$$ \qquad $$ 9. All quadrilaterals are parallelograms. $$ \qquad $$ 10. All parallelograms are quadrilaterals. $$ \qquad $$ 11. A square is a parallelogram. $$ \qquad $$ 12. A parallelogram with a right angle is a square. $$ \qquad $$ 13. All rectangles are parallelograms. $$ \qquad $$ 14. All rhombuses are squares. $$ \qquad $$ 15. All squares are rectangles. $$ \qquad $$ 16. A parallelogram with four congruent sides is a square. $$ \qquad $$ 17. A parallelogram with perpendicular diagonals is a square. For 18-21, find the measure of the numbered angles in the figures. $$ m \angle 1= $$ $$ \qquad $$ 18. $$ A B C D $$ is rectangle 19. RSTV is a rhombus 20. $$ \mathrm{EF} G \mathrm{H} $$ is a square

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UpStudy AI · Accepted Answer

Sure, let's determine the measures of the specified angles for each of the given quadrilaterals. Here's a step-by-step explanation for each problem:

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### **Problem 18:** $$ ABCD $$ is a Rectangle. Find $$ m\angle 1 $$.

**Properties of a Rectangle:**
- A rectangle is a quadrilateral with four right angles.
- Opposite sides are equal and parallel.

**Solution:**
- In a rectangle, all interior angles are right angles.
  
$$
m\angle 1 = 90^\circ
$$

---

### **Problem 19:** $$ RSTV $$ is a Rhombus. Find $$ m\angle $$ (assuming it's one of the angles, say $$ \angle R $$).

**Properties of a Rhombus:**
- A rhombus is a quadrilateral with all four sides of equal length.
- Opposite angles are equal.
- Adjacent angles are supplementary (sum to $$ 180^\circ $$).
- The diagonals bisect each other at right angles.
  
**Solution:**
- Without specific information about the angles, we can provide general properties:
  - Let’s denote one of the acute angles as $$ \alpha $$.
  - Then the obtuse angle is $$ 180^\circ - \alpha $$.
  
However, **if no specific measures are given**, we cannot determine the exact measure of $$ \angle R $$. Typically, in problems like this, additional information (such as the measure of one angle or the relationship between angles) is provided to find the exact measure.

**General Answer:**
- $$ m\angle R = \alpha $$ (where $$ 0^\circ < \alpha < 180^\circ $$)
- Without more information, the exact measure cannot be determined.

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### **Problem 20:** $$ EFGH $$ is a Square. Find $$ m\angle $$ (assuming it's one of the angles, say $$ \angle E $$).

**Properties of a Square:**
- A square is a quadrilateral with all four sides equal in length.
- All interior angles are right angles.
- It has both properties of a rectangle (opposite sides equal and all angles $$ 90^\circ $$) and a rhombus (all sides equal and diagonals bisect each other at right angles).

**Solution:**
- Since all angles in a square are right angles:
  
$$
m\angle E = 90^\circ
$$

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### **Summary of Angle Measures:**

1. **Problem 18 (Rectangle $$ ABCD $$):**
   $$
   m\angle 1 = 90^\circ
   $$
2. **Problem 19 (Rhombus $$ RSTV $$):**
   $$
   \text{Cannot determine without additional information}
   $$
3. **Problem 20 (Square $$ EFGH $$):**
   $$
   m\angle = 90^\circ
   $$

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If you have diagrams or additional information for **Problems 19 and 21**, please provide them so a more precise answer can be given.
- For rectangle $$ ABCD $$, each angle is $$ 90^\circ $$. - For rhombus $$ RSTV $$, each angle is $$ 90^\circ $$ (assuming no additional information). - For square $$ EFGH $$, each angle is $$ 90^\circ $$.

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