Lesson 13

Proofs about Parallelograms

  • Let’s prove theorems about parallelograms.

Problem 1

Conjecture: A quadrilateral with one pair of sides both congruent and parallel is a parallelogram.

  1. Draw a diagram of the situation. 
  2. Mark the given information. 
  3. Restate the conjecture as a specific statement using the diagram.

Problem 2

In quadrilateral \(ABCD\), \(AD\) is congruent to \(BC\), and \(AD\) is parallel to \(BC\). Show that \(ABCD\) is a parallelogram.

Quadrilateral ABCD with line segment AC.

Problem 3

\(ABDE\) is an isosceles trapezoid. Name one pair of congruent triangles that could be used to show that the diagonals of an isosceles trapezoid are congruent.

Isosceles trapezoid A B D E. Angles are marked with tick marks, as follows: D E A, one mark; E A B, two marks; A B D, two tick marks; B D E, one tick mark. Segment A E and B D each have one tick mark.  

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(From Unit 2, Lesson 12.)

Problem 4

Select the conjecture with the rephrased statement of proof to show the diagonals of a parallelogram bisect each other. 

Quadrilateral EFGH. Line segments EG and HF intersect at point K.
A:

In parallelogram \(EFGH\), show triangle \(HEF\) is congruent to triangle \(FGH\).

B:

In parallelogram \(EFGH\), show triangle \(EKH\) is congruent to triangle \(GKF\).

C:

In parallelogram \(EFGH\), show \(EK\) is congruent to \(KG\) and \(FK\) is congruent to \(KH\).

D:

In quadrilateral \(EFGH\) with \(GH\) congruent to \(FE\) and \(EH\) congruent to \(FG\), show \(EFGH\) is a parallelogram.

(From Unit 2, Lesson 12.)

Problem 5

Is triangle \(EJH\) congruent to triangle \(EIH\)?
Explain your reasoning.

\(\overline{HJ} \perp \overline{JE}, \overline{HI} \perp \overline{IE}, \overline{HJ} \cong \overline{HI}\)

Quadrilateral J H I E. A line connects H and E. Sides J H and H I have single tick marks. Angles J and I are marked with small squares.

 

(From Unit 2, Lesson 11.)

Problem 6

Select all true statements based on the diagram.

Quadrilateral ABCD. Line AB is parallel to line DC, both cut by congruent transversals AD and BC. Diagonals AC and DB bisect each other at point E at a right angle. Segment AE is congruent to segment BE.
A:

Segment \(DC\) is congruent to segment \(AB\).

B:

Segment \(DA\) is congruent to segment \(CB\).

C:

Line \(DC\) is parallel to line \(AB\).

D:

Line \(DA\) is parallel to line \(CB\).

E:

Angle \(CBE\) is congruent to angle \(DEA\).

F:

Angle \(CEB\) is congruent to angle \(DEA\).

(From Unit 2, Lesson 10.)

Problem 7

Which conjecture is possible to prove?

A:

If the four angles in a quadrilateral are congruent to the four angles in another quadrilateral, then the two quadrilaterals are congruent.

B:

If the four sides in a quadrilateral are congruent to the four sides in another quadrilateral, then the two quadrilaterals are congruent.

C:

If the three angles in a triangle are congruent to the three angles in another triangle, then the two triangles are congruent.

D:

If the three sides in a triangle are congruent to the three sides in another triangle, then the two triangles are congruent.

(From Unit 2, Lesson 5.)