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Common Core: High School - Geometry : Parallelogram Proofs

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Example Questions

Example Question #21 : Prove Parallelogram Theorems: Ccss.Math.Content.Hsg Co.C.11

Prove that since this quadrilateral has two pairs of opposite congruent angles it is a parallelogram.

Screen shot 2020 08 20 at 9.48.56 am

Possible Answers:

Proof:

Screen shot 2020 08 20 at 9.50.51 am

Proof:

Screen shot 2020 08 20 at 9.51.28 am

Proof:

Screen shot 2020 08 20 at 9.50.12 am

Correct answer:

Proof:

Screen shot 2020 08 20 at 9.50.51 am

Explanation:

Screen shot 2020 08 20 at 9.49.06 am

Thus, we have shown that if a quadrilateral has two pairs of opposite congruent angles, it is a parallelogram.

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Example Question #151 : Congruence

Prove the following parallelogram has two pairs of opposite congruent angles.

Screen shot 2020 08 20 at 10.02.30 am

Possible Answers:

Proof:

ABCD is a parallelogram (given)

We can add the line across the diagonal

Screen shot 2020 08 20 at 10.04.42 am

$\overline{AB}$ is parallel to $\overline{CD}$

$\overline{AC}$ is parallel to $\overline{BD}$ (definition of a parallelogram)

$\angle 1\cong \angle 4$

$\angle 2\cong \angle 3$ (alternate interior angles)

$\overline{CB}$ is a common side between $\Delta ABC$ and $\Delta DCB$

$\Delta ABC\cong \Delta DCB$ (By Angle-Side-Angle Theorem)

$\angle A\cong \angle D$ (corresponding parts of congruent triangles)

$\angle 1= \angle 4, \angle 2= \angle 3$

$\angle 1+\angle 2= \angle 4+\angle 3$ (Addition of equalities)

$\angle 1+\angle 2= \angle C$

$\angle 4+\angle 3= \angle B$ (Angle Addition Postulate)

$\angle C= \angle B$

$\angle C\cong \angle B$

Therefore $\angle A\cong \angle D$ and $\angle C\cong \angle B$

Proof:

ABCD is a parallelogram (given)

We can add the line across the diagonal

Screen shot 2020 08 20 at 10.04.42 am

$\overline{AB}$ is parallel to $\overline{CD}$

$\overline{AC}$ is parallel to $\overline{BD}$ (definition of a parallelogram)

$\angle 1\cong \angle 4$

$\angle 2\cong \angle 3$ (alternate interior angles)

$\overline{CB}$ is a common side between $\Delta ABC$ and $\Delta DCB$

$\Delta ABC\cong \Delta DCB$ (By Angle-Side-Angle Theorem)

Because the triangles are congruent we can assume:

$\angle C\cong \angle B$ , $\angle A\cong \angle D$

Therefore $\angle A\cong \angle D$ and $\angle C\cong \angle B$

Proof:

ABCD is a parallelogram (given)

We can add the line across the diagonal

Screen shot 2020 08 20 at 10.04.42 am

$\overline{AB}$ is parallel to $\overline{CD}$

$\overline{AC}$ is parallel to $\overline{BD}$ (definition of a parallelogram)

$\angle 1\cong \angle 4$

$\angle 2\cong \angle 3$ (corresponding angles)

$\overline{CB}$ is a common side between $\Delta ABC$ and $\Delta DCB$

$\Delta ABC\cong \Delta DCB$ (By Angle-Side-Angle Theorem)

$\angle A\cong \angle D$ (corresponding parts of congruent triangles)

$\angle 1= \angle 4, \angle 2= \angle 3$

$\angle 1+\angle 2= \angle 4+\angle 3$ (Addition of equalities)

$\angle 1+\angle 2= \angle C$

$\angle 4+\angle 3= \angle B$ (Angle Addition Postulate)

$\angle C= \angle B$

$\angle C\cong \angle B$

Therefore $\angle A\cong \angle D$ and $\angle C\cong \angle B$

Correct answer:

