Problem 2.3 Congruent Triangles II
Focus Question: What is the smallest number of side and/or angle measurements needed to conclude that two triangles are congruent??
A. Can you be sure that two triangles are congruent if you have information about one pair of corresponding sides or corresponding angles? Let’s see.
1. Draw a triangle with one side that measures 5cm long. Compare your triangle with other students in your group. Are the triangles congruent?
2. Draw a triangle with one angle that measures 50°. Compare your triangle with other students in your group. Are the triangles congruent?
B. Can you be sure that two triangles are congruent if you have information about two pairs of corresponding sides or corresponding angles? Let’s see.
1. The triangle below has sides of 4cm and 6cm. Can you draw another triangle with sides measuring 4cm and 6cm long that is NOT congruent to the triangle below?
2. The triangle below has angles of 90° and 20°. Can you draw another triangle with angles that measure 90° and 20° that is NOT congruent to the triangle below?
3. The triangle below has an angle measuring 60° and a side that is 5cm long. Can you draw
another triangle with an angle that measures 60° and a side of 5cm that is NOT congruent
to the triangle below?
C. Can you be sure that two triangles are congruent if you have information about three pairs of corresponding sides or corresponding angles? Let’s see.
1. The triangles below have two pair of congruent corresponding angles and one pair of congruent corresponding sides as shown. Are they congruent? ______
If yes, what transformations can you use to show that the triangles are congruent?
2. The triangles below have two pair of congruent corresponding sides and one pair of congruent corresponding angles as shown. Are they congruent? ______
If yes, what transformations can you use to show that the triangles are congruent?
D. Can you be sure that two triangles are congruent if you have information about any three pairs of corresponding sides or corresponding angles? Let’s see.
1. The triangles below have three pairs of congruent corresponding angles as shown. Are they congruent? ______If yes, what transformations can you use to show that the triangles are congruent?
2. What combinations of three congruent corresponding parts will guarantee two triangles are congruent?
There are 4 ways to prove that two triangles are congruent without knowing the measures of all the angles and all the sides.
SSS / If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
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SAS / If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. (The included angle is the angle formed by the sides being used.) /
ASA / If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. (The included side is the side between the angles being used. It is the side where the rays of the angles would overlap.) /
AAS / If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. (The non-included side can be either of the two sides that are not between the two angles being used.) /
The information below is NOT enough to prove that triangles are congruent.
AAA / SSAButterflies, Pinwheels, and Wallpaper Investigation 2: Transformations and Congruence