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
Understand similarity in terms of similarity transformations.
G.SRT.A.3 Use properties of similarity transformations to establish the AA criterion for two
triangles to be similar.
G.SRT.A.3 Task
(This task is an extension of G.CO.C.10 Item 4, but could be done independently. It is meant to
transition between the Third Angle Theorem and Angle-Angle Similarity by examining
invariance and making conjectures.)
Look at the two triangles below.
1. What do you know about the two triangles? Justify your answer.
2. If you changed one of the congruent angle pairs to different measure (but keeping them
congruent), what would change in your answer to part 1? What would stay the same?
3. If you changed the measures of both congruent angle pairs (but keeping the measures in
each pair congruent) what would change in your answer to part 1? What would stay the
same?
4. Make a conjecture about any two triangles with two pairs of congruent angles. State what
will always be true about them. Can you provide any proofs for your conjecture? If not,
name something about the conjecture that you aren’t sure you can prove.
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Possible Solutions:
Teachers should emphasize the fact that the triangles are similar more than the third angles are
congruent.
1. Students should note that both angles will be 40°. They may be able to justify this answer
using the Triangle Sum Theorem, if it has already been covered. It should also be pointed
out that 3 congruent pairs of angles means the triangles are similar. Students can also
verify that the triangles are similar by measure the sides, which should have a 1:2 ratio.
2. Example: if ∠A and ∠A’ were both changed to 120°, ∠B and ∠ B’ would have to both
become 30°. The triangles are still similar and ∠B and ∠B’ are still congruent, just with
different measurements.
3. Changing both ∠A and ∠A’ and ∠C and ∠C’ means that all three angles will have new
values in both triangles. However, there will still be 3 pairs of congruent angles. The
triangles will still be similar.
4. If two triangles have two pairs of congruent angles, the third pair of angles will also be
congruent and the triangles will always be similar. An informal proof of this conjecture
would reference the Third Angle Theorem (either by name, or indirectly) and show that if
all 3 pairs of angles are congruent, the triangles are similar.
G.SRT.A.3 Formative Item 1
Prove the AA Similarity Theorem. You may find the following diagram helpful.
Sample Solution
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
△ABC ~ △A’B’C’
G.SRT.A.3 Item 2
Segments DE, FG, and BC are parallel. Using AA criterion for similarity, solve for measures of
segments AE, AG, and AC.
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
Solution:
AE = 18
AG = 42
AC = 51
G.SRT.A.3 Item 3
ABCD is a trapezoid with parallel sides AB and CD. Tell why it must be true that △ABE ~
△CDE.
Solution:
Since line segments AB and CD are parallel, diagonal AC forms a transversal that makes
alternate interior angles BAE and DCE congruent. For the same reason, BD forms a transversal
that makes alternate interior angles ABE and CDE congruent. Thus, △ABE ~ △CDE by the AA
Similarity Postulate. [Note: Students can also use only one pair of alternate interior angles plus
the fact that ∠AEB is congruent to ∠CED because they are vertical angles to show AA similarity
for the two triangles.]
G.SRT.A.3 Item 4
Why are all three triangles in the diagram below similar to each other (in other words, why is
△ABC ~ △BDC ~ △ADB)?
Solution:
Two of the triangle pairs (ABC & BDC and ABC & ADB) share a common angle and they each
also have a right angle, so all three of their angle measures must be the same. Therefore these
triangles are similar to each other by the AA Triangle Similarity Postulate. More specifically,
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under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
△ABC ~ △BDC because ∠C is congruent to itself, and ∠ABC is congruent to ∠BDC because
they are both right angles. Similarly, △ABC ~ △ADB because ∠ A is congruent to itself, and
∠ABC is congruent to ∠ ADB because they are both right angles. Then, △BDC ~ △ADB
because ∠ BDC and ∠ADB are both right angles, making them congruent; and ∠CBD is
congruent to ∠BAD since they are corresponding parts of similar triangles ABC and BDC that
we have already proven.
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 Item 1
Duane wants to know if the following pair of triangles is similar? What criteria can he used to
justify his answer?
Possible solution:
No, the triangles are not similar. The sides are not proportional:
G.SRT.B.5 Item 2
Chase wants to create a triangle that is similar to his Baltimore Orioles pennant. He knows the
following measurements: angle ABC is 30°, angle ACB is 75°, segment AB is 32 inches, and AC
is 17 inches. How can he create a similar triangle that is smaller? Explain your reasoning.
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:
Students will be able to have multiple solutions depending on their scale factor. One possible
solution is if they choose a scale factor of ½, the new pennant would have segment AB as 16
inches, segment AC as 8.5 inches, and the angle measures would stay the same. The reason it is
similar is by using SAS. Students may also use SAS as a reason for the triangle being similar.
G.SRT.B.5 Item 3
Based on what you know about congruent and similar triangles, draw three different triangles. In
your drawing provide enough information for a classmate to prove that only two of the triangles
are similar. Students may use patty paper, compass, straightedge, dynamic software and/or other
tools.
Possible solution:
Students answers may vary.
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.