Proof:

ABCD is a parallelogram (given)

We can add the line across the diagonal

Screen shot 2020 08 20 at 10.04.42 am

$\overline{AB}$ is parallel to $\overline{CD}$

$\overline{AC}$ is parallel to $\overline{BD}$ (definition of a parallelogram)

$\angle 1\cong \angle 4$

$\angle 2\cong \angle 3$ (alternate interior angles)

$\overline{CB}$ is a common side between $\Delta ABC$ and $\Delta DCB$

$\Delta ABC\cong \Delta DCB$ (By Angle-Side-Angle Theorem)

$\angle A\cong \angle D$ (corresponding parts of congruent triangles)

$\angle 1= \angle 4, \angle 2= \angle 3$

$\angle 1+\angle 2= \angle 4+\angle 3$ (Addition of equalities)

$\angle 1+\angle 2= \angle C$

$\angle 4+\angle 3= \angle B$ (Angle Addition Postulate)

$\angle C= \angle B$

$\angle C\cong \angle B$

Therefore $\angle A\cong \angle D$ and $\angle C\cong \angle B$

Explanation:

Statement Reasoning

ABDC is a parallelogram. This is given in the problem.

We can add the line across the diagonal Connecting any two points make a line, so this is a valid line we can add.

Screen shot 2020 08 20 at 10.03.17 am

$\overline{AB}$ is parallel to $\overline{CD}$

$\overline{AC}$ is parallel to $\overline{BD}$ We know this to be true according to the definition of a parallelogram.

$\angle 1\cong \angle 4$

$\angle 2\cong \angle 3$ Line is a transversal line intersection two parallel lines. We could extend lines and or lines and to make this relationship more clear. Alternate interior angles are formed by this transversal line.

Screen shot 2020 08 20 at 10.03.56 am

$\overline{CB}$ is a common side between $\Delta ABC$ and $\Delta DCB$

$\Delta ABC\cong \Delta DCB$ We are able to use the Angle-Side-Angle Theorem because we have one congruent side between these two triangles, , and two pairs of congruent angles, $\angle 1\cong \angle 4$ , $\angle 2\cong \angle 3$

$\angle A\cong \angle D$ Congruent triangles have congruent corresponding parts by definition

$\angle 1= \angle 4, \angle 2= \angle 3$ Because these angles are congruent they are also equal

$\angle 1+\angle 2= \angle 4+\angle 3$ Since $\angle 2= \angle 3$ we can add one of these angles to each side and still keep the equation balanced and equal

$\angle 1+\angle 2= \angle C$

$\angle 4+\angle 3= \angle B$ The Angle Addition Postulate says that two side by side angles create a new angle whose measure is equal to their sum

$\angle C= \angle B$ We are simply substituting these equalities into the equation $\angle 1+\angle 2= \angle 4+\angle 3$

$\angle C\cong \angle B$ Equal angles are congruent

Therefore $\angle A\cong \angle D$ and $\angle C\cong \angle B$

Thus, we have proven that this parallelogram has two pairs of opposite congruent angles.

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Example Question #11 : Parallelogram Proofs

Prove the following parallelogram has diagonals that bisect each other.

Screen shot 2020 08 20 at 10.34.27 am

Possible Answers:

Proof:

Screen shot 2020 08 20 at 10.36.23 am

Proof:

Screen shot 2020 08 20 at 10.35.33 am

Proof:

Screen shot 2020 08 20 at 10.35.58 am

Correct answer:

Proof:

Screen shot 2020 08 20 at 10.35.58 am

Explanation:

Screen shot 2020 08 20 at 10.34.58 am

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Common Core: High School - Geometry : Parallelogram Proofs

Study concepts, example questions & explanations for Common Core: High School - Geometry

All Common Core: High School - Geometry Resources

Example Questions

Example Question #21 : Prove Parallelogram Theorems: Ccss.Math.Content.Hsg Co.C.11

Example Question #151 : Congruence

Example Question #11 : Parallelogram Proofs

All Common Core: High School - Geometry Resources

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