Question

According to the given information, segment AB is parallel to segment DC and segment BC is parallel to segment AD.. Using a straightedge, extend segment AB and place point P above point B. By the same reasoning, extend segment AD and place point T to the left of point A. Angles BCD and PBC are congruent by the Alternate Interior Angles Theorem. Angles PBC and BAD are congruent by the ____________. By the Transitive Property of Equality, angles BCD and BAD are congruent. Angles ABC and BAT are congruent by the _____________. Angles BAT and CDA are congruent by the Corresponding Angles Theorem.

Answer 1

User: Edna Butterfield
In Course: Geometry V15 ( 3940)
Instructor: Wendy Bowerman





03.04 Parallelogram Proofs

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Question 1 (Multiple Choice Worth 4 points)

(03.04 MC)

The following is an incomplete paragraph proving that the opposite sides of parallelogram ABCD are congruent:

Parallelogram ABCD is shown where segment AB is parallel to segment DC and segment BC is parallel to segment AD.

According to the given information, segment AB is parallel to segment DC and segment BC is parallel to segment AD.. Construct diagonal A C with a straightedge. It is congruent to itself by the Reflexive Property of Equality. Angles BAC and DCA are congruent by the Alternate Interior Angles Theorem. Angles BCA and DAC are congruent by the same theorem. __________. By CPCTC, opposite sides AB and CD, as well as sides BC and DA, are congruent.

Which sentence accurately completes the proof?

Triangles BCA and DAC are congruent according to the Angle-Angle-Side (AAS) Theorem.

Triangles BCA and DAC are congruent according to the Angle-Side-Angle (ASA) Theorem.

Angles ABC and CDA are congruent according to a property of parallelograms (opposite angles congruent).

Angles BAD and ADC, as well as angles DCB and CBA, are supplementary by the Same-Side Interior Angles Theorem.





Question 2 (Multiple Choice Worth 4 points)

(03.04 MC)

The figure below shows rectangle ABCD:

Rectangle ABCD with diagonals AC and BD passing through point E

The two-column proof with missing statement proves that the diagonals of the rectangle bisect each other.




Statement

Reason

ABCD is a rectangle. Given
Line segment AB and Line segment CD are parallel Definition of a Parallelogram
Line segment AD and Line segment BC are parallel Definition of a Parallelogram
∠CAD ≅ ∠ACB Alternate interior angles theorem
Line segment BC is congruent to line segment AD Definition of a Parallelogram
Alternate interior angles theorem
Triangle ADE is congruent to triangle CBE Angle-Side-Angle (ASA) Postulate
Line segment BE is congruent to line segment DE CPCTC
Line segment AE is congruent to line segment CE CPCTC
Line segment ACbisects Line segment BD Definition of a bisector


Which statement can be used to fill in the blank space?
∠ADB ≅ ∠CBD

∠ABE ≅ ∠ADE

∠ACD ≅ ∠ACE

∠ACE ≅ ∠CBD





Question 3 (Multiple Choice Worth 4 points)

(03.04 MC)

The following is an incomplete two-column proof that rectangle ABCD is a parallelogram with congruent diagonals:

Rectangle ABCD is shown with four right angles shown in each corner.




Statements

Reasons

m∠DAB = 90°
m∠ABC = 90°
m∠BCD = 90°
m∠CDA = 90° Definition of a Rectangle
segment AB is parallel to segment DC and segment BC is parallel to segment AD Converse of the Same-Side Interior Angles Theorem
Quadrilateral ABCD is a parallelogram Definition of a Parallelogram
Draw segment AC and segment BD by Construction with a straightedge
segment BA is congruent to segment CD Property of Parallelograms
(opposite sides are congruent)
∠BAD ≅ ∠CDA
segment AD is congruent to segment AD Reflexive Property of Equality
ΔBAD ≅ ΔCDA Side-Angle-Side (SAS) Theorem
segment AC is congruent to segment BD Corresponding Parts of Congruent Triangles are Congruent (CPCTC)


What reason completes the proof?
Alternate Interior Angles Theorem

Definition of Congruence

Property of Parallelograms

Same-Side Interior Angles Theorem





Question 4 (Multiple Choice Worth 4 points)

(03.04 MC)

In quadrilateral ABCD, diagonals AC and BD bisect one another:

Quadrilateral ABCD is shown with diagonals AC and BD intersecting at point P.

What statement is used to prove that quadrilateral ABCD is a parallelogram?

Angles ABC and BCD are congruent.

Sides AB and BC are congruent.

Triangles BPA and DPC are congruent.

Triangles BCP and CDP are congruent.





Question 5 (Multiple Choice Worth 4 points)

(03.04 MC)

The following is an incomplete paragraph proving that the opposite angles of parallelogram ABCD are congruent:

Parallelogram ABCD is shown where segment AB is parallel to segment DC and segment BC is parallel to segment AD.