Question

find the measures of all the angles in the rectangle.

Explanation:

Step1: Recall Rectangle Angle Property

In a rectangle, all interior angles are right angles (90°). So angles 1, 3, 4 are 90°.

Step2: Find Angle 5

Angles on a straight line sum to 180°. Given one angle is 68°, so \( \angle 5 = 180^\circ - 90^\circ - 68^\circ = 22^\circ \)? Wait, no, in the triangle, angle at the bottom left: the triangle is a right triangle? Wait, rectangle has right angles, so the triangle formed by the diagonal is a right triangle? Wait, no, the angle with 68° and angle 5: since the side is a straight line, \( 68^\circ + \angle 5 = 90^\circ \)? Wait, no, in the rectangle, angle 1 is 90°, so the triangle with angle 1 (90°), 68°, and angle opposite? Wait, maybe better: in the rectangle, adjacent angles are 90°, so the angle between the side and the diagonal: \( \angle 5 = 90^\circ - 68^\circ = 22^\circ \).

Step3: Find Angle 2

In the triangle, angles sum to 180°. The triangle has a right angle? Wait, no, the rectangle's angle is 90°, so the triangle formed by the diagonal: angle 1 is 90°, so the triangle with angles 68°, angle 5 (22°), and angle 2? Wait, no, angle 2 is in the other triangle. Wait, maybe: in the rectangle, opposite sides are equal and parallel, so the triangles formed by the diagonal are congruent? Wait, angle 5 and the angle in the other triangle (let's say angle at the top right) are equal? Wait, maybe simpler: angle 1, 3, 4 are 90° (rectangle angles). Angle 5: since the angle with 68° is part of a right angle (90°), so \( \angle 5 = 90^\circ - 68^\circ = 22^\circ \). Then, in the triangle, angle 2: since it's a triangle, angles sum to 180°, and if one angle is 90° (rectangle), wait no, angle 1 is 90°, so the triangle with angle 1 (90°), 68°, and angle 2: \( 90^\circ + 68^\circ + \angle 2 = 180^\circ \)? No, that can't be. Wait, maybe I messed up. Let's re-express:

• Angles 1, 3, 4: 90° (rectangle interior angles).
• \( \angle 5 \): adjacent to 68° in a right angle, so \( \angle 5 = 90^\circ - 68^\circ = 22^\circ \).
• Angle 2: in the triangle, since it's a rectangle, the diagonal splits it into two triangles. The triangle with angle 3 (90°), angle 5 (22°), so \( \angle 2 = 180^\circ - 90^\circ - 22^\circ = 68^\circ \)? Wait, no, maybe angle 2 is equal to 68°? Wait, maybe the triangles are congruent, so angle 2 is 68°, angle 5 is 22°, angles 1,3,4 are 90°.

Wait, let's list all angles:

• \( \angle 1 = 90^\circ \) (rectangle angle)
• \( \angle 3 = 90^\circ \) (rectangle angle)
• \( \angle 4 = 90^\circ \) (rectangle angle)
• \( \angle 5 = 90^\circ - 68^\circ = 22^\circ \) (complementary angles in right angle)
• \( \angle 2 = 68^\circ \) (since the triangles are congruent, or alternate interior angles)

Wait, maybe the correct angles:

• \( \angle 1 = 90^\circ \)
• \( \angle 3 = 90^\circ \)
• \( \angle 4 = 90^\circ \)
• \( \angle 5 = 22^\circ \) (because \( 68^\circ + 22^\circ = 90^\circ \))
• \( \angle 2 = 68^\circ \) (since the triangle has angles 90°, 22°, so 68°)

Answer:

\( \angle 1 = 90^\circ \), \( \angle 2 = 68^\circ \), \( \angle 3 = 90^\circ \), \( \angle 4 = 90^\circ \), \( \angle 5 = 22^\circ \)