Illustrative Mathematics, CC BY 4.0
Subject: Geometry

23 questions

Classroom:
Due:
Student Name:
Date Submitted:
##### Problem 1

1) Select ﻿$\textbf{all}$﻿ quadrilaterals that have 180 degree rotational 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

Lin wrote a proof to show that diagonal ﻿$EG$﻿ is a line of symmetry for rhombus ﻿$EFGH$﻿. Fill in the blanks to complete her proof:

Because ﻿$EFGH$﻿ is a rhombus, the distance from ﻿$E$﻿ to ﻿$\underline{\quad\quad 1 \quad\quad}$﻿ is the same as the distance from ﻿$E$﻿ to ﻿$\underline{\quad\quad 2 \quad\quad}$﻿. Since ﻿$E$﻿ is the same distance from ﻿$\underline{\quad\quad 3 \quad\quad}$﻿ as it is from ﻿$\underline{\quad\quad 4 \quad\quad}$﻿, it must lie on the perpendicular bisector of segment ﻿$\underline{\quad\quad 5 \quad\quad}$﻿. By the same reasoning, ﻿$G$﻿ must lie on the perpendicular bisector of ﻿$\underline{\quad\quad 6 \quad\quad}$﻿. Therefore, line ﻿$\underline{\quad\quad 7 \quad\quad}$﻿ is the perpendicular bisector of segment ﻿$FH$﻿. So reflecting rhombus ﻿$EFGH$﻿ across line ﻿$\underline{\quad\quad 8 \quad\quad}$﻿ will take ﻿$E$﻿ to ﻿$\underline{\quad\quad 9 \quad\quad}$﻿ and ﻿$G$﻿ to ﻿$\underline{\quad\quad 10 \quad\quad}$﻿ (because ﻿$E$﻿ and ﻿$G$﻿ are on the line of reflection) and ﻿$F$﻿ to ﻿$\underline{\quad\quad 11 \quad\quad}$﻿ and ﻿$H$﻿ to ﻿$\underline{\quad\quad 12 \quad\quad}$﻿ (since ﻿$FH$﻿ is perpendicular to the line of reflection, and ﻿$F$﻿ and ﻿$H$﻿ are the same distance from the line of reflection, on opposite sides). Since the image of rhombus ﻿$EFGH$﻿ reflected across ﻿$EG$﻿ is rhombus ﻿$EHGF$﻿ (the same rhombus!), line ﻿$EF$﻿ must be a line of symmetry for rhombus ﻿$EFGH$﻿.

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

Bank of Terms: ﻿$E$﻿, ﻿$F$﻿, ﻿$G$﻿, ﻿$H$﻿, ﻿$EG$﻿, ﻿$FH$﻿

2) Blanks 1 and 2, separated by a comma

3) Blanks 3 and 4, separated by a comma

4) Blank 5

5) Blank 6

6) Blank 7

7) Blank 8

8) Blank 9

9) Blank 10

10) Blank 11

11) Blank 12

##### Problem 3

In quadrilateral ﻿$ABCD$﻿, ﻿$AD$﻿ is congruent to ﻿$BC$﻿, and ﻿$AD$﻿ is parallel to ﻿$BC$﻿. Andre has written a proof to show that ﻿$ABCD$﻿ is a parallelogram. Fill in the blanks to complete the proof.

Since ﻿$AD$﻿ is parallel to ﻿$\underline{\quad\quad 1 \quad\quad}$﻿, alternate interior angles ﻿$\underline{\quad\quad 2 \quad\quad}$﻿ and ﻿$\underline{\quad\quad 3 \quad\quad}$﻿ are congruent. ﻿$AC$﻿ is congruent to ﻿$\underline{\quad\quad 4 \quad\quad}$﻿ since segments are congruent to themselves. Along with the given information that ﻿$AD$﻿ is congruent to ﻿$BC$﻿, triangle ﻿$ADC$﻿ is congruent to﻿$\underline{\quad\quad 5 \quad\quad}$﻿ by the ﻿$\underline{\quad\quad 6 \quad\quad}$﻿ Triangle Congruence. Since the triangles are congruent, all pairs of corresponding angles are congruent, so angle ﻿$DCA$﻿ is congruent to ﻿$\underline{\quad\quad 7 \quad\quad}$﻿. Since those alternate interior angles are congruent, ﻿$AB$﻿ must be parallel to ﻿$\underline{\quad\quad 8 \quad\quad}$﻿. Since we define a parallelogram as a quadrilateral with both pairs of opposite sides parallel, ﻿$ABCD$﻿ is a parallelogram.

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

Bank of Terms: ﻿$AC$﻿, ﻿$BC$﻿, ﻿$CD$﻿, ﻿$BAC$﻿, ﻿$BCA$﻿, ﻿$CBA$﻿, ﻿$DAC$﻿, Side-Angle-Side

12) Blank 1

13) Blanks 2 and 3, separated by a comma

14) Blank 4

15) Blank 5

16) Blank 6

17) Blank 7

18) Blank 8

##### Problem 4

19) Select the statement that ﻿$\textbf{must}$﻿ be true.

a)

Parallelograms have at least one right angle.

b)

If a quadrilateral has opposite sides that are both congruent and parallel, then it is a parallelogram.

c)

Parallelograms have congruent diagonals.

d)

The height of a parallelogram is greater than the lengths of the sides.

e)

f)

##### Problem 5

20) ﻿$EFGH$﻿ is a parallelogram and angle ﻿$HEF$﻿ is a right angle. Select ﻿$\textbf{all}$﻿ statements that ﻿$\textbf{must}$﻿ be true. Write each corresponding letter in the answer box and separate letters with commas.

a) ﻿$EFGH$﻿ is a rectangle. ﻿$\quad\quad$﻿ b) Triangle ﻿$HEF$﻿ is congruent to triangle ﻿$GFH$﻿. ﻿$\quad\quad$﻿ c) Triangle ﻿$HEF$﻿ is congruent to triangle ﻿$FGH$﻿. ﻿$\quad\quad$﻿ d) ﻿$ED$﻿ is congruent to ﻿$HD$﻿, ﻿$DG$﻿, and ﻿$DF$﻿. ﻿$\quad\quad$﻿ e) Triangle ﻿$EDH$﻿ is congruent to triangle ﻿$HDG$﻿.

##### Problem 6

Figure ﻿$ABCD$﻿ is a parallelogram.

Given: ﻿$\overline{AB} \cong \overline{CD}$﻿, ﻿$\angle ADB \cong \angle CBD$﻿

21) Is triangle ﻿$ADB$﻿ congruent to triangle ﻿$CBD$﻿?

True or false? Write below.

23) Figure ﻿$KLMN$﻿ is a parallelogram. Prove that triangle ﻿$KNL$﻿ is congruent to triangle ﻿$MLN$﻿.