Find the measures of the numbered angles in each rectangle. 7. 21 61° 4. 3. 5. Find the measures of the numbered angles in each rectangle. Angle 1 Angle 2 Angle 3 Angle 4 Angle 5 
1. A rectangle is a parallelogram. sometimes A rhombus is a square. 2. 3. A parallelogram is a rhombus. never 4. A rhombus is a rectangle. never 5. A square is a rhombus. always 6. A rectangle is a quadrilateral always Find the measures of the numbered angles in each rectangle. 7. 5 8. 61 3 4 2 4 1 2 m2 m23 m24 m25 m21= m42 =
The Correct Answer and Explanation is:
Given:
- The shape is a rectangle.
- One of the angles (Angle 4) is given as 61°.
- The rectangle has a diagonal line that splits it into two right triangles.
Key Geometry Concepts:
- All interior angles of a rectangle are 90°.
- Diagonals of a rectangle are congruent and divide the rectangle into two congruent right triangles.
- The sum of angles in a triangle is 180°.
- In a triangle, if one angle is known, the other two can be found using the 180° total.
Solving the Angles:
We are told ∠4 = 61°.
Since this triangle lies within a right triangle (half the rectangle), one of the angles (Angle 3) must be 90°, because all corners of a rectangle are 90°.
Let’s now analyze triangle formed by angles ∠3, ∠4, and ∠5:
- ∠3 = 90° (corner of rectangle)
- ∠4 = 61° (given)
- ∠5 = 180° – (90° + 61°) = 29°
So:
- ∠5 = 29°
Now let’s look at the triangle on the other side (containing angles 1 and 2):
Since diagonals in a rectangle are congruent and opposite triangles are congruent, angles ∠1 and ∠5 are congruent, and ∠2 and ∠4 are congruent.
So:
- ∠1 = ∠5 = 29°
- ∠2 = ∠4 = 61°
Final Answers:
- m∠1 = 29°
- m∠2 = 61°
- m∠3 = 90°
- m∠4 = 61°
- m∠5 = 29°
Explanation
In geometry, understanding the properties of rectangles is essential to solve problems involving angle measures. A rectangle is a quadrilateral with four right angles (each 90°). One critical property is that its diagonals are equal in length and divide the rectangle into two congruent right triangles. This means each triangle will have a 90° angle at the corner of the rectangle, and the remaining two angles will sum to 90°, because the total of any triangle’s angles is 180°.
In this specific problem, we’re given that angle ∠4 is 61°. Since it’s in a triangle formed by drawing a diagonal in the rectangle, and one angle in a rectangle’s corner is always 90°, we can use the triangle angle sum theorem. Subtracting the given angle and the right angle from 180° gives us the third angle: 180° – 90° – 61° = 29°. This angle is ∠5.
Due to the symmetry and congruency of the triangles created by the diagonal, angle ∠1 must be equal to angle ∠5 (29°), and angle ∠2 must be equal to angle ∠4 (61°). The corner angle ∠3 is a right angle and measures 90°, as all rectangle corners do.
Thus, applying basic angle properties, triangle rules, and congruency principles, we determined all unknown angle measures within the rectangle.
