# Congruence for Quadrilaterals

ID: kuzav-gomol
Illustrative Mathematics
Subject: Geometry

23 questions

# Congruence for Quadrilaterals

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##### Problem 1

1) Select all quadrilaterals that have 180 degree rotational symmetry. Write each corresponding letter in the answer box and separate letters with commas.

a) trapezoid b) isosceles trapezoid c) parallelogram d) rhombus e) rectangle 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 1 is the same as the distance from ﻿$E$﻿ to 2 . Since ﻿$E$﻿ is the same distance from 3 as it is from 4 , it must lie on the perpendicular bisector of segment 5 . By the same reasoning, ﻿$G$﻿ must lie on the perpendicular bisector of 6 . Therefore, line 7 is the perpendicular bisector of segment ﻿$FH$﻿. So reflecting rhombus ﻿$EFGH$﻿ across line 8 will take ﻿$E$﻿ to 9 and ﻿$G$﻿ to 10 (because ﻿$E$﻿ and ﻿$G$﻿ are on the line of reflection) and ﻿$F$﻿ to 11 and ﻿$H$﻿ to 12 (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 1 , alternate interior angles 2 and 3 are congruent. ﻿$AC$﻿ is congruent to 4 since segments are congruent to themselves. Along with the given information that ﻿$AD$﻿ is congruent to ﻿$BC$﻿, triangle ﻿$ADC$﻿ is congruent to 5 by the 6 Triangle Congruence. Since the triangles are congruent, all pairs of corresponding angles are congruent, so angle ﻿$DCA$﻿ is congruent to 7 . Since those alternate interior angles are congruent, ﻿$AB$﻿ must be parallel to 8 . 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 must be true.

a) $\text{Parallelograms have at least one right angle.}$b) $\text{If a quadrilateral has opposite sides that are both congruent and parallel, then it is a parallelogram. }$c) $\text{Parallelograms have congruent diagonals. }$d) $\text{The height of a parallelogram is greater than the lengths of the sides. }$
##### Problem 5

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

a) ﻿$EFGH$﻿ is a rectangle. b) Triangle ﻿$HEF$﻿ is congruent to triangle ﻿$GFH$﻿.

c) Triangle ﻿$HEF$﻿ is congruent to triangle ﻿$FGH$﻿. d) ﻿$ED$﻿ is congruent to ﻿$HD$﻿, ﻿$DG$﻿, and ﻿$DF$﻿.

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.

22) Explain your reasoning.

##### Problem 7

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