Question

problem practice
angles and parallel lines

1. ramps a parking garage ramp rises to connect two horizontal levels of a parking lot. the ramp makes a 10° angle with the horizontal. what is the measure of angle 1 in the figure?

image: ramp, level 2, level 1, 10° angle

1. bridges a double decker bridge has two parallel levels connected by a network of diagonal girders. one of the girders makes a 52° angle with the lower level as shown in the figure. what is the measure of angle 1?

image: bridge girders, 52° angle

1. city engineering seventh avenue runs perpendicular to both 1st and 2nd streets, which are parallel. however, maple avenue makes a 115° angle with 2nd street. what is the measure of angle 1?

image: streets, 115° angle

1. podiums a carpenter is building a podium. the side panel of the podium is cut from a rectangular piece of wood. the rectangle must be sawed along the dashed line the figure. what is the measure of angle 1?

image: rectangle, 115° angle, dashed line

Explanation:

Problem 1: RAMPS

Step1: Identify parallel lines and transversal

Level 1 and Level 2 are parallel, ramp is transversal.

Step2: Use alternate exterior angles (or corresponding)

Angle 1 and the 10° angle are same - side? Wait, no, actually, since the two levels are parallel, the angle 1 and the 10° angle: wait, the ramp makes 10° with horizontal (Level 1), and Level 1 || Level 2. So angle 1 and the 10° angle: actually, angle 1 is supplementary? No, wait, the two horizontal lines are parallel, the transversal is the ramp? Wait, no, the two horizontal lines (Level 1 and Level 2) are parallel, and the transversal is the line connecting the two ends of the ramp? Wait, no, the figure shows that the ramp is a line, and the two levels are parallel. So the angle between the ramp and Level 1 is 10°, and angle 1 is on Level 2. Since Level 1 || Level 2, and the transversal is the line (the ramp's extension?), actually, angle 1 and the 10° angle are corresponding angles? Wait, no, maybe it's a triangle? Wait, no, the two horizontal lines are parallel, so the angle 1 is equal to 180° - 10°? No, wait, no. Wait, the ramp makes 10° with the horizontal (Level 1), so the angle between the ramp and Level 1 is 10°, and since Level 1 and Level 2 are parallel, angle 1 and the 10° angle: actually, angle 1 is 180° - 10°? No, that can't be. Wait, maybe it's a corresponding angle. Wait, no, let's think again. The two horizontal lines are parallel, the transversal is the line that is the ramp's "base" (the line connecting the two ends). Wait, the problem says the ramp makes a 10° angle with the horizontal (Level 1). So Level 1 and Level 2 are parallel, so the angle between the ramp and Level 2 (angle 1) should be equal to 180° - 10°? No, that's not right. Wait, no, if two parallel lines are cut by a transversal, alternate interior angles are equal. Wait, maybe the ramp is the transversal. Wait, the ramp connects Level 1 and Level 2, so the two horizontal lines are parallel, and the ramp is a transversal. So the angle between the ramp and Level 1 is 10°, so the angle between the ramp and Level 2 (angle 1) is also 10°? No, that doesn't make sense. Wait, maybe it's a linear pair. Wait, no, the figure: Level 1 and Level 2 are parallel, the ramp is a line that goes from Level 1 to Level 2, making 10° with Level 1. So the angle above Level 2 (angle 1) and the 10° angle: since Level 1 || Level 2, the angle 1 and the 10° angle are same - side interior angles? No, same - side interior angles are supplementary. Wait, no, the ramp is a transversal, so same - side interior angles add up to 180°. Wait, no, the angle between the ramp and Level 1 is 10°, so the angle between the ramp and Level 2 (angle 1) is 180° - 10° = 170°? No, that can't be. Wait, maybe I got the figure wrong. Wait, the problem says "the ramp makes a 10° angle with the horizontal". So horizontal is Level 1, so the ramp is inclined at 10° above Level 1. Then Level 2 is parallel to Level 1, so the angle between the ramp and Level 2 (angle 1) is 180° - 10° = 170°? No, that's not right. Wait, maybe it's a corresponding angle. Wait, no, let's check the basic angle - parallel line theorems. If two parallel lines are cut by a transversal, corresponding angles are equal, alternate interior angles are equal, same - side interior angles are supplementary. So if the ramp is the transversal, and the angle with Level 1 is 10°, then angle 1 (on Level 2) and the 10° angle: if they are same - side interior angles, they add to 180°, but that would be 170°, which seems too big. Wait, maybe the ramp is not the transversal, but the other…

Step1: Identify parallel lines and transversal

The two bridge levels are parallel, the girder is a transversal.

Step2: Use alternate interior angles or corresponding angles

The lower level has a 52° angle with the girder, and since the two levels are parallel, angle 1 and the 52° angle: wait, the figure shows a zig - zag of girders. Wait, the two parallel lines (bridge levels) and the girders form a pattern. The angle 1 and the 52° angle: since the girders are symmetric? Wait, no, the key is that the two parallel lines are cut by a transversal (the girder), and angle 1 and the 52° angle: if we consider the alternate interior angles, or maybe the angle 1 is equal to 52°? No, wait, the sum of angle 1 and 52° and another angle? Wait, no, the two parallel lines, and the girder makes 52° with the lower level. So the angle between the girder and the upper level (angle 1) should be equal to 52°? No, that's corresponding angles. Wait, yes, if the two levels are parallel, and the girder is a transversal, then corresponding angles are equal. So angle 1 = 52°? No, wait, maybe it's a linear pair. Wait, no, the figure: the lower level has a 52° angle, and the upper level has angle 1. Since the two levels are parallel, and the girder is a transversal, angle 1 and the 52° angle are alternate interior angles, so they are equal? No, alternate interior angles are equal. Wait, maybe the answer is 52°? No, wait, the sum of angle 1 and 52° is 180°? No, that's same - side interior angles. Wait, I think I need to recall that in parallel lines cut by a transversal, same - side interior angles are supplementary. But if the girder is a transversal, and the angle with the lower level is 52°, then angle 1 (on the upper level) and 52° are same - side interior angles, so angle 1 = 180° - 52° = 128°? No, that doesn't make sense. Wait, maybe the girder is parallel to another girder? The problem says "a network of diagonal girders", so maybe the girders are parallel. So if one girder makes 52° with the lower level, and another girder (parallel to it) makes angle 1 with the upper level. So by corresponding angles, angle 1 = 52°? I'm not sure. Let's move to problem 3.

Problem 3: CITY ENGINEERING

Step1: Identify parallel lines and transversal

1st St. and 2nd St. are parallel, Maple Ave. is transversal, 7th Ave. is perpendicular to both (so 7th Ave. is a transversal too, making right angles).

Step2: Use consecutive interior angles (same - side interior angles)

Maple Ave. makes 115° with 2nd St. Since 1st St. || 2nd St., the angle between Maple Ave. and 1st St. (angle 1) and the 115° angle are same - side interior angles, so they are supplementary.
So angle 1 = 180° - 115° = 65°

Answer:

65°

Problem 4: PODIUMS