L9 Congruent Triangles Handouts - Weeblymrchowmath.weebly.com/uploads/2/6/3/9/26392168/l9... ·...

L9 – Congruent Triangles Name __________________________________ 9.1 Introduction Per _______ Date _______________________

______________________________________________________________________________ Geometry Q2: L9 - Congruent Triangles Handouts

How Do We Compare?

Using patty paper, compare the lengths and angles of the following triangle pairs. Record what is the same for each pair and what is different.

1. What is common?

What is different? Is there a rigid motion that shows they are congruent?

2. What is common?


L9 – Congruent Triangles Name __________________________________ 9.1 Introduction Per _______ Date _______________________


3. What is common?


4. Based upon your results, what are some conjectures about when triangles are congruent?

L9 – Congruent Triangles Name __________________________________ 9.2 Determining Congruence Per _______ Date _______________________


Grandma’s Garden Boxes Grandma has been watching the garden channel again. She fell in love with some triangular shaped raised garden boxes. She asked Uncle Bobby to build them, but Uncle Bobby did not want to measure all three sides and all three angles of all the triangles. While he was complaining to you, you mentioned that you heard a rumor that in order to be sure that two triangles are congruent, you only need to measure three pieces of information. You just couldn’t remember what three pieces. Let’s investigate in the next activity. When you are finished with the activity, write a note to Uncle Bobby explaining what three measurements he would need to make sure the triangles are congruent. Note to Uncle Bobby:



WHAT DOES IT TAKE TO BE THE SAME?

Scenario 1: Side-Side-Side (SSS) In this scenario you will explore if having three sides of one triangle congruent to three sides of another triangle guarantees that the two triangles are congruent. 1. Draw a scalene triangle on a sheet of tissue paper. 2. Using three other pieces of tissue paper, trace each of the sides of the triangle onto a separate

piece of paper. Mark the ends of each segment to make them easier to see.

3. Slide the three pieces together to make a triangle and copy the new triangle onto another

piece of tissue paper.

4. Is your new triangle congruent to the original? Explain why or why not. 5. Can you rearrange the pieces to create a new triangle that is not congruent to the original?

Explain why the two triangles must be congruent, or why not. 6. State your Claim:

If three sides of one triangle are congruent to three sides of another triangle, then ______________________________________________________________________.(#THM)



Scenario 2: Angle-Angle-Angle (AAA) In this scenario you will explore if having three angles of one triangle congruent to three angles of another triangle guarantees that the two triangles are congruent. 1. Draw a scalene triangle on a sheet of tissue paper. 2. Using three other pieces of tissue paper, trace each of the angles of the triangle onto a

separate piece of paper. Extend the rays of the angles. 3. Slide the three pieces together to make a new triangle and copy the new triangle onto another

piece of tissue paper. Recall that a ray has no end, hence you will only be using a portion of each ray as a side.

4. Is your new triangle congruent to the original? Explain why or why not.

5. Can you rearrange the pieces to create a new triangle that is not congruent to the original?

Explain why the two triangles must be congruent, or why not. 6. State your Claim:

If three angles of one triangle are congruent to three angles of another triangle, then __________________________________________________________________



Scenario 3: Side-Side-Angle (SSA) In this scenario you will explore if having two concurrent sides and the angle adjacent to the second side of one triangle congruent to two concurrent sides and the angle adjacent to the second side of another triangle guarantees that the two triangles are congruent. 1. Draw a scalene triangle on a sheet of tissue paper. 2. Using three other pieces of tissue paper, trace two concurrent sides and the angle adjacent to

the second side (i.e. opposite the first side) of one triangle onto a separate piece of paper. Mark the ends of each segment to make them easier to see, making sure you keep track of which side was the first side, and extend the rays of the angles.

3. Slide the three pieces together to make a new triangle, making sure the angle is still opposite

the first side, and copy the new triangle onto another piece of tissue paper. Recall that a ray has no end, hence you will only be using a portion of each ray as a side.


