# 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.

trapezoid

isosceles trapezoid

parallelogram

rhombus

rectangle

square

### Problem 2

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

### 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.

### Problem 4

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

### 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.

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 \(\underline{\hspace{.5in}9\hspace{.5in}}\).

### Problem 6

Is triangle \(AFE\) congruent to triangle \(ADE\)?

Explain your reasoning.

### 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.

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

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

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

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

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

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