Question

in the diagram, \\( \angle jkm \\) is a straight angle.
which statements about the diagram are true? check
all that apply.
\\( \angle jkl \cong \angle qkm \\)
\\( \overrightarrow{pk} \\) is an angle bisector.
\\( m \angle mkq + m \angle pkq = m \angle pkm \\)
\\( m \angle jkl = 45 ^ { \circ } \\)
\\( \angle lkq \\) is bisected.
\\( \overrightarrow{kq} \\) is an angle bisector.

Explanation:

Step1: Analyze angle - bisector and angle - addition properties

• For the angle - addition property: If we have angles \(\angle MKQ\) and \(\angle PKQ\), then \(m\angle MKQ + m\angle PKQ=m\angle PKM\) (by the definition of adjacent angles forming a larger angle).
• For the congruent angles: \(\angle JKL\) and \(\angle QKM\) are congruent. Since \(\angle JKM\) is a straight angle (\(180^{\circ}\)) and assume the right - angle (the small square) is \(90^{\circ}\). Let the non - right - angle parts be considered. If we assume the two non - right - angle parts (one in \(\angle JKL\) and one in \(\angle QKM\)) are equal. For example, if we assume the sum of angles around point \(K\) and use the fact that the right - angle divides the straight angle \(\angle JKM\) into two parts, and the non - right parts of \(\angle JKL\) and \(\angle QKM\) are equal.

Step2: Analyze incorrect statements

• For the statement \(m\angle JKL = 45^{\circ}\): There is no information given in the diagram to suggest that \(\angle JKL\) is \(45^{\circ}\). We only know that there is a right - angle (\(90^{\circ}\)) and a straight angle (\(180^{\circ}\)), but no indication of equal non - right - angle sub - divisions to get \(45^{\circ}\) for \(\angle JKL\).
• For the statement \(\overrightarrow{PK}\) is an angle bisector: There is no information (such as equal - angle markings) to suggest that \(\overrightarrow{PK}\) divides an angle into two equal parts.
• For the statement \(\angle LKQ\) is bisected: There is no information (such as equal - angle markings) to suggest that \(\overrightarrow{KQ}\) or any other ray bisects \(\angle LKQ\).

Answer:

\(\angle JKL\cong\angle QKM\), \(m\angle MKQ + m\angle PKQ=m\angle PKM\)