Question

your turn
according to theorem 2, label the parallelogram with the appropriate tick marks.
if parallelogram abcd, then the quadrilateral is a parallelogram.
according to theorem 3, label the parallelogram appropriately.
if parallelogram abcd, then the quadrilateral is a parallelogram.
according to theorem 4, label the parallelogram with the appropriate tick marks.
if parallelogram abcd, then the quadrilateral is a parallelogram.

Explanation:

Identify the standard converse theorems for parallelograms

Using the Parallelogram Theorems knowledge point
Standard geometry curricula define the conditions to prove a quadrilateral is a parallelogram:

• Theorem 2 (Converse of Opposite Sides Theorem): If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
• Theorem 3 (Converse of Opposite Angles Theorem): If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
• Theorem 4 (Converse of Consecutive Angles Theorem / Opposite Sides Parallel and Congruent): If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram.

Determine the required markings for Theorem 2

Using the Geometric Tick Marks knowledge point

• For Theorem 2, we must show both pairs of opposite sides are congruent.
• Mark side \(AB\) and side \(CD\) with single tick marks: \(AB \cong CD\).
• Mark side \(AD\) and side \(BC\) with double tick marks: \(AD \cong BC\).
• The statement is: "If \(AB \cong CD\) and \(AD \cong BC\), then the quadrilateral is a parallelogram."

Determine the required markings for Theorem 3

Using the Geometric Tick Marks knowledge point

• For Theorem 3, we must show both pairs of opposite angles are congruent.
• Mark angle \(\angle A\) and angle \(\angle C\) with single angle arcs: \(\angle A \cong \angle C\).
• Mark angle \(\angle B\) and angle \(\angle D\) with double angle arcs: \(\angle B \cong \angle D\).
• The statement is: "If \(\angle A \cong \angle C\) and \(\angle B \cong \angle D\), then the quadrilateral is a parallelogram."

Determine the required markings for Theorem 4

Using the Geometric Tick Marks knowledge point

• For Theorem 4, we must show one pair of opposite sides is both parallel and congruent.
• Mark side \(AB\) and side \(CD\) with congruent tick marks and parallel arrows.
• The statement is: "If \(AB \parallel CD\) and \(AB \cong CD\), then the quadrilateral is a parallelogram."

Answer:

Question 1

According to Theorem 2, label the parallelogram with the appropriate tick marks.

• Markings on Diagram: Place a single tick mark on sides \(AB\) and \(CD\) to show they are congruent. Place a double tick mark on sides \(AD\) and \(BC\) to show they are congruent.
• Statement: If <blank>\(AB \cong CD\) and \(AD \cong BC\)</blank>, then the quadrilateral is a parallelogram.

Question 2

According to Theorem 3, label the parallelogram appropriately.

• Markings on Diagram: Place a single arc on opposite angles \(\angle A\) and \(\angle C\) to show they are congruent. Place a double arc on opposite angles \(\angle B\) and \(\angle D\) to show they are congruent.
• Statement: If <blank>\(\angle A \cong \angle C\) and \(\angle B \cong \angle D\)</blank>, then the quadrilateral is a parallelogram.

Question 3

According to Theorem 4, label the parallelogram with the appropriate tick marks.

• Markings on Diagram: Place a single tick mark and a parallel arrow on side \(AB\). Place a matching single tick mark and parallel arrow on side \(CD\).
• Statement: If <blank>\(AB \parallel CD\) and \(AB \cong CD\)</blank>, then the quadrilateral is a parallelogram.