Question

here are two intersecting lines. find the measures of the unknown angles. find the measure of the unknown angles. angle skw measures degrees. angle hkw measures degrees. angle eks measures degrees.

Explanation:

Step1: Analyze Angle EKS

Lines \( EK \) and \( SK \) are perpendicular (since they are along East - South axes), so the angle between them is \( 90^\circ \). So \( \angle EKS = 90^\circ \).

Step2: Analyze Angle HKW

\( \angle EKH = 92^\circ \), and \( \angle EKH \) and \( \angle HKW \) are supplementary (they form a linear pair along the line \( EW \)). The sum of supplementary angles is \( 180^\circ \). Let \( \angle HKW=x \), then \( 92^\circ + x=180^\circ \), so \( x = 180^\circ - 92^\circ=88^\circ \)? Wait, no, wait. Wait, \( EK \) and \( WK \) are a straight line (East - West), so \( \angle EKW = 180^\circ \). \( \angle EKH = 92^\circ \), so \( \angle HKW=180^\circ - 92^\circ = 88^\circ \)? Wait, no, maybe I made a mistake. Wait, actually, \( EK \) and \( SK \) are perpendicular, \( SK \) and \( WK \) are a straight line? Wait, no, the axes: East (E), West (W) are a straight line, North - South? Wait, no, in the diagram, \( E \) and \( W \) are opposite, \( S \) and... Wait, the diagram has \( E \) (up), \( H \) (right), \( S \) (left), \( W \) (down)? Wait, maybe the lines are \( EW \) (vertical) and \( SH \) (horizontal). So \( EK \) is vertical (E to K to W), \( SK \) is horizontal (S to K to H). So \( \angle EKS \): \( EK \) (vertical) and \( SK \) (horizontal) intersect at K, so they are perpendicular, so \( \angle EKS = 90^\circ \).

\( \angle EKH = 92^\circ \), and \( \angle EKH \) and \( \angle HKW \): \( EK \) and \( WK \) are a straight line (vertical), so \( \angle EKW = 180^\circ \). Wait, no, \( EK \) is from E to K, \( WK \) is from K to W, so \( EK \) and \( WK \) are a straight line (180 degrees). \( \angle EKH = 92^\circ \), so \( \angle HKW=180^\circ - 92^\circ=88^\circ \)? Wait, no, maybe \( \angle EKH \) and \( \angle SKW \): Wait, let's re - examine.

Wait, the two lines are \( SH \) (horizontal: S to K to H) and \( EW \) (vertical: E to K to W). So the intersection is at K.

- For \( \angle EKS \): \( EK \) (vertical) and \( SK \) (horizontal) are perpendicular, so \( \angle EKS = 90^\circ \).

- For \( \angle HKW \): \( \angle EKH = 92^\circ \), and \( \angle EKH \) and \( \angle HKW \) are adjacent angles forming a linear pair along \( EW \)? Wait, no, \( EW \) is a straight line, so \( \angle EKH+\angle HKW = 180^\circ \)? No, \( EK \) is part of \( EW \), \( HK \) is part of \( SH \). Wait, maybe \( \angle EKH \) and \( \angle SKW \) are vertical angles? No, vertical angles are opposite. Wait, \( \angle EKH = 92^\circ \), then \( \angle SKW \) is equal to \( \angle EKH \) if they are vertical angles? Wait, no, let's use the fact that adjacent angles on a straight line sum to 180, and perpendicular lines form 90 - degree angles.

Wait, let's start over:

1. \( EK \perp SK \) (since they are along perpendicular axes, like North - South and East - West, but here E is up, S is left, H is right, W is down). So \( \angle EKS = 90^\circ \).

2. \( \angle EKH = 92^\circ \), and \( \angle EKH \) and \( \angle HKW \): \( EK \) and \( WK \) are a straight line (vertical), so \( \angle EKW = 180^\circ \). \( \angle EKH = 92^\circ \), so \( \angle HKW=180^\circ - 92^\circ = 88^\circ \)? No, that can't be. Wait, maybe \( \angle EKH \) and \( \angle SKW \): Since \( SH \) is a straight line (horizontal), \( \angle SKH = 180^\circ \). \( \angle EKH = 92^\circ \), \( \angle EKS = 90^\circ \), so \( \angle SKW \): Let's see, \( \angle EKH + \angle HKW=180^\circ \) (since \( EW \) is straight). Wait, I think I messed up the axes. Let's assume that \( E \) and \( W \) are a straight line (horizontal), and \( S \) and \( H \) are a straight line (vertical). No, the diagram shows \( E \) (up), \( H \) (right), \( S \) (left), \( W \) (down). So \( EW \) is vertical (up - down), \( SH \) is horizontal (left - right). So intersection at K.

