End-of-Unit Assessment

1.

Select all the true statements.

A.

Two squares with the same side lengths are always congruent.

B.

Two rectangles with the same side lengths are always congruent.

C.

Two rhombuses with the same side lengths are always congruent.

D.

Two parallelograms with the same side lengths are always congruent.

E.

Two quadrilaterals with the same side lengths are always congruent.

Answer: A, B

Teaching Notes

This problem calls upon students to use their understanding of the meaning of congruence and apply it in an abstract situation in which they must think carefully about the taxonomy of quadrilaterals.

Students who select choice C or choice D may think that all rhombuses (or all parallelograms) have the same shape. They may envision one of the pattern blocks, for instance, and forget that different rhombuses can have different angles. Students who do not select choice A or choice B may forget that all squares and rectangles must have four right angles. Students who select choice E may not take into consideration that parallelograms and rhombuses are quadrilaterals that may not have the same shape.

2.

Lines $CE$ and $AD$ intersect at $B$ .

<p>Lines C E and A D intersect at the point B. Angle A B C is labeled 37 degrees.</p><br>

Select all the true statements.

A.

The measure of angle $CBA$ is equal to the measure of angle $DBE$ .

B.

The sum of the measures of angles $CBA$ and $DBE$ is $180^\circ$ .

C.

The measure of angle $CBD$ is equal to the measure of angle $ABE$ .

D.

The sum of the measures of angles $CBD$ and $CBA$ is $180^\circ$ .

E.

The sum of the measures of angles $CBA$ and $DBE$ is $90^\circ$ .

Answer: A, C, D

Teaching Notes

Students identify pairs of angles in a diagram whose measures are equal and whose measures sum to 180 degrees. Students who do not select choice A or choice C may not recognize vertical angles in a diagram, or may not understand that the measures of vertical angles are equal. Students who select choice B may believe angles $CBA$ and $DBE$ must be supplementary rather than have equal measure. Students who do not select choice D may not recognize a linear pair in a diagram, or may not understand that these angles are supplementary. Students who select choice E may believe angles $CBA$ and $DBE$ must be complementary rather than have equal measure.

3.

Diego made the shape on the left, and Elena made the shape on the right. Each shape uses 5 circles.

Select all the true statements.

A.

The smallest circle in Diego's design is congruent to the smallest circle in Elena’s design.

B.

Diego’s design is congruent to Elena’s design.

C.

Elena’s design is a translation of Diego's design.

D.

The largest circle in Elena's design is congruent to the largest circle in Diego's design.

E.

Each circle in Elena's design has a congruent circle within Diego's design.

Answer: A, D, E

Teaching Notes

The key idea in this problem is that distances between all pairs of corresponding points of congruent figures are the same. It is not enough that the individual parts of complex shapes be congruent, as those parts also need to be in the same position relative to one another.

Students who do not select choice A may think congruence requires the same position. Since these two circles are the same size and shape, they are congruent regardless of relative position. Students who select choice B look at the component parts of the shape, which are each congruent, rather than the shape itself. Likewise, students who select choice C probably imagine translating each circle separately from one figure to the other. Students who select choice D and choice E may understand the individual components of each design are congruent even though the design/shape itself is not congruent.

4.

Describe a sequence of transformations that shows that Polygon A is congruent to Polygon B.

Answer:

Answers vary. Sample response: Polygon A can be rotated 90 degrees counterclockwise around its furthermost point on the right shown in the picture and then translated 4 units to the right.

<p>Transformation of Polygon A to B.</p>

Minimal Tier 1 response:

Work is complete and correct.
Sample: Rotate 90 degrees counterclockwise around the rightmost point of Polygon A, translate 4 units right.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: A drawing showing the intermediate transformation (the green polygon in the sample response), but no verbal descriptions; incomplete verbal descriptions (such as reference to a rotation without specifying a center point); the sequence of transformations contains a small, easily identifiable error (such as saying to rotate clockwise when the counterclockwise direction is the one that works); sequence of transformations is correct but does not use proper vocabulary (“turn” instead of rotate; “move” instead of translate).

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Explanation without reference to rigid transformations, or a sequence of transformations that does not take Polygon A to Polygon B (with no obvious small mistakes responsible for this error); descriptions of transformations are very unclear and the intended meaning is not evident.

Teaching Notes

Students show multistep congruence on a grid.

5.

For each pair of shapes, decide whether or not Shape A is congruent to Shape B. Explain your reasoning.

First pair:
Second pair:

Answer:

Congruent. If Shape A is reflected over its right side, then rotated 90 degrees counterclockwise around the lower vertex, it can be placed on top of Shape B with a translation down and to the right.
Not congruent. The shapes look congruent, but when Shape A is moved on top of Shape B with a 90-degree counterclockwise rotation and a translation, they do not match up.

Minimal Tier 1 response:

Work is complete and correct.
Acceptable errors: Omitting reference to lines of reflection, centers of rotation, angles of rotation, and distance of translation, provided the visual makes these things clear.
Sample:

(with accompanying accurate drawing) Congruent, because I can reflect Shape A, rotate it, and then translate it onto Shape B.
(with accompanying accurate drawing) Not congruent, because when I rotate Shape A and then translate it, it still doesn’t match up with Shape B. Alternate response: I measured the angles, and they are not the same in the two shapes.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Transformations are shown, but with no written descriptions; in part b, transformations are done mostly correctly but enough accuracy was lost that the shapes appear to coincide.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Work states that shapes are or are not congruent with no justification; descriptions of transformations are unclear and the intended meaning is not evident; vague explanations, such as “the shapes look the same.”

