Question

15 construct the angle bisector of each given angle. a (angle with vertex x) b (angle with vertex a)

Explanation:

Step1: Recall Angle Bisector Construction

To construct an angle bisector, we can use a compass and straightedge. For an angle with vertex \( X \) (part a) or \( A \) (part b):

1. Place the compass tip at the vertex (e.g., \( X \) or \( A \)). Draw an arc that intersects both sides of the angle. Let the intersection points be \( B \) (on one side) and \( C \) (on the other side).
2. From each intersection point (\( B \) and \( C \)), draw arcs of equal radius (using the compass) inside the angle. These arcs will intersect at a point, say \( D \).
3. Draw a ray from the vertex (\( X \) or \( A \)) through \( D \). This ray is the angle bisector.

Step2: Apply to Given Angles

For part (a) (Angle at \( X \)):

• Vertex is \( X \), with two rays. Follow the steps:
• Arc from \( X \) intersects both rays at, say, \( P \) (on the left ray) and \( Q \) (on the right ray).
• Arcs from \( P \) and \( Q \) intersect inside the angle at \( R \).
• Ray \( XR \) bisects \( \angle PXQ \).

For part (b) (Angle at \( A \)):

• Vertex is \( A \), with two rays. Follow the steps:
• Arc from \( A \) intersects both rays at, say, \( M \) (on the lower ray) and \( N \) (on the right ray).
• Arcs from \( M \) and \( N \) intersect inside the angle at \( O \).
• Ray \( AO \) bisects \( \angle MAN \).

(Note: Since this is a construction, the actual drawing requires physical tools, but the steps outline the process.)

Answer:

To construct the angle bisector for each angle:

• For angle at \( X \) (part a) and angle at \( A \) (part b), use a compass and straightedge to:

1. Draw an arc from the vertex intersecting both sides of the angle.
2. Draw intersecting arcs from these two intersection points inside the angle.
3. Draw a ray from the vertex through the intersection of the inner arcs (this is the bisector).

(The bisector will be a ray splitting the angle into two equal smaller angles.)