secondary ii 2 chapter 5 & 6 1 secondary ii ... complete the graphic organizer by providing an...

Secondary 2 Chapter 5 & 6

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Secondary II

Chapter 5– Congruence Through Transformations

Chapter 6 – Using Congruence Theorems 2015/2016

Date Section Assignment Concept

A: 10/12 B: 10/13

5.1-5.6 - Worksheet 5.1/5.2 Congruent Triangles SSS, SAS, ASA, AAS Congruence Theorems

A: 10/14

5.7 - Worksheet Proofs Using Congruent Triangles

10/15-16 Fall Break

B: 10/19 5.7 - Worksheet Proofs Using Congruent Triangles

A: 10/20 B: 10/21

6.1 & 6.2 - Worksheet 6.1 & 6.2 Right Triangle Congruence Theorems Corresponding Parts of Congruent Triangles are Congruent

A: 10/22 B: 10/23

6.3 & 6.4 - Worksheet 6.3 & 6.4 Isosceles Triangle Theorems Inverse, contrapositive, Direct Proof, and Indirect Proof

A: 10/26 B: 10/27

Review

A: 10/28 B: 10/29

TEST - After Chapter 5/6 Worksheet (End of 1st Term)

Late and absent work will be due on the day of the review (absences must be excused). The review assignment must be

turned in on test day. All required work must be complete to get the curve on the test.

Remember, you are still required to take the test on the scheduled day even if you miss the review, so come prepared. If

you are absent on test day, you will be required to take the test in class the day you return. You will not receive the curve

on the test if you are absent on test day unless you take the test prior to your absence.


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5.1 -5.6 – Congruent Triangles & SSS, SAS, ASA, AAS Congruence Theorems

Example 1: Understanding congruence.

Graph triangle ABC by plotting the points A (8, 10), B (1, 2), and C (8, 2).

a. Classify triangle ABC.

b. Calculate the length of side 𝐴𝐵̅̅ ̅̅ .

c. Translate triangle ABC 10 units to the left to form

triangle DEF. Graph triangle DEF and list the

coordinates of points D, E, and F.

d. Predict the side lengths of triangle DEF.

e. Verify that the side lengths and angles are the same.

Example 2: Statements of Triangle Congruence.

Consider the congruence statement ∆JRB ≅ ∆MNS .

a. Identify the congruent angles. b. Identify the congruent sides.

Chapter 5: Congruence through Transformations


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The Side-Side-Side Congruence Theorem states: “If three sides of one triangle are congruent to the corresponding sides of another triangle, then the triangles are congruent.”

Example 3: Graph triangle ABC by plotting the points A (8, -5), B (4, -12), and C (12, -8).

1. How can you determine the length of each side of this

triangle?

2. Calculate the length of each side of triangle ABC. Record the

measurements in the table.

3. Translate line segments 𝐴𝐵̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅ , 𝑎𝑛𝑑 𝐴𝐶̅̅ ̅̅ up 7 units to form triangle A’B’C’.

4. Calculate the length of each side of triangle A’B’C’. Record the measurements in the table.


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5. Are the corresponding sides of the pre-image and image congruent? Explain your reasoning.

6. Do you need to determine the measures of the angles to verify that the triangles are congruent? Explain

why or why not.

The Side-Angle-Side Congruence Theorem states: “If two sides and the included angle of one triangle are congruent to the corresponding sides and the included angle of the second triangle, then the triangles are congruent.”

Example 4: Use the Side-Angle-Side (SAS) Congruence Theorem and a protractor to determine if the two triangles

drawn on the coordinate plane shown are congruent. Use a protractor to determine the measures of the included

angles.

Example 5: Determine if there is enough information to prove that the two triangles are congruent by SSS or SAS.

Write the congruence statements to justify your reasoning.


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The Angle-Side-Angle Congruence Theorem states: “If two angles and the included side of one triangle are congruent to the corresponding two angles and the included side of another triangle, then the triangles are congruent.”

Example 6: Analyze triangles ABC and DEF.

a) Measure the angles and calculate the side lengths of both triangles.

b) Describe the possible transformation(s) that could have occurred to transform pre-image ABC into image

DEF.

c) Identify two pairs of corresponding angles and a pair of corresponding included sides that could be used to

determine congruence through the ASA Congruence Theorem.

d) Determine if triangles DEF and GHJ are congruent.


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The Angle-Angle-Side Congruence Theorem states: “If two angles and a non-included side of one triangle are congruent to the corresponding angles and the corresponding non-included side of a second triangle, then the triangles are congruent.”

Example 7: Use the previous graph to show that ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 using AAS.

Example 8: Determine if there is enough information to prove that the two triangles are congruent by ASA or

AAS. Write the congruence statements to justify your reasoning.

a) b)


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Example 9: This chapter focused on four methods that you can use to prove that two triangles are congruent.

