Question

your turn
show that each quadrilateral is a parallelogram for the given values of the variables. step 2 is done for you.

1. \\(z = 11\\) and \\(w = 4.5\\)

step 1 find \\(\angle f\\) and \\(\angle h\\)
step 2 find \\(\angle e\\) and \\(\angle g\\)
\\\angle e = 12(4.5) + 8\\
\\= 54 + 8\\
\\= 62\\
\\\angle g = 14(4.5) - 1\\
\\= 63 - 1\\
\\= 62\\
so, \\(m\angle f = m\angle\\) and \\(m\angle e = m\angle\\) . efgh is a parallelogram since

1. \\(a = 2.4\\) and \\(b = 9\\)

Explanation:

Calculate angle F and angle H

Using the Solving Linear Equations knowledge point

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Identify equal opposite angles

Using the Converse of Parallelogram Theorems knowledge point

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State the parallelogram condition

Using the Converse of Parallelogram Theorems knowledge point

$$ \text{Both pairs of opposite angles are congruent.} $$

Calculate side lengths and angles for Question 2

Using the Solving Linear Equations knowledge point

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Verify the parallelogram condition for Question 2

Using the Converse of Parallelogram Theorems knowledge point

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Answer:

Question 1

So, \(m\angle F = m\angle\) <blank>\(H\)</blank> and \(m\angle E = m\angle\) <blank>\(G\)</blank>. \(EFGH\) is a parallelogram since <blank>both pairs of opposite angles are congruent</blank>.

Question 2

For \(a = 2.4\) and \(b = 9\):

• \(QP = 7(2.4) = 16.8\)
• \(RS = 2(2.4) + 12 = 16.8\)
• \(m\angle Q = 10(9) - 16 = 74^\circ\)
• \(m\angle R = 9(9) + 25 = 106^\circ\)

Since \(QP = RS = 16.8\), one pair of opposite sides is congruent. Since \(m\angle Q + m\angle R = 74^\circ + 106^\circ = 180^\circ\), the consecutive interior angles are supplementary, which proves that \(QP \parallel RS\). Therefore, \(PQRS\) is a parallelogram because <blank>one pair of opposite sides is both parallel and congruent</blank>.