Question

ladders what is the measure of the angle formed by the top of the ladder and the side of the house? 105° the measure of the angle is

Explanation:

Step1: Identify angle relationship

The house wall is vertical (perpendicular to the ground), so the angle between the ground and the wall is \(90^\circ\). The angle between the ladder and the ground is supplementary to \(105^\circ\)? Wait, no—wait, the angle at the ground between the ladder and the horizontal is \(180^\circ - 105^\circ = 75^\circ\)? Wait, no, let's correct. The angle between the ladder and the ground (the acute angle) is \(180^\circ - 105^\circ = 75^\circ\)? Wait, no, actually, the angle between the ladder and the house: since the wall is vertical (right angle with ground), the triangle formed is a right triangle? Wait, no, the angle at the ground between the ladder and the horizontal is \(180 - 105 = 75^\circ\), and then the angle between the ladder and the house (vertical wall) would be \(90^\circ - 75^\circ = 15^\circ\)? Wait, no, let's think again.

Wait, the angle given is \(105^\circ\) between the ladder and the ground (the straight line). So the adjacent angle (between ladder and the ground's horizontal) is \(180 - 105 = 75^\circ\). Then, since the house wall is perpendicular to the ground (so the angle between wall and ground is \(90^\circ\)), the angle between the ladder and the wall is \(90^\circ - 75^\circ = 15^\circ\)? Wait, no, maybe I messed up. Wait, the angle between the ladder and the house: the house is vertical, ground is horizontal. So the ladder, ground, and house form a right triangle? Wait, no, the angle at the ground between the ladder and the ground is \(180 - 105 = 75^\circ\) (since the straight line is \(180^\circ\)). Then, in the right triangle (wall, ground, ladder), the two acute angles sum to \(90^\circ\). So the angle between ladder and wall is \(90^\circ - (180^\circ - 105^\circ)\)? Wait, no, let's do it step by step.

1. The angle between the ladder and the ground (the obtuse angle is \(105^\circ\)), so the acute angle between ladder and ground is \(180^\circ - 105^\circ = 75^\circ\).
2. The wall is perpendicular to the ground, so the angle between wall and ground is \(90^\circ\).
3. In the triangle formed by ladder, wall, and ground, the angles are: \(90^\circ\) (wall-ground), \(75^\circ\) (ladder-ground), and the angle between ladder and wall. Let's call that angle \(x\).
4. In a triangle, angles sum to \(180^\circ\), but wait, this is a right triangle? Wait, no, the ladder, wall, and ground: the wall is vertical, ground is horizontal, so they are perpendicular (right angle). So the triangle is right-angled at the base (where wall meets ground). So the two acute angles are \(75^\circ\) (ladder-ground) and \(x\) (ladder-wall), so \(75 + x = 90\), so \(x = 15^\circ\)? Wait, no, that can't be. Wait, maybe the angle between the ladder and the house is supplementary? No, wait, let's draw it mentally: the ladder is leaning against the house. The ground is horizontal, house is vertical. The angle between the ladder and the ground (the straight line) is \(105^\circ\), so the angle between the ladder and the horizontal ground is \(180 - 105 = 75^\circ\). Then, the angle between the ladder and the vertical house is \(90 - 75 = 15^\circ\)? Wait, no, maybe I made a mistake. Wait, the angle between the ladder and the house: the house is vertical, so the angle between ladder and house is \(180 - 90 - (180 - 105)\)? No, that's confusing. Wait, another approach: the angle between the ladder and the house is equal to \(180^\circ - 105^\circ - 90^\circ\)? Wait, no, the sum of angles in a triangle is \(180^\circ\). The right angle is \(90^\circ\), the angle at the ground between ladder and ground is \(180 - 105 = 75^\circ\), so the third angle (between ladder and house) is \(180 - 90 - 75 = 15^\circ\)? Wait, that seems low, but maybe. Wait, let's check: if the angle between ladder and ground is \(75^\circ\), then the angle between ladder and house (vertical) is \(90 - 75 = 15^\circ\). Yes, because in the right triangle, the two acute angles are complementary. So \(75 + 15 = 90\), which makes sense. So the angle between the top of the ladder and the side of the house is \(15^\circ\)? Wait, no, wait, maybe I got the angle wrong. Wait, the given angle is \(105^\circ\) between the ladder and the ground (the external angle). So the internal angle at the ground is \(180 - 105 = 75^\circ\). Then, the angle between the ladder and the house (vertical) is \(90 - 75 = 15^\circ\). So the measure is \(15^\circ\)? Wait, no, maybe I messed up. Wait, let's think again. The angle between the ladder and the house: the house is vertical, so the angle between ladder and house is \(180 - 105 - 90 = -15\)? No, that can't be. Wait, no, the triangle is ladder, ground, house. Ground and house are perpendicular (90 degrees). The angle between ladder and ground is \(180 - 105 = 75\) degrees (since the straight line is 180). Then, the angle between ladder and house is \(90 - 75 = 15\) degrees. Yes, that's correct. So the angle is \(15^\circ\).

Wait, no, maybe I made a mistake. Let's re-express:

- The ground is a straight line (180 degrees).
- The angle between the ladder and the ground (the obtuse angle) is 105 degrees. So the acute angle between the ladder and the ground (the horizontal part) is \(180 - 105 = 75\) degrees.
- The house wall is perpendicular to the ground, so the angle between the wall and the ground is 90 degrees.
- Therefore, the angle between the ladder and the wall (house) is \(90 - 75 = 15\) degrees.

Yes, that makes sense. So the measure of the angle is \(15^\circ\).

Step2: Verify the calculation

- Supplementary angle: \(180^\circ - 105^\circ = 75^\circ\) (angle between ladder and ground, acute).
- Complementary angle (since wall is perpendicular to ground): \(90^\circ - 75^\circ = 15^\circ\) (angle between ladder and house).

Answer:

\(15^\circ\)