cferreras.weebly.com · Web viewIn #4, we were given 2 pairs of sides and an included angle of one triangle are congruent to 2 pairs of sides and an included angle of another triangle

Honors Math 2 Unit 5 Class Packet Sanderson High SchoolDay 1: CPCTC

Warm-Up:What can you conclude about two triangles that are congruent?

When you know that two triangles are congruent, you can make conclusions about the sides and angles of the triangles.

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Honors Math 2 Unit 5 Class Packet Sanderson High School

2.

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CPCTC and Naming Congruent Triangles3

Honors Math 2 Unit 5 Class Packet Sanderson High SchoolI. Draw and label a diagram. Then solve for the variable and the missing measure or length.

1. If ∆ BAT ≅ ∆ DOG, and m∠B=14 ,m∠G=29 ,∧m∠O=10x+7. Find the value of x m∠O.

x = ___________

m∠O= _________

2. If ∆ COW ≅ ∆ PIG, and CO=25 ,CW=18 , IG=23 ,∧PG=7 x−17. Find the value of x and PG.

x = ___________

PG=___________

3. If ∆≝≅ ∆PQR andDE=3 x−10 ,QR=4 x−23 ,∧PQ=2 x+7. Find the value of x and EF.

x = ___________

EF = __________

II. Use the given information and triangle congruence statement to complete the following.

1. ∆ ABC ≅∆GEO, AB = 4, BC = 6, and AC = 8. What is the length of GO? How do you know?

2. ∆ BAD≅ ∆ LUK ,m∠D=52° ,m∠B=48° ,∧m∠ A=80° .

a. What is the largest angle of ∆ LUK ?

b. What is the smallest angle of ∆ LUK ?

3. ∆ SUN ≅ ∆ HOT . ∆ SUN is isosceles. Is there enough information to determine if ∆ HOT is isosceles? Explain why or why not.

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Honors Math 2 Unit 5 Class Packet Sanderson High SchoolDay 2: Proving Triangle Congruence

1. What does it mean to say two triangles are congruent?

2. List the ways to justify that triangles are similar.

3. Examine the triangles with all side lengths labeled.

a. How do we justify that the two triangles are similar? _________

b. Complete the similarity statement: ∆ ABC ∆¿.

c. What is the scale factor? _______ : _______

d. What do we know about the corresponding angles of similar triangles?

e. What does this tell us about the pair of triangles?

4. Examine the triangles with two sides lengths and an included angle labeled.


b. Complete the similarity statement: ∆ PIG ∆¿.


d. Since the triangles are similar, what do we know about ∠P and its corresponding angle?

What do we know about ∠ I and its corresponding angle?

e. Use the scale factor you gave in part c to determine the length of OH .

f. What does this tell us about the pair of triangles?

5. Examine the triangles with two angle pairs marked congruent.5

3 cm2 cm

4 cm4 cm

3 cm2 cm

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4 cm

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4 cm

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b. Complete the similarity statement: ∆ KLM ∆¿.


d. Since the triangles are similar, what do we know about ∠K and ∠Y ?

e. Use the scale factor you gave in part c to determine the lengths of KL and YZ.

f. What does this tell us about the pair of triangles?

6. Think back to the three situations we examined. In #3, we were given 3 pairs of sides of one triangle are congruent to 3 pairs of sides of another triangle. We

were able to conclude that the triangles are ____________________.

In #4, we were given 2 pairs of sides and an included angle of one triangle are congruent to 2 pairs of sides and

an included angle of another triangle. We were able to conclude that the triangles are

_________________________.

In #5, we were given 2 pairs of angles and an included side of one triangle are congruent to 2 pairs of angles and

an included side of another triangle. We were able to conclude that the triangles are

_______________________.

This investigation illustrates 3 of the shortcuts to prove that triangles are ___________________________.

Those shortcuts are _______________, _______________, and _______________.

