Deck 4: Quadrilaterals

Question 1

Suppose that you have already proved that "The diagonals of an isosceles trapezoid are congruent." Use this theorem to prove that "A pair of base angles of an isosceles trapezoid are congruent." Supply missing statements and missing reasons for the proof.
Given: Trapezoid ABCD; and Prove: S1. R1.
S2. Draw and R2.
S3. R3.
S4. R4. Identity
S5. R5.
S6. R6.

Answer 1

S1. Trapezoid ABCD; and R1. Given
R2. Through 2 points, there is exactly one line.
R3. The diagonals of an isosceles trapezoid are congruent.
S4. R5. SSS
R6. CPCTC

Question 2

Supply missing statements and missing reasons for the following proof.
Given: Rectangle MNPQ with diagonals and Prove: S1. R1.
S2. and are rt. R2.
S3. R3.
S4. R4. The diagonals of a rectangle are congruent.
S5. R5.

Answer 2

S1. Rectangle MNPQ with diagonals and R1. Given
R2. All angles of a rectangle are right angles.
R3. Identity
S4. S5. R5. HL

Question 3

Provide missing reasons for the proof of the theorem, "A diagonal of a parallelogram separates it into two congruent triangles."
Given: with diagonal Prove: S1. with diagonal R1.
S2. R2.
S3. R3.
S4. R4.
S5. R5.
S6. R6.
S7. R7.

Answer 3

R1. Given
R2. The opposite sides of a parallelogram are parallel (definition).
R3. If 2 lines are cut by a trans, the alternate interior angles are congruent.
R4. Same as reason 2
R5. Same as reason 3.
R6. Identity
R7. ASA

Question 4

Given the rhombus RSTV, diagonals and intersect at point W. Explain why it follows that .

Question 5

Use the drawing shown to explain the following theorem.
"The length of the median of a trapezoid equals one-half the sum of the lengths of the two bases."
Given: Trapezoid ABCD with median Prove: [Hint: X is the midpoint of auxiliary diagonal .]

Question 6

Provide the missing statements abd nissing reasons for the proof of this theorem.
"If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram."
Given: Quad. MNPQ; and Prove: MNPQ is a parallelogram
S1. R1. Given
S2. Draw diagonal R2.
S3. R3. Identity
S4. R4. SSS
S5. R5.
S6. R6. If 2 lines are cut by a trans. so that alternate interior angles
are congruent, these lines are parallel.
S7. R7.
S8. R8.
S9. R9.

Question 7

Supply missing statements and missing reasons for this proof.
Given: Kite MNPQ with diagonal ; and Prove: bisects S1. R1.
S2. R2. In a kite, one pair of opposite angles are congruent.
S3. R3.
S4. R4.
S5. R5.

Question 8

Provide missing statements and missing reasons for the proof of this theorem.
"In a kite, one pair of opposite angles are congruent."
Given: Kite ABCD with and Prove: S1. R1.
S2. Draw R2.
S3. R3.
S4. R4. SSS
S5. R5.

Question 9

In quadrilateral ABCD, the midpoints of the sides are joined in order.
Use the auxiliary diagonals to explain why the resulting quadrilateral MNPQ must be a parallelogram.

Question 10

Supply all statements and all reasons for the following proof.
Given: ; M is the midpoint of and N is the midpoint of Prove: MNAB is a trapezoid

Question 11

Provide missing statements and missing reasons for the following proof.
Given: ; diagonals and intersect at point P
Prove: and S1. R1. Given
S2. R2.
S3. and R3.
S4. R4.
S5. R5. ASA
S6. R6.

Question 12

Consider the drawing provided. Using the auxiliary diagonals (shown dashed), explain why the "opposite angles and opposite sides of a parallelogram must be congruent."

Question 13

Provide missing statements and missing reasons for the proof of the following theorem.
"The diagonals of a rhombus are perpendicular."
Given: Rhombus ABCD; diagonals and intersect at point X
Prove: S1. R1.
S2. R2. The diagonals of a rhombus ( ) bisect each other.
S3. R3.
S4. R4. All sides of a rhombus are congruent.
S5. R5. SSS
S6. R6.
S7. R7.

Question 14

Provide all statements and all reasons for the following proof.
Given: , , , and Prove: Quad. ABCD is a parallelogram