Download Geometry Unit 2 Formative Items Part 1

Geometry Items to Support Formative Assessment
Unit 2: Triangles, Proof, and Similarity
Part A: Triangle Relationships, Congruence, and Proof
Prove theorems involving similarity.
G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
G.SRT.B.5 Task
The math club is planning to paint a tangram on the wall to decorate the math hall. They need to
know the area of each of the seven pieces so they plan the materials they need. The tangram is
formed by dissecting a square by connecting various midpoints.
The area of △AEC is 25 square feet and the area of△ AJF is 6.25 square feet. What is the area of
the other five pieces?
Solution:
The area of △DEC is 25 square feet.
The area of △HEI is 6.25 square feet.
The area of △FBG, FHEJ, and DGHI is 12.5 square feet each.
Prove geometric theorems.
G.CO.C.11 Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other,
and conversely, rectangles are parallelograms with congruent diagonals. (Note: Revisit this
Unit 1 standard. Use triangle theorems and postulates to prove these theorems.)
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
G.CO.C.11 Item 1
Given parallelogram ABCD explain how to prove that the opposite sides are congruent .
Possible solution:
Angle ABD is congruent to angle BDC because of alternate interior angles. Angle ADB and angle
DBC are congruent because of alternate interior angles. Line segment DB is reflexive. △DAB is
congruent to △ BCD because of ASA. Segment AB is congruent to segment DC and segment BC
is congruent to segment DA because corresponding parts of congruent triangles are congruent.
Therefore segment AB is congruent to segment DC and segment BC is congruent to segment DA.
G.CO.C.11 Item 2
In the figure below, ÐPQR is congruent to ÐPSR and QPS is congruent to ÐQRS . Prove
that PQRS is a parallelogram.

Source: http://blog.harbormagic.com/breakfast-with-the-dolphins-at-the-baltimore-aquarium/
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Solution:
G.CO.C.11 Item 3
In math today, the teacher showed the picture of the National Aquarium in Baltimore below and
told the class that in the diagram segment QR is both parallel and congruent to segment SP. She
wanted to know if that was enough information to prove that PQRS is a parallelogram.
Source: http://blog.harbormagic.com/breakfast-with-the-dolphins-at-the-baltimore-aquarium/
Darius provided the following as his proof.
Since segment QR is parallel to segment SP, angle QPR is congruent to angle PRS because they
are alternate interior angles. Segment PR is congruent to segment PR by the reflexive property
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
and we were given that segment QR is congruent to segment SP. Triangle QPR is congruent to
triangle PRS by SAS. Angle QRP is congruent to angle RPS because corresponding parts of
congruent figures are congruent so segment PQ is parallel to segment RS. PQRS is a
parallelogram because both pairs of opposite sides are parallel
Do you agree with Darius’ reasoning? Explain your answer.
Solution:
The student should indicate that Darius’ approach was correct but that he has the angles wrong.
Since segment QR is parallel to segment SP, angle QRP is congruent to angle RPS because they
are alternate interior angles.
Angle QPR is congruent to angle RPS because corresponding parts of congruent figures are
congruent so segment PQ is parallel to segment RS.
G.CO.C.11 Item 4
Luis wants to make a picture frame as a gift for his mother. He carefully cut the wood for the
sides so that they are congruent and cut the top piece congruent to the bottom. Luis knows that
when he puts the pieces together he will have a parallelogram, but he wants to make sure it is a
rectangle. He asked his friend Carlos for help.
Carlos told him that as long as he makes sure that the diagonals are congruent, he will have a
rectangle.
Is Carlos correct? Explain your answer.
Solution:
Yes, Carlos is correct. If the diagonals are congruent the two overlapping triangles formed by the
sides, the diagonals, and the bottom (or top) will be congruent by SSS congruence. The angles
formed by each side and the bottom (or top) must be congruent because they are corresponding
parts of congruent figures, but they must also be supplementary since the frame is a
parallelogram. If the angles are congruent and supplementary then they must be right angles
making the frame a rectangle.
G.CO.C.11 Item 5
The puzzle shown above is made from small square pieces of different colors. Can you prove
that the shape of this figure is a parallelogram? Explain why or why not.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Possible solution
By counting the squares the top and bottom are congruent and the sides are congruent. If you
draw a diagonal it would be congruent to itself by the reflexive property. You would have two
congruent triangles by SSS congruence. Two of the pairs of angles that are congruent because
they are corresponding parts of congruent figures can be used to prove that the opposite sides are
parallel.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.