5. Can you rearrange the pieces to create a new triangle that is not congruent to the original,

where the angle is still opposite the first side? Explain why the two triangles must be congruent, or why not.

6. State your Claim:

If two concurrent sides and the angle adjacent to the second side of one triangle are congruent to two concurrent sides and the angle adjacent to the second side of another triangle, then



Scenario 4: Side-Angle-Side (SAS) In this scenario you will explore if having two sides and the angle between them of one triangle congruent to two sides and the angle between them of another triangle guarantees that the two triangles are congruent. 1. Draw a scalene triangle on a sheet of tissue paper. 2. Using three other pieces of tissue paper, trace two sides and the angle between them of the

triangle onto a separate piece of paper. Mark the ends of each segment to make them easier to see and extend the rays of the angles. Recall that a ray has no end, hence you will only be using a portion of each ray as a side.

3. Slide the three pieces together to make a new triangle, making sure the angle is still between

the two sides, and copy the new triangle onto another piece of tissue paper. Recall that a ray has no end, hence you will only be using a portion of each ray as a side.



where the angle is still between the two sides? Explain why the two triangles must be congruent, or why not.


If two sides and the angle between them of one triangle are congruent to two sides and the angle between them of another triangle, then ___________________________________________________________________. (#THM)



Scenario 5: Side-Angle-Angle (SAA) In this scenario you will explore if having two angles and the side not between them of one triangle congruent to two angles and one of the sides not between them, of another triangle guarantees that the two triangles are congruent. 1. Draw a scalene triangle on a sheet of tissue paper. 2. Using three other pieces of tissue paper, trace two angles and one of the sides not between

them of the triangle onto a separate piece of paper. Mark the ends of each segment to make them easier to see and extend the rays of the angles.

3. Slide the three pieces together to make a new triangle, making sure the side is still not

between the two angles, and copy the new triangle onto another piece of tissue paper. Recall that a ray has no end, hence you will only be using a portion of each ray as a side.



making sure the side is still not between the two angles? Explain why the two triangles must be congruent, or why not.


If two angles and the side not between them of one triangle are congruent to two angles and the side not between them of another triangle, then

___________________________________________________________________.(#THM)



Scenario 6: Angle-Side-Angle (ASA) In this scenario you will explore if having two angles and the side between them of one triangle congruent to two angles and the side between them of another triangle guarantees that the two triangles are congruent. 1. Draw a scalene triangle on a sheet of tissue paper. 2. Using three other pieces of tissue paper, trace two angles and the side between them of the

triangle onto a separate piece of paper. Mark the ends of each segment to make them easier to see and extend the rays of the angles.

3. Slide the three pieces together to make a new triangle, making sure the side is still between

the two angles, and copy the new triangle onto another piece of tissue paper. Recall that a ray has no end, hence you will only be using a portion of each ray as a side.



making sure the side is still between the two angles? Explain why the two triangles must be congruent, or why not.


If two angles and the side between them of one triangle are congruent to two angles and the side between them of another triangle, then ___________________________________________________________________.(#THM)



Summary Complete the following. Use this sheet as a summary for your class and homework. 1. List the four Congruence Theorems here. Write the acronym and then describe what that acronym means. Be specific and clear when describing an angle or side. 2. By definition, congruent triangles have ______________________________ and _____________________________________. 3. Thus, we can say “Corresponding parts of ____________________ triangles are _______________________.” (#THM) We use this statement very often in geometry. When we use it, we use an acronym, CPCTC. Congruence Statement If ∆𝐴𝐵𝐶 is congruent to ∆𝐷𝐸𝐹, then we write ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹. So, if ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 then complete the following: ∠𝐸 ≅ ______ 𝐴𝐵 ≅______ ∠𝐶 ≅______ 𝐸𝐹 ≅______ ∠𝐷 ≅______ 𝐴𝐶 ≅______ Now go back to Grandma’s Garden Boxes and write your note to Uncle Bobby.

