Lesson 14

Bisect It

  • Let’s prove that some constructions we conjectured about really work.

Problem 1

Select all quadrilaterals for which a diagonal is also a line of symmetry.

A:

trapezoid

B:

isosceles trapezoid

C:

parallelogram

D:

rhombus

E:

rectangle

F:

square

Problem 2

Show that diagonal \(EG\) is a line of symmetry for rhombus \(EFGH\).

Rhombus E F G H. Line E G is drawn in the middle of the rhombus.

Problem 3

\(ABDE\) is an isosceles trapezoid. Priya makes a claim that triangle \( AEB\) is congruent to triangle \(DBE\). Convince Priya this is not true. 

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.  

 

(From Unit 2, Lesson 13.)

Problem 4

In quadrilateral \(ABCD\), triangle \(ADC\) is congruent to \(CBA\). Show that \(ABCD\) is a parallelogram.

Quadrilateral ABCD with line segment AC.

 

(From Unit 2, Lesson 13.)

Problem 5

Priya is convinced the diagonals of the isosceles trapezoid are congruent. She knows that if she can prove triangles congruent that include the diagonals, then she will show that diagonals are also congruent. Help her complete the proof.

\(ABDE\) is an isosceles trapezoid. 

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.  

Draw auxiliary lines that are diagonals \(\underline{\hspace{.5in}1\hspace{.5in}}\) and \(\underline{\hspace{.5in}2\hspace{.5in}}\).  \(AB \) is congruent to \(\underline{\hspace{.5in}3\hspace{.5in}}\)because they are the same segment. We know angle \(B\)  and \(\underline{\hspace{.5in}4\hspace{.5in}}\) are congruent. We know \(AE\) is congruent to  \(\underline{\hspace{.5in}5\hspace{.5in}}\). Therefore, triangle \(ABE \) and \(\underline{\hspace{.5in}6\hspace{.5in}}\) are congruent because of \(\underline{\hspace{.5in}7\hspace{.5in}}\). Finally, diagonal \(BE\) is congruent to \(\underline{\hspace{.5in}8\hspace{.5in}}\) because \(\underline{\hspace{.5in}9\hspace{.5in}}\). 

(From Unit 2, Lesson 12.)

Problem 6

Is triangle \(AFE\) congruent to triangle \(ADE\)?
Explain your reasoning.

\(\overline{AF} \cong \overline{AD}, \angle F \cong \angle D\)

Triangle A F D. Segment A E, with E on F D. A F and A D have single tick marks. Angles F and D have arcs with single tick marks.
(From Unit 2, Lesson 11.)

Problem 7

Triangle \(DAC\) is isosceles with congruent sides \(AD\) and \(AC\). Which additional given information is sufficient for showing that triangle \(DBC\) is isosceles? Select all that apply.

Triangle ACD with point B near the center. Line segment AC is congruent to AD.
A:

Segment \(DB\) is congruent to segment \(BC\). 

B:

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

C:

Angle \(ABD\) is congruent to angle \(ABC\).

D:

Angle \(ADC\) is congruent to angle \(ACD\).

E:

\(AB\) is an angle bisector of \(DAC\).

F:

Triangle \(BDA\) is congruent to triangle \(BDC\).

(From Unit 2, Lesson 6.)