Question

find the measures of all the angles in the rectangle.
1
2
3
5
18°
4

Explanation:

Step1: Identify Rectangle Angles

In a rectangle, all interior angles are right angles (90°). So, ∠1, ∠3, ∠4, ∠5's adjacent right angle:
∠1 = 90°, ∠3 = 90°, ∠4 = 90°.

Step2: Analyze Triangle Angles

The triangle with 18°: in a triangle, angles sum to 180°. For the triangle with 18° and right angle (∠5's part? Wait, ∠5: in the right triangle, one angle is 18°, so ∠5 = 90° - 18° = 72°? Wait, no: the triangle has angles 18°, ∠5, and 90°? Wait, the rectangle's angle is 90°, so the triangle formed by the diagonal is a right triangle. So in the right triangle (e.g., the one with 18°), angles are 18°, ∠5, and 90°? Wait, no: ∠5 + 18° + 90°? No, wait, the rectangle's angle is 90°, so the triangle inside (split by diagonal) has angles: let's see, ∠1 is 90°, the triangle with ∠1, ∠2, and the 18°-related angle. Wait, maybe better:

In the right triangle (the one with 18°), one angle is 18°, another is 90° (since rectangle), so the third angle (∠5) is 180° - 90° - 18° = 72°? Wait, no, ∠5 is adjacent to the 18° angle in the rectangle's corner. Wait, the angle labeled 18°: let's say in the triangle, angle at the bottom left is 18°, so the other acute angle (∠5) is 90° - 18° = 72°? Wait, no, sum of angles in triangle is 180°. So if one angle is 90° (rectangle's angle), one is 18°, then the third is 180 - 90 - 18 = 72°. So ∠5 = 72°? Wait, no, ∠5 is part of the rectangle's angle? Wait, the rectangle's angle is 90°, so ∠5 + 18° = 90°? Yes! Because in the rectangle, the corner is 90°, so the two angles (18° and ∠5) are complementary (sum to 90°). So ∠5 = 90° - 18° = 72°.

Then, ∠2: in the other triangle, angles: ∠2, ∠3 (90°), and ∠5 (72°)? Wait, no, the diagonal splits the rectangle into two congruent triangles? Wait, no, unless it's a square, but it's a rectangle. Wait, the two triangles formed by the diagonal are congruent? Wait, no, unless it's a square, but in a rectangle, the triangles are congruent (SAS: sides of rectangle, diagonal common). Wait, but the angle 18°: maybe the angle between the side and diagonal is 18°, so the other angle (∠2) would be 18°? No, wait, let's re-express:

- ∠1: 90° (rectangle)
- ∠3: 90° (rectangle)
- ∠4: 90° (rectangle)
- ∠5: 90° - 18° = 72° (complementary to 18° in right angle)
- ∠2: In the triangle with ∠1 (90°), ∠5 (72°), so ∠2 = 180° - 90° - 72° = 18°? Wait, no, maybe the triangle with 18°: the angle ∠2 is equal to 18°? Wait, no, let's list all angles:

1. ∠1: 90° (rectangle angle)
2. ∠2: Let's see, the triangle with ∠1, ∠2, and the angle equal to ∠5 (72°)? Wait, no, the diagonal creates two triangles. In the left triangle (with ∠1, ∠5, and ∠2), angles sum to 180°: ∠1=90°, ∠5=72°, so ∠2=180-90-72=18°.
3. ∠3: 90° (rectangle angle)
4. ∠4: 90° (rectangle angle)
5. ∠5: 72° (as 90-18=72, since 18° and ∠5 are complementary in the right angle)

Wait, let's correct:
- Rectangle angles: ∠1, ∠3, ∠4 are 90° (since rectangle has four right angles).
- The triangle formed by the diagonal: it's a right triangle (because rectangle's angle is 90°). So in that right triangle, one acute angle is 18°, so the other acute angle (∠5) is 90° - 18° = 72°.
- Then, in the other triangle (the one with ∠2), since the triangles are congruent? Wait, no, but angles: ∠2 should be equal to 18° (alternate interior angles? Or because the triangles are congruent? Wait, in a rectangle, diagonals are equal but not necessarily bisecting angles unless it's a square. Wait, no, the angle of 18°: let's assume the angle between the side and diagonal is 18°, so the other angle (∠2) is also 18°? Wait, no, let's use angle sum:

For ∠5: in the right triangle, angles are 18°, ∠5, and 90°? No, sum is 180, so 18 + ∠5 + 90 = 180 → ∠5 = 72°. Then, in the left triangle (with ∠1=90°, ∠5=72°, ∠2=18° (180-90-72=18)).

So all angles:
∠1 = 90°, ∠2 = 18°, ∠3 = 90°, ∠4 = 90°, ∠5 = 72°.

Answer:

∠1 = 90°, ∠2 = 18°, ∠3 = 90°, ∠4 = 90°, ∠5 = 72°