7. What if different parts of the 2 triangles are given congruent? Suppose rather than being given 2 pairs of angles and

an included side of one triangle congruent to the corresponding parts of the 2nd triangle as in #5, we were instead

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52 52

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4 cm

3.2 cm 2.15 cm

4 cm L M

K Y

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k'60 60

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Honors Math 2 Unit 5 Class Packet Sanderson High Schoolgiven 2 pairs of angles and a non-included side of triangle congruent to 2 pairs of angles and a non-included side of

another triangle. Could the triangles still be proven congruent? Examine the diagram below where this is the given

information.

a. What is m∠M ? ____________

b. What is m∠Z? ____________

c. Do you now have enough information to prove the triangles congruent by one of the shortcut methods from

earlier in this investigation? _____________

If so, which one? ________________

8. Since we have now seen that SSS, SAS, ASA, and AAS are shortcuts to proving that triangles are congruent (without

needing all 6 pairs of corresponding parts congruent to prove the triangles congruent by definition), one might

wonder if there are additional shortcuts that will work. Students often wonder if AAA will work since SSS does or if

SSA will work since AAS does.

a. Let’s start by thinking about whether AAA would work.

Using the diagram at the right, you can see that 3 angles

of ∆ ABC are congruent to 3 angles of ∆≝¿.

These triangles are drawn to scale. (They really are equilateral triangles.)

Are their corresponding sides congruent? ____________

Are the triangles congruent? ____________

Is AAA an appropriate shortcut to prove triangles congruent? ______________

(If it were, all equilateral triangles would be congruent to one another, and we know that isn’t the case!)

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CD

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CB

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Honors Math 2 Unit 5 Class Packet Sanderson High Schoolb. Now let’s examine whether SSA is sufficient to guarantee congruent triangles. The 3 diagrams below show

∆ ABC and ∆ ADC drawn to scale. The first diagram is the original figure and shows that the triangles are

overlapping. In the 2nd and 3rd diagrams, they are separated out so it is easier to see the parts that are marked

congruent on each separate triangle.

Notice that two sides and a non-included angle of one of the triangles (∆ ABC) are congruent to two sides and a

non-included angle of the other triangle (∆ ADC). That means that the given congruent parts are a SSA

situation.

i. What do you know about ∆ ABC and ∆ ADC from the first diagram?

ii. Based on your answer to part b, is it possible for ∆ ABC and ∆ ADC to be congruent? __________

iii. Therefore, does SSA guarantee congruent triangles? __________

iv. If you need more convincing, use patty paper to trace ∆ ABC and ∆ ADC from their separate diagrams

above and compare to those triangles in the 1st overlapping diagram. This should make it clear that we

should never use SSA to try to prove congruent triangles. This does not mean that when given these parts,

triangle cannot be congruent. Certainly if we had congruent triangles, we could correctly mark 2 sides and a

non-included angle of one triangle to the corresponding parts of the other triangle. It does mean, however,

that if we only have those parts available, we cannot conclude definitively that the triangles are congruent.

They might be, but they might not be. You will study the various possibilities when these corresponding

parts are known to be congruent in more depth in a later course.

9. You have now examined 6 different situations in this investigation. Before investigating one last situation, think (or

look) back. Instead of needing all 6 pairs of corresponding parts congruent to prove triangles congruent by

definition, what are 4 valid shortcuts for proving triangles congruent?

_______________, _______________, _______________, and _______________.

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Honors Math 2 Unit 5 Class Packet Sanderson High School10. In #8, we established that SSA does not guarantee congruent triangles. Look at when 2 right triangles are used in a

similar situation where the two pairs of congruent sides are a pair of legs and the hypotenuses and the congruent angle pair is the right angles as shown in the diagram.

a. What theorem do you know that relates the 3 sides of a right triangle?

b. Use that theorem to write an equation relating the sides of ∆ PKM . Then solve that equation for the length of side p.

c. Use that same theorem to write an equation relating the sides of ∆ XYZ. Do not solve it!

d. Based on the given information in the diagram, what can you substitute in place of y in the equation above? What can you substitute in place of z? Rewrite your equation from part c above using these substitutions.

e. Now solve the equation in part d above for the length of side x.

f. How do the lengths of side p and side x relate to one another?

Use this idea to appropriately mark the sides in the diagram above.

g. Based on the markings in the diagram above, what shortcut from earlier in this investigation can you use to prove that ∆ PKM and ∆ XYZ are congruent?

h. Since we can use the Pythagorean Theorem to find the 3rd side of a right triangle when 2 sides are known, we were able to determine that the 2 triangles above are congruent. This illustrates why a fifth shortcut to prove triangles congruent exists. It is called HL, which stands for Hypotenuse-Leg Theorem. It states that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent. Remember that only right triangles can be proven using this theorem!

