29. building the roof and exterior walls of a house represent intersecting planes. using the image, name all the lines that are formed by the intersecting planes. 30. if the surface of a lake represents a plane, what geometric term is represented by the intersection of a fishing line and the lake’s surface? 31. art perspective drawing is a method that artists use to create paintings and drawings of three-dimensional objects. the artist first draws the horizon line and two vanishing points along the horizon. buildings or other objects are created by drawing receding lines and vertical lines. a. where do the receding lines and horizon lines intersect? b. identify examples of planes within this picture.

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### Question 29
# Brief Explanations:
To determine the lines formed by intersecting planes (roof and exterior walls), we analyze the house diagram. The roof (a plane) and each exterior wall (another plane) intersect along an edge. For the given house, the lines formed are $AB$, $AH$, $HG$, $GF$, $FB$ (for the roof and upper walls) and $CD$, $DH$, $HF$, $FE$, $ED$ (for the roof and lower walls)? Wait, no, more accurately, looking at the house with vertices $A, B, C, D, E, F, G, H$, the roof planes (like $ABG$, $BFG$ etc.?) and the wall planes (like $BCHD$, $DHEF$ etc.). Wait, the key is that the intersection of two planes is a line. So the roof (e.g., the triangular or slanted part) and the vertical walls (rectangular) intersect along their common edges. From the diagram, the lines formed by the roof (a plane) and exterior walls (other planes) would be $AB$, $BF$, $FG$, $GA$ (for the upper roof and upper walls) and $CD$, $DE$, $EF$, $FC$? No, better to look at the labeled points. The house has a roof (e.g., the top part with $A, B, F, G$) and walls: upper wall $B, C, D, H$ and lower wall $D, E, F, H$. So the intersection of roof (e.g., plane $ABFH$ or $BFGH$?) and wall $BCHD$ is line $BH$? Wait, maybe the diagram shows the roof as the triangular or pentagonal top, and the walls as the vertical rectangles. So the lines where roof and walls meet: for example, $AB$ (where roof meets the left upper wall), $BF$ (where roof meets the front upper wall), $FG$ (where roof meets the right upper wall), $GA$ (where roof meets the back upper wall), and similarly for the lower part? Wait, the problem says "the roof and exterior walls of a house represent intersecting planes". So each exterior wall is a plane, and the roof is a plane. Their intersection is a line (the edge where roof meets wall). Looking at the labeled points: $A, B, C, D, E, F, G, H$. Let's list the lines: $AB$, $BH$, $HF$, $FG$, $GA$? No, maybe the correct lines are $AB$, $BF$, $FG$, $GA$ (for the upper roof) and $CD$, $DE$, $EF$, $FC$ (for the lower roof)? Wait, the diagram shows a house with two floors: upper floor with windows $B, C, H, D$ and lower floor $D, E, F, H$, and roof $A, B, F, G$. So the roof (plane $ABFG$) intersects with wall $BCHD$ along $BH$, with wall $DHEF$ along $HF$, with wall $BCDE$? No, maybe the answer is the edges where roof and walls meet, like $AB$, $BF$, $FG$, $GA$ (if the roof is the top triangle/quadrilateral) and $CD$, $DE$, $EF$, $FC$ (if there's a lower roof), but more likely, from the diagram, the lines are $AB$, $BH$, $HF$, $FG$, $GA$ or similar. Wait, the example 5 says "Yes, because plane... all lie on the plane". For question 29, using the image (the house with points $A, B, C, D, E, F, G, H$), the lines formed by the intersection of roof (a plane) and exterior walls (other planes) are $AB$, $BF$, $FG$, $GA$ (upper roof) and $CD$, $DE$, $EF$, $FC$ (lower roof)? Or maybe $AB$, $BH$, $HF$, $FG$, $GA$ and $CD$, $DH$, $HF$, $FE$, $ED$? Wait, perhaps the correct lines are $AB$, $BF$, $FG$, $GA$, $CD$, $DE$, $EF$, $FC$ (but that might be too many). Alternatively, looking at the house, the roof (e.g., the top slanted part) and the vertical walls (front, back, left, right) intersect along their common edges. So for the upper part: $AB$ (left wall and roof), $BF$ (front wall and roof), $FG$ (right wall and roof), $GA$ (back wall and roof). For the lower part: $CD$ (left wall and lower roof), $DE$ (front wall and lower roof), $EF$ (right wall and lower roof), $FC$ (back wall and lower roof). But maybe the diagram has a simpler structure. Alternatively, the lines are $AB$, $BH$, $HF$, $FG$, $GA$ (if $H$ is the middle point). Wait, the diagram shows two vanishing points, horizontal line, receding lines, and the house with $A, B, C, D, E, F, G, H$. So the roof planes (e.g., $ABG$ and $BFG$) and wall planes (e.g., $BCHD$ and $DHEF$) intersect along $BH$ and $HF$ respectively. But maybe the answer is the edges of the roof that meet the walls, like $AB$, $BF$, $FG$, $GA$ (upper roof) and $CD$, $DE$, $EF$, $FC$ (lower roof). However, a more precise answer would be the lines where the roof (a plane) and each exterior wall (a plane) meet, which are the edges of the roof connected to the walls. So for the given house, the lines are $AB$, $BF$, $FG$, $GA$ (upper roof) and $CD$, $DE$, $EF$, $FC$ (lower roof), or maybe $AB$, $BH$, $HF$, $FG$, $GA$ and $CD$, $DH$, $HF$, $FE$, $ED$ (if $H$ is the intersection point). But perhaps the correct lines are $AB$, $BF$, $FG$, $GA$, $CD$, $DE$, $EF$, $FC$ (assuming the roof is a pentagon or two triangles) and the walls are rectangles. Alternatively, looking at the labeled points: $A$ is the top of the roof, $B$ and $F$ are the front corners, $G$ is the other top, $C$ and $D$ are the upper left and right, $E$ is the lower front. So the roof (plane $ABFG$) intersects with wall $BCHD$ along $BH$, with wall $DHEF$ along $HF$, with wall $BCDE$? No, maybe the answer is $AB$, $BF$, $FG$, $GA$ (upper roof edges) and $CD$, $DE$, $EF$, $FC$ (lower roof edges), as these are the lines where the roof (a plane) and the exterior walls (planes) intersect.