L9 – Congruent Triangles Name __________________________________ 9.3 Identifying Congruent Triangles Per _______ Date _______________________


Are We Identical Twins?

Which of the following pairs of triangles are congruent? Explain which criteria for triangle congruence you used to determine your answer.

1. Are the triangles congruent? Explain why or why not.




L9 – Congruent Triangles Name __________________________________ 9.3 Identifying Congruent Triangles Per _______ Date _______________________






L9 – Congruent Triangles Name __________________________________ 9.4 Homework 1 Per _______ Date _______________________


Congruent Triangles Homework There are four pairs of congruent triangles. State how you know they are congruent using the measurements of the sides and/or angles. You have four methods to use to prove they are congruent: SSS, SAS, ASA, and AAS. Once you use one method, you may not use it again. Thus, you must use a different method for each pair. Label each triangle and write a congruence statement for each. 1. Congruence Statement ______________________ Reason _______________ 2. Congruence Statement ______________________ Reason _______________



3. Congruence Statement ______________________ Reason _______________

4. Congruence Statement ______________________ Reason _______________



B

CD

G

H

F𝑥 + 5

2𝑥 − 3

11.5 𝑥 + 3

CPCTC means _________________________________________________________ ______________________________________________________________________ Using the congruent statement, solve for x. Then, give the length of the sides for each triangle. ∆𝐵𝐶𝐷 ≅ ∆𝐹𝐻𝐺 𝐵𝐶 =_________ 𝐹𝐻 =_________ 𝐵𝐷 =_________ 𝐹𝐺 =_________ 𝐷𝐶 =_________ 𝐺𝐻 =_________ Using the above triangles, we are given that 𝑚∠𝐷 = (4𝑦 + 12)° and 𝑚∠𝐶 = (5𝑦 − 8)° and 𝑚∠𝐹 = (6𝑦 − 34)°. Solve for y and use y to find all the angle measures. 𝑦 = ___________ 𝑚∠𝐷 = ___________° 𝑚∠𝐹 = ___________° 𝑚∠𝐶 = ___________° 𝑚∠𝐻 = ___________° 𝑚∠𝐵 = ___________° 𝑚∠𝐺 = ___________°

L9 – Congruent Triangles Name __________________________________ 9.5 Proving Congruent Triangles Per _______ Date _______________________


Now we will use SSS, SAS, ASA, AAS and CPCTC to prove statements involving congruent triangles. 1. Given: A is the midpoint of CE

A is the midpoint of BD Prove: ΔBCA ≅ ΔDEA What transformation could take ΔBCA onto ΔDEA ? __________________________

Statement Reason

L9 – Congruent Triangles Name __________________________________ 9.5 Proving Congruent Triangles Per _______ Date _______________________


2. Given: GH ! JI ∠G ≅ ∠I

Prove: GJ ≅ IH What transformation could take ΔJGH onto ΔHIJ ? __________________________

Statement Reason



Prove the following statements. 1. Given: OR and SP bisect each other Prove: ΔONP ≅ ΔRNS

Statement Reason

2. Given: ∠JKM ≅ ∠LKM

∠JMK ≅ ∠LMK Prove: JM ≅ LM

Statement Reason



3. Given: R is the midpoint of QS

QT ≅ ST Prove: ΔQRT ≅ ΔSRT

Statement Reason

4. Given: ∠U and ∠X are right angles.

∠W ≅ ∠Y UW ≅ YX

Prove: ∠V ≅ ∠Z

Statement Reason

L9 Congruent Triangles Handouts - Weeblymrchowmath.weebly.com/uploads/2/6/3/9/26392168/l9... ·...

Documents

Transcript of L9 Congruent Triangles Handouts - Weeblymrchowmath.weebly.com/uploads/2/6/3/9/26392168/l9... ·...