11. List the 5 shortcuts you have explored that can be used to prove triangles congruent:

_______________, _______________, _______________, _______________, and _______________

Triangle Congruence Picture Questions9


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Day 3: Intro to Triangle Congruence Proof11


Ex 1) Given: AD≅ DC AC⊥BD Prove: ΔABD ≅ ΔCBD

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J K

Honors Math 2 Unit 5 Class Packet Sanderson High School Given Given Reflexive Prop ≅

Ex 2) Given: <E ≅ <H G isthe midpoint of EH Prove: ΔGFE ≅ ΔGIH

Ex 3) Given: JK /¿ ML JK≅ ML

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G

F

E

H

I

G isthe midpoint of EH<FGE ≅ <IGH

J K

DA C

B

Honors Math 2 Unit 5 Class Packet Sanderson High School Prove: ∡J ≅∡L

Reflexive Prop ≅

Ex 4) Given: AB≅ BC AC⊥BD Prove: ΔABD ≅ ΔCBD

Reflexive prop ≅

Def of right Δ

Use separate paper (if needed) to complete the following.

Ex 5) Given: AB /¿CD AB≅ CD Prove: AE≅ EC

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JK /¿ ML JK≅ ML

Δ ≅Δ

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M P

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Ex 6) Given: IG bisects<FIJ IF≅ IH Prove: ∡F ≅ ∡H

Ex 7) Given: KL /¿JM KJ /¿ LM Prove: KJ ≅ LM

Day 4: Practice with Triangle CongruenceEx 1) Given: MQ≅ PR , <M and <P are right angles. N is the midpoint of MP Prove: <MQN ≅ <PRN MQ≅ PR M and <P are right angles N is the midpoint of MP

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I

F HG

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JM

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D C

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F

IH

GK

J

<MQN ≅

AD≅ BC


MN ≅ NP

Δ_______________ ≅ Δ ________________

Ex 2) Given: AD≅ BC <ADC ≅ < BCD Prove: AC ≅ BD

Reflexive Prop ≅ <ADC ≅ < BCD

Δ_______________ ≅ Δ ________________

Ex 3) Given: <I ≅ <G <1 ≅ <2 JI ≅ KG

Prove: <1 ≅ < 2 Given

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Honors Math 2 Unit 5 Class Packet Sanderson High School <I ≅ <G JI ≅ KG

Δ_______________ ≅ Δ ________________

Ex 4) Given: AB /¿CD AB≅ CD <AEB ≅ <DFC Prove: BE ≅ DF

AB /¿CD AB≅ CD <AEB ≅ <DFC

<1 ≅ < 2

Δ_________________ ≅ Δ ________________

BE ≅ DF

Ex 5) Given: PI ≅ DI RI ≅ EI Prove: ∠R≅∠E

Given Given Reflexive Prop ≅17

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CPCTC

Ex 6) Given: KM⊥JL M is the midpoint of JL Prove: ∆ JKM ≅ ∆ LKM

Day 5: Isosceles Triangle Theorem (ITT)

Warm-Up:

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< 1 and < 2 are right angles

21M LJ

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Isosceles Triangle: triangle with _______________________________________________________________________

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Isosceles Triangle Theorem: If 2 sides of a triangle are congruent, then

__________________________________________________________________________________________________.

Isosceles Triangle Theorem Converse: If 2 angles of a triangle are congruent, then

__________________________________________________________________________________________________.

1. If CM ≅ EM , then _________ ≅ _________ by ________________________________________________________.

2. If ∠2≅∠3, then _________ ≅ _________ by

_________________________________________________________.

Isosceles Triangle Practice

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Honors Math 2 Unit 5 Class Packet Sanderson High School1. In triangle ABC, mA = x, mB = x + 10, and mC = 3x + 20. Find the number of degrees in A.

2. In triangle DEF, mE = x + 10, mD = 3x + 30, and mF = 5x + 50. How many degree are there in F?

3. The measure of each base angle of an isosceles triangle is 20o. Find the measure of the vertex angle.

4. Two angles of a triangle are equal in measure and the third angle is 110o. Find the number of degrees in each of the two equal angles.

