Question

lesson 9.1 homework
complete problems 1-5 below for independent practice.
when you are finished, check the solutions with your teacher.

1. solve for (x) in the parallelogram below.

1. solve for (x) in the parallelogram below. then, find the measure of (overline{qr}).

1. solve for (x) in the parallelogram below. then, find the measure of (angle y).

1. solve for (x) in the parallelogram below. then, find the measure of (angle t).

1. (abcd) is a parallelogram. determine whether each statement must be true. select the correct answer for each lettered part.

a. the perimeter of (abcd) is (2ab + 2bc). yes no
b. (de = \frac{1}{2} db) yes no
c. (overline{bc} cong overline{dc}) yes no
d. (angle dac cong angle bca) yes no
e. (delta aed cong delta ceb) yes no
f. (angle dac cong angle bac) yes no

Explanation:

Solve for x in Question 1

Using the Consecutive Angles of Parallelogram and Solving Linear Equations knowledge points

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Solve for x and QR in Question 2

Using the Parallelogram Theorems and Solving Linear Equations knowledge points

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LATEXBLOCK1
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Solve for x and angle Y in Question 3

Using the Opposite Angles of Parallelogram and Solving Linear Equations knowledge points

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Solve for x and angle T in Question 4

Using the Consecutive Angles of Parallelogram and Solving Linear Equations knowledge points

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Analyze statements for Question 5

Using the Parallelogram Theorems and Parallelogram Diagonal Properties knowledge points

• A. Perimeter of ABCD is \(2AB + 2BC\): Since opposite sides are equal (\(AB = CD\) and \(BC = DA\)), the perimeter is \(AB + BC + CD + DA = 2AB + 2BC\). (Yes)
• B. \(DE = \frac{1}{2}DB\): Diagonals of a parallelogram bisect each other, so \(E\) is the midpoint of \(DB\). (Yes)
• C. \(\overline{BC} \cong \overline{DC}\): Consecutive sides are not necessarily congruent in a general parallelogram. (No)
• D. \(\angle DAC \cong \angle BCA\): Since \(AD \parallel BC\), alternate interior angles are congruent. (Yes)
• E. \(\Delta AED \cong \Delta CEB\): By SAS or ASA using bisected diagonals and vertical angles, the opposite triangles are congruent. (Yes)
• F. \(\angle DAC \cong \angle BAC\): Diagonals do not necessarily bisect the angles in a general parallelogram. (No)

Answer:

Question 1

\(x = 6\)

Question 2

\(x = 4\), \(QR = 6\)

Question 3

\(x = 10\), \(m\angle Y = 75^\circ\)

Question 4

\(x = 11\), \(m\angle T = 120^\circ\)

Question 5

• A. The perimeter of ABCD is \(2AB + 2BC\).
• Yes (Correct answer)
• No

• B. \(DE = \frac{1}{2}DB\)
• Yes (Correct answer)
• No

• C. \(\overline{BC} \cong \overline{DC}\)
• Yes
• No (Correct answer)

• D. \(\angle DAC \cong \angle BCA\)
• Yes (Correct answer)
• No

• E. \(\Delta AED \cong \Delta CEB\)
• Yes (Correct answer)
• No

• F. \(\angle DAC \cong \angle BAC\)
• Yes
• No (Correct answer)