Exercises: Prove Theorems About Parallelograms

The midpoints of segment $\overline{AC}$ and segment $\overline{BD}$ are both the point $(3, 4)$. What can you conclude?

Lines $\ell$ and $m$ are parallel, and line $t$ is a transversal crossing both. Which statement about the alternate interior angles is true?

In $\triangle ABC \cong \triangle DEF$, which reason justifies concluding that $\overline{AB} \cong \overline{DE}$?

Parallelogram $ABCD$ has diagonal $\overline{AC}$ drawn. To prove $\triangle ABC \cong \triangle CDA$ using ASA, a student uses the fact that $AB \parallel DC$. Which angle pair does this parallel relationship justify as congruent?

In parallelogram $ABCD$, it is given that $AB = 7$ cm and $BC = 4$ cm. What are the lengths of $CD$ and $DA$?

In parallelogram $ABCD$, $\angle A = 65^\circ$. Find $m\angle C$ in degrees.

In parallelogram $PQRS$, $\angle P = 112^\circ$. Find $m\angle Q$ in degrees.

In quadrilateral $WXYZ$, the diagonals $\overline{WY}$ and $\overline{XZ}$ intersect at point $M$, with $WM = MY = 6$ and $XM = MZ = 4$. Which conclusion is justified?

In the proof that opposite sides of parallelogram $ABCD$ are congruent, diagonal $\overline{AC}$ is drawn, creating $\triangle ABC$ and $\triangle CDA$. What is the correct order of justifications that establishes $AB = CD$?

In parallelogram $KLMN$, $\angle K = 74^\circ$.

$m\angle M = \underline{\hspace{5em}}^\circ$ because opposite angles are   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

$m\angle L = \underline{\hspace{5em}}^\circ$ because consecutive angles are   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

Rhombus $ABCD$ has all four sides equal to 5 cm and $\angle A = 60^\circ$. Which statement is true?

The diagram shows two parallelograms side by side. In parallelogram $ABCD$ (left), the diagonals have lengths $AC = 8$ and $BD = 6$. In parallelogram $EFGH$ (right), the diagonals have lengths $EG = 7$ and $FH = 7$. Which classification is correct?

Which statement is always true for any parallelogram?

In quadrilateral $PQRS$, the diagonals $\overline{PR}$ and $\overline{QS}$ intersect at point $T$. A surveyor finds that $PT = 12$ m, $TR = 12$ m, $QT = 8$ m, and $TS = 8$ m. The surveyor concludes that $PQRS$ is a parallelogram. What theorem justifies this conclusion?

Enter the number of the correct justification:
1 — Both pairs of opposite sides are parallel (definition)
2 — Both pairs of opposite sides are congruent
3 — The diagonals bisect each other (converse theorem)
4 — One pair of opposite sides is both parallel and congruent

A carpenter is building a rectangular frame and wants to verify it is truly rectangular without measuring angles. She measures both diagonals and finds they are the same length. Her assistant says: "That alone doesn't prove it's a rectangle — it just proves it's a parallelogram." Is the assistant correct?

Is the student's claim correct? Identify the error and explain the correct relationship.

Is the student's reasoning valid? Identify the error and state the correct condition needed.

In parallelogram $ABCD$, diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$. A student claims: "Since the diagonals bisect each other, $AE = BE$ — the half-diagonal segments from the same vertex are equal."

Explain whether the student's claim is correct. If it is incorrect, describe what "bisect each other" actually means and give an example of specific segment equalities that do hold.

The following theorem has been proved in class: "If a parallelogram has congruent diagonals, then it is a rectangle."

A student says: "The converse must also be true without any extra proof — if it is a rectangle, its diagonals must be congruent, because the theorem goes both ways."

Explain whether the student is correct. State clearly which direction is the theorem, which direction is the converse, and whether each requires its own proof.