5. Triangle ABC is an equilateral triangle. Fill in the measures of all the numbered angles.

1: 2: 3:

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Honors Math 2 Unit 5 Class Packet Sanderson High School4: 5:

6. Find the measure of A, if C is a right angle and mABD = 130:

7. In triangle ABC, AB CB and mCBD = 124. Find the measure of A.

8. In triangle ABC, mBCD = 100 and mBAC = 35. Find the measure of B.

9. In isosceles triangle ABC, AB = AC. mC = 6x + 10 and mB = 3x + 40. Find the measure of the exterior angle at the vertex angle A.

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10. Find the value of x:

a.

b.

c.

Day 6: Angle Bisectors and Perpendicular Bisectors

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Theorems Involving Perpendicular Bisectors and Angle Bisectors24

Honors Math 2 Unit 5 Class Packet Sanderson High SchoolInvestigation 1: Using dynamic software (GSP, Geogebra, etc.):

1. Construct a segment. 2. Construct its midpoint.3. Select the segment and the midpoint and construct a perpendicular line.4. Construct a point on the perpendicular line.5. Draw a segment whose endpoints are the point on the perpendicular line and one of the endpoints of the

original segment.6. Draw a segment from the same point on the perpendicular line to the other endpoint of the original

segment.7. Measure the length of segments drawn from the point on the perpendicular line that you created in steps

5 and 6. What do you notice about their lengths?8. Select and drag the point along the perpendicular line. What do you notice about the 2 segment lengths

from step 7 as you drag the point to various places along the perpendicular line?

Investigation 2: Using dynamic software (start a new sketch):

1. Construct an angle by constructing 2 rays with the same endpoint. To construct a ray, click and hold the straight edge (segment) tool. You will have the option to the right to select the ray.

2. Select the angle by selecting the three points in order as you would trace them to draw the angle. (That means the 2nd point you select must be the vertex of the angle.)

3. Construct an angle bisector.4. Construct a point on the angle bisector.5. Select the point and one side of the angle and construct the perpendicular line. 6. Select the angle bisector and the perpendicular line from step 5 and construct the intersection.7. Select the point of intersection constructed in step 6 and the point on the angle bisector you constructed

earlier (in step 4). Measure the distance.8. Repeat steps 5 – 7 using the other side of the angle.9. The two distances you measured is the distance from the point to the side of the angle. (To measure

distance between a point and a line you have to measure the perpendicular distance from the point to the line.)

10. Select and drag the point on the angle bisector. Because of how you constructed it, the point will stay on the angle bisector as you drag it. Notice that the perpendicular line travel with the point so that the distances measured remain the distance between the point and each side. What do you notice about those distances as you drag the point along the angle bisector?

11. Now select one of the first three points in the diagram that were from the 2 rays that formed the angle. It can be the vertex of the angle or one of the other 2 points on the rays from step 1. What do you notice about the distances as you drag one or more of these points?

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DB C

A

Honors Math 2 Unit 5 Class Packet Sanderson High SchoolIt is possible to prove the findings from Investigations 1 and 2. Use your knowledge of congruent triangle proofs to complete the following flow proofs.

1. Given: AD is the⊥bisector of BC

Prove: AB=AC

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AB=AC

∆ ABD≅ ∆ ACD∠ ADB≅∠ ADC

Reflexive prop of ≅

D isthe midpoint of BCAD⊥BC

AD is the⊥bisector of BC

Theorem: If a point is on the perpendicular bisect of a segment,

it is __________________________________ from the _________________________

of the segment.

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Honors Math 2 Unit 5 Class Packet Sanderson High School2. Given: D is onthe bisector of ∠ ABC

DE⊥ BA, DF⊥ BC

Prove: DE=DF

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reflexive prop of ≅

DE⊥ BA

DF⊥ BC

D is onthe bisector of ∠ ABC

Theorem: If a point is on the bisector of an angle, it is

__________________________________ from the _________________________

of the angle.

Honors Math 2 Unit 5 Class Packet Sanderson High SchoolDay 7: Unit Review

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Documents

cferreras.weebly.com · Web viewIn #4, we were given 2 pairs of sides and an included angle of one triangle are congruent to 2 pairs of sides and an included angle of another triangle