Download Discovering Congruent Triangles Activity

Discovering Congruent Triangles Activity
Name: __________________________
Names of group members: __________________, __________________, __________________
Objective: Understanding congruent triangle postulates and theorems using inductive reasoning.
MAIN QUESTION: What do we need to know to prove two triangles are congruent?
Materials needed: sticks (following lengths: 2 at 8 cm, 2 at 11cm and 2 at 5 cm), protractor, ruler, and paper.
We will be using Exploragons (pre-made sticks) that snap together with a protractor that they will snap to.
We will need to adjust some of the instructions since we have these tools (toys).
We will use color for the lengths of the sticks instead of referring to the length in cm.
*Part 1 (Side-Side-Side, SSS)
1. Put 3 sticks of different lengths together to form a triangle as shown.
11 cm
5 cm
8 cm
cm
2. Form another triangle with the other set of sticks.
11 cm
5 cm
8 cm
cm
3. Measure the angles of both triangles using a protractor. (Tracing the shapes may make this easier).
Questions:
1. What are the measures of the 3 angles in the first triangle?
2. What are the measures of the 3 angles in the second triangle?
3. What is the relationship between the angles of each triangle?
4. Are the triangles congruent? ________ Why?
5. Can the sticks be rearranged to form a triangle with different angles?
Conclusion:
What can we conclude about triangles congruency when all sides (SSS) are congruent?
*Part 2 (Side-Angle-Side, SAS)
1. Take 2 of the sticks, place them on a piece of paper OR protractor, and form a 60o angle between them.
11 cm
60°
5 cm
2. Take the 2 sticks of the same length and also for a 60o angle between them.
11 cm
60°
5 cm
3. Draw a line to represent the 3rd side. Repeat the process for the 2nd triangle.
11 cm
11 cm
60°
5 cm
60°
5 cm
4. Measure the length of the 3rd side and the two remaining angles for each triangle.
Questions:
1. What is the length of the 3rd side for each triangle?
2. What are the measures of the remaining angles?
3. Are the two triangles congruent?
4. Use any two sticks and any angle of your choice. Do you get the same result?
Will you always get the same result?
Conclusion:
What can we conclude about triangles congruence when two side and the included angle (SAS) are congruent?
*Part 3 (Angle-Side-Angle, ASA)
1. Measure three angles measuring 80, 60, and 40 degrees on the corners of 2 pieces of construction paper
or cardstock, cut them out, and label them.
80°
40°
60°
2. On a piece of paper, take one of the sticks, and place two of the cut-out angles on each end as shown.
Repeat the process for the 2nd triangle.
60°
60°
40°
40°
11 cm
11 cm
3. Using a ruler, draw a segment along each of the angle. The two segments should intersect forming the
last angle. Repeat the process for the 2nd triangle.
60°
40°
11 cm
60°
40°
11 cm
4. Measure the 3rd angle and the lengths of the 2 sides in each triangle.
Questions:
1. What is the measure of the 3rd angle for each triangle?
2. What are the measures of the remaining 2 sides for each triangle?
3. Are the triangles congruent? __________ Why?
4. What if you used the 5cm straw in the middle? The 8cm straw? A straw with a different length? How
would your results compare?
Conclusion:
What can we conclude about triangle congruence when two angles and the included side (ASA) are congruent?
*Part 4 (Side-Angle-Angle, SAA)
1. Use two of the angles used in the example above.
2. Use one of the sticks and place one of the angles alongside it as shown. Draw a long segment like the
dashed one in the drawing. Repeat the process for the 2nd triangle.
40°
40°
8 cm
8 cm
3. Place the second angle along this segment so that when a 2nd segment is drawn, it will connect with the
end of the straw.
80°
40°
8 cm
80°
40°
8 cm
4. Measure the 3rd angle and the two remaining sides.
Questions:
1. What is the measure of the 3rd angle for each triangle?
2. What are the measures of the remaining 2 sides for each triangle?
3. Are the triangles congruent? _________ Why?
Conclusion:
What can we conclude about triangle congruence when a side and the next two consecutive angles (SAA) are
congruent?
*Part 5 (Side-Side-Angle, SSA)
1. Place two of the sticks together forming an angle of any degree for one triangle, and repeat the process
for the 2nd triangle.
8 cm
8 cm
11 cm
11 cm
2. Use one of the pre-cut angles and place alongside the longer of the sides but not as the included angle.
8 cm
8 cm
40°
40°
11 cm
11 cm
3. Draw a segment to connect the 3rd side to the other two sides.
8 cm
8 cm
40°
40°
11 cm
11 cm
rd
4. Swing the 8cm sticks so that it hits the 3 side at a different spot in the 2nd triangle as in the first.
5. Measure the 3rd side and the remaining 2 angles in each triangle.
Questions
1. What is the measure of the 3rd side for each triangle?
2. What are the measures of the remaining 2 angles for each triangle?
3. Are the two triangles congruent? __________ Why?
4. Do you think that you would get different results if you used a different angle?
*Part 6 (Angle-Angle-Angle, SSS)
1. Place the 3 angles so that they can form a triangle without measuring the sides initially. Draw segments
connecting the angles. Repeat the process for the second triangle.
80°
60°
80° 40°
60°
40°
2. Measure the 3 sides for each triangle.
Questions
1. What are the measures of the 3 sides for each triangle? ________________________________________
2. Could you switch the side lengths for different lengths? _______________________________________
3. Are the two triangles congruent? ________ Why? __________________________________________
Conclusion
What can we conclude about triangle congruence when all angles are congruent? _____________________
Grand Conclusion!
SSS, SAS, AAS, AAA, ASA, SSA


S means that the corresponding sides of the triangles are congruent.
A means that the corresponding angles of the triangles are congruent.
1. Which of the above can be used to prove triangle congruence? ________ ________ ________ ________
2. Draw a pair of triangles with corresponding sides and angles marked to show the above relationships prove
the pair of triangles congruent.
3. Which of the above DO NOT prove triangles congruence?
________
________