# Answer:
The lines formed by the intersecting planes (roof and exterior walls) are $AB$, $BF$, $FG$, $GA$, $CD$, $DE$, $EF$, $FC$ (or other relevant edges based on the diagram, e.g., $AB$, $BH$, $HF$, $FG$, $GA$, $CD$, $DH$, $HF$, $FE$, $ED$ depending on the house's structure).

### Question 30
# Brief Explanations:
A plane is a flat, two - dimensional surface. The intersection of two planes is a line. The surface of the lake is a plane, and the fishing line is also a plane (or we can consider the fishing line as a line, but the problem says the fishing line represents a plane? Wait, no: "the surface of a lake represents a plane, what geometric term is represented by the intersection of a fishing line and the lake’s surface?". Wait, the fishing line: if the lake's surface is a plane, and the fishing line (assuming it's a straight line, or a plane? No, a fishing line is a line. Wait, the intersection of a line and a plane: if the line is not parallel to the plane and not in the plane, they intersect at a point. But if the fishing line is a plane (maybe a vertical plane), then the intersection of two planes is a line. Wait, the problem says "the surface of a lake represents a plane" (so lake surface: plane $L$), "a fishing line" – maybe the fishing line is a plane (e.g., the vertical plane of the fishing line in the water). Then the intersection of two planes is a line. But if the fishing line is a line (a one - dimensional figure), then the intersection of a line and a plane is a point, but that seems less likely. Wait, maybe the fishing line is considered as a plane (the plane containing the fishing line, which is vertical), so the intersection of two planes (lake surface, a horizontal plane; and fishing line plane, a vertical plane) is a line. But let's recall: the intersection of two planes is a line. So if the lake is a plane (horizontal) and the fishing line is part of a vertical plane, their intersection is a line. Alternatively, if the fishing line is a line (one - D), intersecting a plane (two - D) at a point. But the problem says "the surface of a lake represents a plane" and "the fishing line" – maybe the fishing line is a plane (e.g., the plane of the fishing line in the water, like a vertical plane). So the intersection of two planes is a line.

# Answer:
The geometric term representing the intersection of the fishing line (considered as a plane) and the lake’s surface (a plane) is a line. If the fishing line is considered as a line, the intersection with the lake’s surface (a plane) is a point, but more likely, since both are considered planes (lake surface as a horizontal plane, fishing line as a vertical plane), the intersection is a line.

### Question 31a
# Brief Explanations:
In perspective drawing, receding lines are lines that appear to converge towards a vanishing point, and horizon lines are horizontal lines at eye level. The intersection of receding lines and horizon lines occurs at the vanishing points. In the given perspective drawing, there are two vanishing points (labeled "Vanishing point" in the diagram). So the receding lines (which are lines going back into the distance) and the horizon line (the horizontal line) intersect at the vanishing points.

# Answer:
The receding lines and horizon lines intersect at the vanishing points.

### Question 31b
# Brief Explanations:
To identify planes in the perspective drawing, we look for flat, two - dimensional surfaces. The buildings (or objects) in the drawing are made up of planes. For example, the front face of a building, the side face of a building, the roof of a building. In the diagram, the horizontal line, receding lines, and vertical lines form the sides, front, back, and roof of the buildings. So examples of planes could be the front wall of a building (a rectangular plane), the side wall of a building (another rectangular plane), the roof of a building (a slanted or rectangular plane), and the ground plane (implied by the horizon line).

# Answer:
Examples of planes within the picture include the front wall of a building, the side wall of a building, the roof of a building, and the ground plane (or other flat surfaces formed by the intersection of receding, horizontal, and vertical lines).

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