Linear Algebra Wednesday August 27

Answers for homework

Answers for homework Question 2 Check it Out The graph is a straight line at an angle. The line starts at 0,0. The information on the x-axis could be the time the car has traveled. The information on the y-axis could be the distance covered. You might have mentioned that the car is moving at a constant rate.

Learning Target Students will investigate what is the smallest number of side and/or angle measurements needed to conclude that two triangles are congruent.

Connect to Prior Understanding What do we remember about Problem 2.2 Were all the triangles congruent? What did we have to know at a minimum to decide if the triangles were congruent?

2.3 Minimum Measurement Congruent Triangles In Problem 2.2 you might have noticed that it is not necessary to move one triangle onto the other to determine whether two triangles are congruent.

2.3 Minimum Measurement Congruent Triangles p. 35 in your book

Problem 2.3 A-E Consider the conditions described in Questions A-C. For each case, give an argument to support your answer. If the conditions are not enough to determine two triangles are congruent, give a counterexample.

Problem 2.3 A Can you be sure that two triangles are congruent if you know only 1. one pair of congruent corresponding sides? 2. one pair of congruent corresponding angles? Counterexamples

Problem 2.3 B1 B. Can you be sure that two triangles are congruent if you know only: 1. two pairs of congruent sides? 2. two pairs of congruent angles? 3. one pair of congruent corresponding sides and one pair of congruent corresponding angles?

Problem B2 B. Can you be sure that two triangles are congruent if you know only: 1. two pairs of congruent sides? 2. two pairs of congruent angles? 3. one pair of congruent corresponding sides and one pair of congruent corresponding angles?

Problem 2.3 B3 B. Can you be sure that two triangles are congruent if you know only: 1. two pairs of congruent sides? 2. two pairs of congruent angles? 3. one pair of congruent corresponding sides and one pair of congruent corresponding angles?

Problem 2.3 C Can you be sure that two triangles are congruent if you know two pairs of congruent corresponding angles and one pair of congruent corresponding sides as shown? Use your understanding of transformation to justify your answer.

Problem 2.3 C 2. Can you be sure that two triangles are congruent if you know two pairs of congruent corresponding sides and one pair of congruent corresponding angles as shown?

Problem 2.3 D Amy and Becky have different ideas about how to decide whether the condition in Question C, Part (2) are enough to show triangles are congruent. Amy flips triangle GHI as shown. She says you can translate the triangle so that HK and GJ. So all of the measures in triangle GHI match measure in triangle JKL. Do you agree with Amy’s reasoning? Explain. Amy does not give a full explanation. She does not talk about HI and KL and she doesn’t talk about angles at all.

Problem 2.3 D Amy and Becky have different ideas about how to decide whether the condition in Question C, Part (2) are enough to show triangles are congruent. 2. Becky thinks Amy should also explain why the translation matches all the sides and angles. She says that if you translate triangle GHI so that GJ, there will be two parallelograms in the figure. These parallelograms show her which corresponding angles and sides congruent. What parallelogram does she see? How do these parallelograms help identify congruent corresponding sides and angles? GLJI and HKLI are parallelograms. The sides of a parallelogam are parallel and congruent, so HI is congruent to KL and angle H = angle K. Lastly, angle I = 180 – the measure of angle G + angle H AND angle L = 180- angle K + angle L, so they match also.

Problem 2.3 E 1. Can you be sure that two triangles are congruent if you know three pairs of congruent corresponding angles? Explain. Use tracing paper to see if this works. No

Problem 2.3 E 2. Are there any other combinations of three congruent corresonding parts what will guarantee two triangles are congruent? Make sketches to justify your answer. No

Problem 2.3 E 3. Suppose two triangles appear to be NOT congruent. What is the minimum number of measures you should check to show they are NOT congruent? No

Summarize We know that triangles are congruent if we know: All sides are the same SSS (Side, Side, Side) Two sides with an angle in between SAS (Side, Angle, Side Two angles with a side in between ASA (Angle, Side, Angle) Two angles and one side AAS (Angle, Angle, Side)

Rate your understanding Students will investigate what is the smallest number of side and/or angle measurements needed to conclude that two triangles are congruent.

Homework ACE questions starting on page 38 #7-12 and page 3 of Mathematics warm-ups for CCSS, grade 7