Back to Exercise: Prove theorems about parallelograms

Exercises: Prove Theorems About Parallelograms

Work through each section in order. Show all proof steps and justifications where indicated.

Grade 9·21 problems·~30 min·Common Core Math - HS Geometry·standard·hsg-co-c-11

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Warm-Up: Review What You Know

These problems review skills from earlier in the course that you will need for parallelogram proofs.

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

$\overline{AC}$ and $\overline{BD}$ bisect each other.

$\overline{AC}$ and $\overline{BD}$ are congruent.

$\overline{AC}$ and $\overline{BD}$ are perpendicular.

$\overline{AC}$ and $\overline{BD}$ are parallel.

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

They are supplementary (they sum to $180^\circ$ ).

They are congruent.

They are complementary (they sum to $90^\circ$ ).

They are vertical angles.

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

Definition of congruent triangles

CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

SAS (Side-Angle-Side)

Reflexive Property

Fluency Practice

$Parallelogram ABCD with diagonal AC drawn as a dashed line from top-left vertex A to bottom-right vertex C.$

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?

$\angle BAC \cong \angle DCA$ (alternate interior angles, transversal $\overline{AC}$ , lines $AB \parallel DC$ )

$\angle ABC \cong \angle CDA$ (corresponding angles, transversal $\overline{AC}$ , lines $AB \parallel DC$ )

$\angle BAD \cong \angle BCD$ (alternate interior angles, transversal $\overline{AC}$ , lines $AB \parallel DC$ )

$\angle BAC \cong \angle DAC$ (alternate interior angles, transversal $\overline{AC}$ , lines $AB \parallel DC$ )

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

$CD = 4$ cm, $DA = 7$ cm

$CD = 7$ cm, $DA = 4$ cm

$CD = 7$ cm, $DA = 7$ cm

$CD = 4$ cm, $DA = 4$ cm

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?

$WXYZ$ is a parallelogram because its diagonals bisect each other.

$WXYZ$ is a parallelogram because $WY = XZ$ .

$WXYZ$ is a parallelogram because one pair of opposite sides is congruent.

$WXYZ$ is a rectangle because the diagonals have different lengths.

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