Complete the graphic organizer by providing an illustration of each theorem.


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Additional Notes


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5.7 – Proofs

Steps to complete a triangle congruence proof:

a. Mark the picture based on the given information.

b. Decide what else you know for a fact is congruent (reflexive, vertical angles, etc.)

c. Decide which theorem to use based on what is congruent (SSS, SAS, ASA, AAS)

d. Fill in the five lines of your proof.

Example 1: Fill in the missing information for the proofs.

a. Given: 𝐴𝐷̅̅ ̅̅ ⊥ 𝐷𝐶̅̅ ̅̅ ; 𝐶𝐵̅̅ ̅̅ ⊥ 𝐴𝐵̅̅ ̅̅ ; ∠𝐴𝐶𝐷 ≅ ∠𝐶𝐴𝐵

Prove: ∆𝐴𝐷𝐶 ≅ ∆𝐶𝐵𝐴


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b. Given: C bisects 𝐵𝐸̅̅ ̅̅ and 𝐴𝐷̅̅ ̅̅

Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐶

c. Given: 𝐺𝐸̅̅ ̅̅ ⊥ 𝐷𝐹̅̅ ̅̅ ; 𝐷𝐺̅̅ ̅̅ ≅ 𝐺𝐹̅̅ ̅̅

Prove: ∆𝐷𝐸𝐺 ≅ ∆𝐹𝐸𝐺


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Additional Notes


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Chapter 6: Using Congruence Theorems

6.1/6.2 – Right Triangle Congruence Theorems & Corresponding Parts of Congruent Triangles are Congruent (Standard: G.CO.10)

List all of the triangle congruence theorems you have explored previously.

The congruence theorems apply to all triangles. There are also theorems that only apply to right triangles. Methods

for proving that two right triangles are congruent are somewhat shorter. You can prove that two right triangles are

congruent using only two measurements. Explain why.

The Hypotenuse-Leg (HL) Congruence Theorem states: “If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent.”

Example 1:

Statement Reason

The Leg-Leg (LL) Congruence Theorem states: “If two legs of one right triangle are congruent to two legs of another right triangle, then the triangles are congruent.”

The Hypotenuse-Angle (HA) Congruence Theorem states: “If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and acute angle of another right triangle, then the triangles are congruent.”

The Leg-Angle (LA) Congruence Theorem states: “If a leg and an acute angle of one right triangle are congruent to a leg and an acute angle of another right triangle, then the triangles are congruent.”


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Determine if there is enough information to prove that the two triangles are congruent. If so, name the

congruence theorem used.

Example 2: If 𝐶𝑆̅̅̅̅ ⊥ 𝑆𝐷̅̅ ̅̅ , 𝑊𝐷̅̅ ̅̅ ̅ ⊥ 𝑆𝐷̅̅ ̅̅ , and P is the midpoint of 𝐶𝑊̅̅ ̅̅ ̅ , is ∆𝐶𝑆𝑃 ≅ ∆𝑊𝐷𝑃?

Example 3: Pat always trips on the third step and she thinks that step may be a

different size. The contractor told her that all the

treads and risers are perpendicular to each other. Is

that enough information to state that the steps

are the same size?

In other words, if 𝑊𝑁̅̅ ̅̅ ̅ ⊥ 𝑁𝑍̅̅ ̅̅ and 𝑍𝐻̅̅ ̅̅ ⊥ 𝐻𝐾̅̅ ̅̅ ,

is ∆𝑊𝑁𝑍 ≅ ∆ 𝑍𝐻𝐾?

Example 4: If 𝐽𝐴̅̅ ̅ ⊥ 𝑀𝑌̅̅̅̅̅ and 𝐽𝑌̅̅ ̅ ≅ 𝐴𝑌̅̅ ̅̅ , is ∆ 𝐽𝑌𝑀 ≅ ∆𝐴𝑌𝑀?


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Example 5: If 𝑆𝑇̅̅̅̅ ⊥ 𝑆𝑅̅̅̅̅ , 𝐴𝑇̅̅̅̅ ⊥ 𝐴𝑅̅̅ ̅̅ , and ∠𝑆𝑇𝑅 ≅ ∠𝐴𝑇𝑅, is ∆𝑆𝑇𝑅 ≅ ∆𝐴𝑇𝑅?

Which triangle congruence theorem is most closely related to the

LL Congruence Theorem? Explain your reasoning.

HA Congruence Theorem? Explain your reasoning.

LA Congruence Theorem? Explain your reasoning.

HL Congruence Theorem? Explain your reasoning .

If two triangles are congruent, then each part of one triangle is congruent to the corresponding part of the other

triangle. “Corresponding parts of congruent triangles are congruent,” abbreviated as CPCTC, is often used as a

reason in proofs. CPCTC states that corresponding angles or sides in two congruent triangles are congruent. This

reason can only be used after you have proven that the triangles are congruent.


