Quadrilaterals And Proof Worksheet With Answers (Page 2 of 6) in pdf

For part (b), we can continue the previous proof, again using congruent parts of congruent

triangles, to justify that ∠ADC ≅ ∠CBA. That gives one pair of opposite angles congruent.

To get the other pair, we need to draw in the other diagonal.

As before, the alternate interior angles are congruent,

∠ADB ≅ ∠CBD and ∠ABD ≅ ∠CDB, because the opposite

sides are parallel. Using the Reflexive Property, BD ≅ BD .

Therefore, ΔABD ≅ ΔCDB by the ASA ≅ conjecture. Now

that we know that the triangles are congruent, we can

conclude that the corresponding parts are also congruent.

Therefore, ∠DAB ≅ ∠BCD. We have just proven that the

opposite angles in the parallelogram are congruent.

Lastly, we will prove that the diagonals bisect each other. To

begin, we need a picture with both diagonals included. There

are many triangles in the figure now, so our first task will be

deciding which ones we should prove congruent to help us with

the diagonals. To show that the diagonals bisect each other we

will show that AE ≅ CE and BE ≅ DE since “bisect” means to

cut into two equal parts.

We have already proven facts about the parallelogram that we can use here. For instance, we

know that the opposite sides are congruent, so AD ≅ CB . We already know that the alternate

interior angles are congruent, so ∠ADE ≅ ∠CBE and ∠DAE ≅ ∠BCE. Once again we have

congruent triangles: ΔADE ≅ ΔCBE by ASA ≅. Since congruent triangles give us congruent

corresponding parts, AE ≅ CE and BE ≅ DE , which means the diagonals bisect each other.

Example 2

PQRS at right is a rhombus. Do the diagonals bisect

each other? Justify your answer. Are the diagonals

perpendicular? Justify your answer.

The definition of a rhombus is a quadrilateral with four sides of equal length. Therefore,

PQ ≅ QR ≅ RS ≅ SP . By the Reflexive Property, PR ≅ RP . With sides congruent, we can use

the SSS ≅ conjecture to write ΔSPR ≅ ΔQRP. Since the triangles are congruent, all

corresponding parts are also congruent. Therefore, ∠SPR ≅ ∠QRP and ∠PRS ≅ ∠RPQ. The

first pair of congruent angles means that SP QR . (If the alternate interior angles are congruent,

the lines are parallel.) Similarly, the second pair of congruent angles means that PQ RS . With

both pairs of opposite sides congruent, this rhombus is a parallelogram. Since it is a

parallelogram, we can use what we have already proven about parallelograms, namely, that the

diagonals bisect each other. Therefore, the answer is yes, the diagonals bisect each other.

Core Connections Geometry

Quadrilaterals And Proof Worksheet With Answers Page 2