# Bisect It

ID: livir-dabik
Illustrative Mathematics, CC BY 4.0
Subject: Geometry

15 questions

# Bisect It

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##### Problem 1

1) Select ﻿$\textbf{all}$﻿ quadrilaterals for which a diagonal is also a line of symmetry. Write each corresponding letter in the answer box and separate letters with commas.

a) trapezoid ﻿$\quad\quad$﻿ b) isosceles trapezoid ﻿$\quad\quad$﻿ c) parallelogram ﻿$\quad\quad$﻿ d) rhombus ﻿$\quad\quad$﻿ e) rectangle ﻿$\quad\quad$﻿ f) square

##### Problem 2

2) Show that diagonal ﻿$EG$﻿ is a line of symmetry for rhombus ﻿$EFGH$﻿.

##### Problem 3

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

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.

Given: ﻿$ABCD$﻿ is an isosceles trapezoid.

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

Fill in the blanks using items from the Bank of Terms below. Some items might be used more than once.

Bank of Terms: Side-Angle-Side Triangle Congruence Theorem, Corresponding parts of congruent figures are congruent, Triangle ﻿$BAD$﻿, Angle ﻿$A$﻿, ﻿$AD$﻿, ﻿$BA$﻿, ﻿$BD$﻿, ﻿$BE$﻿.

5) Blanks 1 and 2, answers separated by a comma.

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##### Problem 6

Given: ﻿$\overline{AF} \cong \overline{AD}$﻿, ﻿$\angle F \cong \angle D$﻿

13) Is there enough information in the diagram to prove that triangle ﻿$AFE$﻿ is congruent to triangle ﻿$ADE$﻿?

True or false? Write below.

15) Triangle ﻿$DAC$﻿ is isosceles with congruent sides ﻿$AD$﻿ and ﻿$AC$﻿. Which additional given information is sufficient for showing that triangle ﻿$DBC$﻿ is isosceles? Select ﻿$\textbf{all}$﻿ that apply. Write each corresponding letter in the answer box and separate letters with commas.
a) Segment ﻿$DB$﻿ is congruent to segment ﻿$BC$﻿. ﻿$\quad\quad$﻿ b) Segment ﻿$AB$﻿ is congruent to segment ﻿$BD$﻿.
c) Angle ﻿$ABD$﻿ is congruent to angle ﻿$ABC$﻿. ﻿$\quad\quad$﻿ d) Angle ﻿$ADC$﻿ is congruent to angle ﻿$ACD$﻿.
e) ﻿$AB$﻿ is an angle bisector of ﻿$DAC$﻿. ﻿$\quad\quad$﻿ f) Triangle ﻿$BDA$﻿ is congruent to triangle ﻿$BDC$﻿.