Quadrilaterals Exercises

Rectangle PQRS has a perimeter equal to 34 units. What is x, the length of the diagonal PR?

Hint

Is ∠PQR a right angle? What's the length of PQ?

What is the length of the diagonal x?

Hint

Are the two quadrilaterals congruent? Are you sure?

Prove that if the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Hint

Show ΔPQS ≅ ΔQPR. What can we can say about angles that are both congruent and supplementary?

WXYZ is a rectangle. Prove that ΔWQZ is isosceles.

Hint

Show WQ ≅ ZQ.

The plan for a truss calls for cross beams BD and CF. How long are these beams together?

Hint

You're looking at a rectangle. Maybe you should be looking at two triangles.

If both diagonals of a rectangle total 34 meters and one of the sides is 8 meters long, what is the perimeter of the rectangle?

Hint

You may want to take a second to recognize your oldest and most loyal frenemy: Pythagoras.

Statements	Reasons
1. PQRS is a parallelogram	Given
2. PR ≅ QS	Given
3. PS ≅ RQ	Opposite sides of parallelograms are congruent (1)
4. PQ ≅ PQ	Reflexive property
5. ΔPQS ≅ ΔQPR	SSS Postulate (2, 3, 4)
6. ∠SPQ and ∠PQR are supplementary	Consecutive angles of parallelograms are supplementary (1)
7. ∠SPQ and ∠PQR are congruent	CPCTC (5)
8. m∠SPQ + m∠PQR = 180°	Definition of supplementary angles (6)
9. m∠SPQ = m∠PQR	Definition of congruent angles (7)
10. 2(m∠SPQ) = 180°	Substitution (8, 9)
11. m∠SPQ = 90°	Division property of equality (10)
12. ∠SPQ is a right angle	Definition of a right angle (11)
13. PQRS is a rectangle	A parallelogram with at least one right angle is a rectangle (1, 12)