Teaching Notes

Students determine if two shapes are congruent without the use of a grid. Tracing paper would be useful for this task. The description of the transformations when there is a congruence does not have the same precision as a description aided by a grid. That is, students may talk about translating to the left rather than specifying the exact distance on a grid. Similarly, they may talk about a vertical or horizontal reflection or a rotation without necessarily drawing the line of reflection or providing the measure of the angle of rotation.

6.

Lines $AB$ and $CD$ are parallel. Find the measures of the three angles in triangle $ABF$ .

<p>Parallel lines A B and C D cut by transversals A D and B C intersecting at point F. Point E is on A D and above the parallel lines. Angle C D E is 157 degrees. Angle D C F is 42 degrees.</p><br>

Answer:

$B: 42^\circ$ , $A: 23^\circ$ , $F:115^\circ$

Teaching Notes

Students can use the following to calculate angles:

Supplementary angles add to $180^\circ$ .
Alternate interior angles made by parallel lines cut by a transversal are congruent.
The three angles of a triangle add to $180^\circ$ .

Here they are asked to use this information to find the angles of a triangle.

7.

Triangle $CDA$ is the image of triangle $ABC$ after a $180^\circ$ rotation around the midpoint of segment $AC$ . Triangle $ECB$ is the image of triangle $ABC$ after a $180^\circ$ rotation around the midpoint of segment $BC$ .

The measure of angle $D$ is $52^\circ$ and the measure of angle $E$ is $94^\circ$ .

<p>Quadrilateral A B D E. Point C lies on D E. Segments A C and A B are drawn inside the quadrilateral and each have a point marked.</p><br>

Identify at least two pairs of congruent angles in the figure and explain how you know they are congruent.
What is the measure of angle $CBE$ ? Explain how you know.
Name a triangle congruent to triangle $CBE$ . Describe a rigid transformation from triangle $CBE$ to that triangle.

Answer:

Any two pairs: Angle $D$ , angle $ECB$ , and angle $CBA$ are congruent; angle $DCA$ , angle $E$ , and angle $BAC$ are congruent; angle $DAC$ , angle $BCA$ , and angle $CBE$ are congruent. Sample reasoning: Triangle $CDA$ and triangle $ECB$ are the images of triangle $ABC$ after a rotation, so there is a rigid transformation from any one of the triangles to the other, and they must be congruent. Since the triangles are congruent, their corresponding angles are also congruent.
$34^\circ$ . Angle $ECB$ is $52^\circ$ since it is congruent to angle $D$ . Angle $DCA$ is $94^\circ$ since it is congruent to angle $E$ . Since the angles around point $C$ form a straight angle, they must be $180^\circ$ . Angle $ACB$ is $34^\circ$ since $180-94-52=34$ . Since angle $ACB$ is congruent to angle $CBE$ , angle $CBE$ is also $34^\circ$ .
Sample responses:
- Triangle $DAC$ . Translate left by the directed line segment from point C to point D.
- Triangle $BCA$ . Rotate $180^\circ$ around the midpoint of $BC$ .

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample:

Angle $D$ and angle $ECB$ . Angle $E$ , and angle $DCA$ . Triangle $ACD$ is the image of triangle $BEC$ after a translation, so their corresponding angles are congruent.
$34^\circ$ . Angle $ECB$ is $52^\circ$ since it is congruent to angle $D$ . Angle $DCA$ is $94^\circ$ since it is congruent to angle $E$ . Since the angles around point $C$ form a straight angle, they must be $180^\circ$ . Angle $ACB$ is $34^\circ$ since $180-94-52=34$ . Since angle $ACB$ is congruent to angle $CBE$ , angle $CBE$ is also $34^\circ$ .
Triangle $DAC$ . Translate left by the directed line segment from point C to point D.

Acceptable errors: An argument in part c that relies on congruent pairs of angles named in part a without restating the congruence in part c.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Explanations in parts b and c appeal to the diagram and are logically sequenced but do not appeal specifically to transformations; good, complete explanations for parts b and c with incorrect angles identified in part b; no mention that the rotations are $180^\circ$ in part b; one incorrect angle pair in part a; work for part c mentions that the sum of three angle measures is $180^\circ$ but does not justify this by saying they are in a triangle or a straight angle.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Work for part a does not appeal to rotations; response to part c does not mention congruent angle pairs (see note under Tier 1 response); response to part c does not mention angles adding to $180^\circ$ ; two incorrect angle pairs in part a without excellent parts b and c; three or more error types under Tier 2 response.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Little progress on any of the problem parts; justification with many errors or no justification for parts b and c.

Teaching Notes

Students understand that a $180^\circ$ rotation using a point not on line $L$ takes it to a line parallel to $L$ . Using this knowledge and a construction that students have seen, students argue that the sum of the interior angles of a quadrilateral is $360^\circ$ .