- \( \angle EKH \): between \( EK \) (up) and \( HK \) (right), is \( 92^\circ \).

- \( \angle EKS \): between \( EK \) (up) and \( SK \) (left). Since \( SK \) and \( HK \) are a straight line (left - right), \( \angle SKH = 180^\circ \). So \( \angle EKS=180^\circ - 92^\circ = 88^\circ \)? Wait, no, that contradicts the perpendicular idea. Wait, maybe the lines \( EW \) and \( SH \) are not perpendicular? But usually, in such diagrams, the axes are perpendicular. Wait, the problem says "two intersecting lines", not necessarily perpendicular. Oh! Maybe I assumed they are perpendicular, but they are just two intersecting lines.

So, let's correct:

- \( \angle EKH = 92^\circ \)

- \( \angle EKS \): \( \angle EKH \) and \( \angle EKS \) are supplementary (since \( SH \) is a straight line, \( \angle SKH = 180^\circ \)), so \( \angle EKS=180^\circ - 92^\circ = 88^\circ \)

- \( \angle HKW \): \( \angle EKH \) and \( \angle HKW \) are vertical angles? No, vertical angles are opposite. Wait, \( \angle EKH \) and \( \angle SKW \) are vertical angles? Wait, when two lines intersect, vertical angles are equal. The two lines are \( EW \) and \( SH \), intersecting at K. So \( \angle EKH \) and \( \angle SKW \) are vertical angles, so \( \angle SKW=\angle EKH = 92^\circ \)

- \( \angle HKW \): \( \angle EKS \) and \( \angle HKW \) are vertical angles, so \( \angle HKW=\angle EKS = 88^\circ \)

Wait, let's use the property of intersecting lines:

1. When two lines intersect, adjacent angles are supplementary (sum to \( 180^\circ \)), and vertical angles are equal.

Lines \( EW \) and \( SH \) intersect at K.

- \( \angle EKH \) and \( \angle EKS \): adjacent, so \( \angle EKH+\angle EKS = 180^\circ \). Given \( \angle EKH = 92^\circ \), so \( \angle EKS=180 - 92 = 88^\circ \)

- \( \angle EKH \) and \( \angle SKW \): vertical angles, so \( \angle SKW=\angle EKH = 92^\circ \)

- \( \angle EKS \) and \( \angle HKW \): vertical angles, so \( \angle HKW=\angle EKS = 88^\circ \)

Wait, now I see my initial mistake: I thought \( EK \) and \( SK \) are perpendicular, but they are not necessarily. The diagram just shows two intersecting lines, with \( \angle EKH = 92^\circ \).

So:

- Angle EKS: \( 180^\circ - 92^\circ=88^\circ \)

- Angle HKW: \( 88^\circ \) (vertical angle with EKS)

- Angle SKW: \( 92^\circ \) (vertical angle with EKH)

Wait, let's verify:

- \( \angle EKH + \angle EKS=92 + 88 = 180^\circ \) (supplementary, correct)

- \( \angle SKW + \angle HKW=92 + 88 = 180^\circ \) (supplementary, correct)

- Vertical angles: \( \angle EKH=\angle SKW = 92^\circ \), \( \angle EKS=\angle HKW = 88^\circ \) (correct)

Step1: Calculate Angle EKS

Since \( \angle EKH \) and \( \angle EKS \) are supplementary (they form a linear pair along line \( SH \)), we use the formula \( \angle EKS=180^\circ-\angle EKH \).
Given \( \angle EKH = 92^\circ \), so \( \angle EKS = 180^\circ - 92^\circ=88^\circ \).

Step2: Calculate Angle HKW

\( \angle EKS \) and \( \angle HKW \) are vertical angles (opposite angles formed by intersecting lines \( EW \) and \( SH \)). Vertical angles are equal, so \( \angle HKW=\angle EKS = 88^\circ \).

Step3: Calculate Angle SKW

\( \angle EKH \) and \( \angle SKW \) are vertical angles (opposite angles formed by intersecting lines \( EW \) and \( SH \)). Vertical angles are equal, so \( \angle SKW=\angle EKH = 92^\circ \).

Answer:

Angle EKS measures \( \boldsymbol{88} \) degrees.

Angle HKW measures \( \boldsymbol{88} \) degrees.

Angle SKW measures \( \boldsymbol{92} \) degrees.