Question

parallel lines and shapes
2.
5.
6.

Explanation:

Step1: Identify angle - sum property for trapezoid

For the trapezoid in the first figure, the sum of adjacent angles between parallel lines is 180 degrees. So, \(y + 3y=180\).

Step2: Solve the equation for \(y\)

Combining like - terms, we get \(4y = 180\). Then, dividing both sides by 4 gives \(y=\frac{180}{4}=45\).

Step3: Use angle - parallel line relationship for triangle

In the second figure, since \(HJ\parallel KL\), the corresponding angles are equal. So, \(3x = 43\) (corresponding angles), then \(x=\frac{43}{3}\approx14.33\). Also, using the angle - sum property of a triangle (\(90+(y - 9)+43 = 180\)), substitute \(y = 45\) into it: \(90+(45 - 9)+43=90 + 36+43=169
eq180\). There is a mistake in using the triangle's angle - sum property here. Since \(HJ\parallel KL\), the alternate interior angle gives \(3x=43\), \(x = \frac{43}{3}\), and for the other angle relationship, we know that the non - right non - \(43^{\circ}\) angle in the smaller right - triangle and \((y - 9)^{\circ}\) are corresponding angles. So \(y-9 = 43\), then \(y=43 + 9=52\).

Answer:

For the trapezoid, \(y = 45\). For the triangle, \(x=\frac{43}{3}\), \(y = 52\)