Question

1. (10 pts) in the figure below, \\(\overleftrightarrow{tf} \parallel \overleftrightarrow{vl}\\), \\(\overleftrightarrow{ds} \parallel \overleftrightarrow{yq}\\), \\(m\angle rhb = 2x + 25\\), \\(m\angle myf = 5x + 113\\).

a. find the measures of \\(\angle rhb\\) and \\(\angle myf\\). justify your answer and each step you take by stating which theorem, postulate, fact, or definition you used.

Explanation:

Identify the geometric relationship

We are given that line \(\overleftrightarrow{TF} \parallel \overleftrightarrow{VH}\) and line \(\overleftrightarrow{DS} \parallel \overleftrightarrow{YQ}\).
Let us analyze the angles \(\angle RHB\) and \(\angle MYF\):

• \(\angle RHB\) is formed by the transversal line \(\overleftrightarrow{VH}\) and the parallel line \(\overleftrightarrow{DS}\) (or \(\overleftrightarrow{DR}\)).
• \(\angle MYF\) is formed by the transversal line \(\overleftrightarrow{TF}\) and the parallel line \(\overleftrightarrow{YQ}\) (or \(\overleftrightarrow{MQ}\)).

Since \(\overleftrightarrow{TF} \parallel \overleftrightarrow{VH}\) and \(\overleftrightarrow{DS} \parallel \overleftrightarrow{YQ}\), the quadrilateral \(SHBY\) is a parallelogram.

• \(\angle RHB\) and \(\angle SHB\) are supplementary (linear pair), so \(m\angle SHB = 180^\circ - m\angle RHB\).
• Since \(SHBY\) is a parallelogram, opposite angles are equal: \(m\angle SYB = m\angle SHB\).
• \(\angle MYF\) and \(\angle SYB\) are vertical angles, so \(m\angle MYF = m\angle SYB\).

Therefore, \(m\angle MYF = m\angle SHB = 180^\circ - m\angle RHB\).
This means \(\angle RHB\) and \(\angle MYF\) are supplementary:

$$m\angle RHB + m\angle MYF = 180^\circ$$

Set up the algebraic equation

Using the given expressions:

$$(2x + 25) + (5x + 113) = 180$$

Combine like terms:

$$7x + 138 = 180$$

Solve for x

Using the Linear Equation Solving knowledge point

$$
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$$

Calculate the angle measures

Substitute \(x = 6\) back into the angle expressions:

• \(m\angle RHB = 2(6) + 25 = 12 + 25 = 37^\circ\)
• \(m\angle MYF = 5(6) + 113 = 30 + 113 = 143^\circ\)

Check: \(37^\circ + 143^\circ = 180^\circ\).

Answer:

\(m\angle RHB = 37^\circ\) and \(m\angle MYF = 143^\circ\)