Question

use this diagram for items 5 - 8.

1. write a statement from the diagram that can be justified by using the angle addition postulate.
2. write a statement from the diagram that can be justified by using the definition of a right angle.
3. given that k is the midpoint of fj, write a statement that can be justified by using the definition of a midpoint.
4. given that ∠ghf ≅ ∠jfh, write a statement that can be justified by using the definition of congruent angles.

Explanation:

Item 5

Step1: Recall Angle Addition Postulate

The Angle Addition Postulate states that if a point lies in the interior of an angle, then the sum of the two smaller angles formed is equal to the measure of the larger angle. In the diagram, point \( K \) is on \( FJ \), and \( \angle GFH \), \( \angle HFK \), and \( \angle GFJ \) (or other angle combinations with a common vertex and a point on a side) can be considered. For example, with vertex \( F \), and \( K \) on \( FJ \), we can say \( m\angle GFK + m\angle KFH = m\angle GFH \) (assuming appropriate angle labels from the rectangle: \( GFJ \) is a right angle? Wait, the diagram is a rectangle \( GFJH \) with \( FJ \) as the base, \( GF \) and \( HJ \) as vertical sides, and \( FH \) as a diagonal. So angles at \( F \): \( \angle GFJ \) is a right angle (since it's a rectangle, \( GF \perp FJ \)), and \( FH \) is the diagonal, so \( \angle GFH \) and \( \angle JFH \) are angles formed by the diagonal with the sides. Wait, maybe better: The Angle Addition Postulate applies when a point is on a side of an angle. So at point \( F \), the angle \( \angle GFJ \) is a right angle (90 degrees), and if we consider the diagonal \( FH \), then \( K \) is the midpoint of \( FJ \), but for Angle Addition, let's take the angle at \( F \) between \( GF \) and \( FH \), and between \( FH \) and \( FJ \). Wait, maybe the angles are \( \angle GFH \) and \( \angle JFH \) with \( F \) as the vertex, and \( K \) on \( FJ \). Wait, perhaps the correct statement is \( m\angle GFK + m\angle KFH = m\angle GFH \) but maybe more accurately, since \( FJ \) is a straight line (the base), and \( K \) is on \( FJ \), then for angle \( \angle GFJ \) (right angle), but no, \( GF \) is vertical, \( FJ \) is horizontal, so \( \angle GFJ = 90^\circ \). The diagonal \( FH \) splits \( \angle GFJ \) into two angles: \( \angle GFH \) and \( \angle JFH \). Wait, but the Angle Addition Postulate would say that \( m\angle GFH + m\angle JFH = m\angle GFJ \), since \( H \) is a point such that \( FH \) is between \( GF \) and \( FJ \)? Wait, no, \( GF \) is vertical, \( FJ \) is horizontal, so \( \angle GFJ \) is 90 degrees, and \( FH \) is the diagonal, so \( \angle GFH \) is the angle between \( GF \) (vertical) and \( FH \) (diagonal), and \( \angle JFH \) is the angle between \( FH \) (diagonal) and \( FJ \) (horizontal). Then by Angle Addition Postulate, \( m\angle GFH + m\angle JFH = m\angle GFJ \), because \( FH \) is a ray that lies in the interior of \( \angle GFJ \) (since it's the diagonal of the rectangle, so it's inside the right angle \( \angle GFJ \)). So that's a valid statement justified by the Angle Addition Postulate: the sum of the measures of \( \angle GFH \) and \( \angle JFH \) is[SSE Completed, Client Connection Error][SSE Completed, Client Connection Error][LLM SSE On Failure]

Answer:

Step1: Recall Angle Addition Postulate

The Angle Addition Postulate states that if a point lies in the interior of an angle, then the sum of the two smaller angles formed is equal to the measure of the larger angle. In the diagram, point \( K \) is on \( FJ \), and \( \angle GFH \), \( \angle HFK \), and \( \angle GFJ \) (or other angle combinations with a common vertex and a point on a side) can be considered. For example, with vertex \( F \), and \( K \) on \( FJ \), we can say \( m\angle GFK + m\angle KFH = m\angle GFH \) (assuming appropriate angle labels from the rectangle: \( GFJ \) is a right angle? Wait, the diagram is a rectangle \( GFJH \) with \( FJ \) as the base, \( GF \) and \( HJ \) as vertical sides, and \( FH \) as a diagonal. So angles at \( F \): \( \angle GFJ \) is a right angle (since it's a rectangle, \( GF \perp FJ \)), and \( FH \) is the diagonal, so \( \angle GFH \) and \( \angle JFH \) are angles formed by the diagonal with the sides. Wait, maybe better: The Angle Addition Postulate applies when a point is on a side of an angle. So at point \( F \), the angle \( \angle GFJ \) is a right angle (90 degrees), and if we consider the diagonal \( FH \), then \( K \) is the midpoint of \( FJ \), but for Angle Addition, let's take the angle at \( F \) between \( GF \) and \( FH \), and between \( FH \) and \( FJ \). Wait, maybe the angles are \( \angle GFH \) and \( \angle JFH \) with \( F \) as the vertex, and \( K \) on \( FJ \). Wait, perhaps the correct statement is \( m\angle GFK + m\angle KFH = m\angle GFH \) but maybe more accurately, since \( FJ \) is a straight line (the base), and \( K \) is on \( FJ \), then for angle \( \angle GFJ \) (right angle), but no, \( GF \) is vertical, \( FJ \) is horizontal, so \( \angle GFJ = 90^\circ \). The diagonal \( FH \) splits \( \angle GFJ \) into two angles: \( \angle GFH \) and \( \angle JFH \). Wait, but the Angle Addition Postulate would say that \( m\angle GFH + m\angle JFH = m\angle GFJ \), since \( H \) is a point such that \( FH \) is between \( GF \) and \( FJ \)? Wait, no, \( GF \) is vertical, \( FJ \) is horizontal, so \( \angle GFJ \) is 90 degrees, and \( FH \) is the diagonal, so \( \angle GFH \) is the angle between \( GF \) (vertical) and \( FH \) (diagonal), and \( \angle JFH \) is the angle between \( FH \) (diagonal) and \( FJ \) (horizontal). Then by Angle Addition Postulate, \( m\angle GFH + m\angle JFH = m\angle GFJ \), because \( FH \) is a ray that lies in the interior of \( \angle GFJ \) (since it's the diagonal of the rectangle, so it's inside the right angle \( \angle GFJ \)). So that's a valid statement justified by the Angle Addition Postulate: the sum of the measures of \( \angle GFH \) and \( \angle JFH \) is[SSE Completed, Client Connection Error][SSE Completed, Client Connection Error][LLM SSE On Failure]