → x+3 = 4x → x = 1, BD = BE+DE = 4+4 = 8.)

2) BD = 18

(Reason: In a rectangle diagonals are congruent and bisect each other → AE = ED. So 3x = −4x+21 → x = 3, ED = 9, BD = 2·9 = 18.)

3) Filled two-column proof entries
- Statement 2: ∠5 ≅ ∠13 and ∠9 ≅ ∠13
Reason: Corresponding Angles Theorem
- Reason for statement 4: Substitution Property of Equality
- Statement 5 (proved by Transitive Property of Equality): m∠13 + m∠6 = m∠13 + m∠11
- Reason for statement 6 (m∠6 = m∠11): Subtraction Property of Equality

4) m∠X = 150°

(Adjacent angles in a parallelogram are supplementary: 180° − 30° = 150°.)

5) DF

(Reason: Diagonals of a parallelogram bisect each other, so DF bisects EG.)