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Example 7: Create a proof of the following.

Given: 𝐶𝑊̅̅ ̅̅ ̅ and 𝑆𝐷̅̅̅̅ bisect each other

Prove: 𝐶𝑆̅̅̅̅ ≅ 𝑊𝐷̅̅ ̅̅ ̅

Statement Reason

Example 8: Mark the given information and state the theorem used if you were to write a proof.

Given: 𝑆𝑈̅̅̅̅ ≅ 𝑆𝐾̅̅̅̅ , 𝑆𝑅̅̅̅̅ ≅ 𝑆𝐻̅̅ ̅̅

Prove: ∠𝑈 ≅ ∠𝐾


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CPCTC makes it possible to prove other theorems.

Example 9: How wide is the horse’s pasture?

The Isosceles Triangle Base Angle Theorem states: “If two sides of a

triangle are congruent, then the angles opposite these sides

are congruent.”

The Isosceles Triangle Base Angle Converse Theorem states: “If two angles

of a triangle are congruent, then the sides opposite these angles are

congruent.”


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Example 10: Calculate AP if the perimeter of ∆𝐴𝑌𝑃 is 43 cm.

Example 11: Lighting booms on a Ferris wheel consist of four steel beams that have cabling with light

bulbs attached. These beams, along with three shorter beams, form the edges of three congruent

isosceles triangles, as shown. Maintenance crews are installing new lighting along the four beams.

Calculate the total length of lighting needed.

Example 12: Calculate 𝑚∠𝑇.


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Additional Notes


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6.3/6.4 – Isosceles Triangle Theorems (Standard: G.CO.10)

You will prove theorems related to isosceles triangles. These proofs involve altitudes, perpendicular bisectors,

angle bisectors, and vertex angles. A vertex angle of an isosceles triangle is the angle formed by the two congruent

legs in an isosceles triangle.

The Isosceles Triangle Base Theorem states: “The altitude to the base of an isosceles triangle bisects the base.”

Example 1: Given: Isosceles ∆𝐴𝐵𝐶 with 𝐶𝐴̅̅̅̅ ≅ 𝐶𝐵̅̅ ̅̅ .

a. Construct altitude CD from the vertex angle to the

base.

The Isosceles Triangle Vertex Angle Theorem states: “The altitude to the base of an isosceles triangle bisects the vertex angle.”

Example 2: Label a diagram you can use to help you prove the

Isosceles Triangle Vertex Angle Theorem. State the

“Given” and “Prove” statements.

The Isosceles Triangle Perpendicular Bisector Theorem states: “The altitude from the vertex angle of an isosceles triangle is the perpendicular bisector of the base.”

Example 3: Label a diagram you can use to help you prove the

Isosceles Triangle Perpendicular Bisector Theorem.

State the “Given” and “Prove” statements.


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The Isosceles Triangle Altitude to Congruent Sides Theorem states: “In an isosceles triangle, the altitudes to the congruent sides are congruent.”

Example 4: Label a diagram you can use to help you prove this

theorem. State the “Given” and “Prove”

statements.

The Isosceles Triangle Angle Bisector to Congruent Sides Theorem states: “In an isosceles triangle, the angle bisectors to the congruent sides are congruent.”

Example 5: Draw and label a diagram you can use to help you

prove this theorem. State the “Given” and “Prove”

statements.

Example 6: Solve for the width of the dog house.

𝐶𝐷̅̅ ̅̅ ⊥ 𝐴𝐵̅̅ ̅̅

𝐴𝐶̅̅̅̅ ≅ 𝐵𝐶̅̅ ̅̅

𝐶𝐷 = 12"

𝐴𝐶 = 20"


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The Hinge Theorem states: “If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first pair is larger than the included angle of the second pair, then the third side of the first triangle is longer than the third side of the second triangle.”

Example 8: In the two triangles shown, notice that 𝑅𝑆 = 𝐷𝐸,

𝑆𝑇 = 𝐸𝐹, and ∠𝑆 > ∠𝐸 . The Hinge Theorem says that 𝑅𝑇 > 𝐷𝐹 .

Example 9: In the two triangles shown, notice that

𝑅𝑇 = 𝐷𝐹, 𝑅𝑆 = 𝐷𝐸, and 𝑆𝑇 > 𝐸𝐹. The Hinge Converse Theorem

guarantees that 𝑚∠𝑅 > 𝑚∠𝐷.

Example 10: Matthew and Jeremy’s families are going camping for the weekend. Before

heading out of town, they decide to meet at Al’s Diner for breakfast. During

breakfast, the boys try to decide which family will be further away from the

diner “as the crow flies.” “As the crow flies” is an expression based on the fact

that crows, generally fly straight to the nearest food supply.

Matthew’s family is driving 35 miles due north and taking an exit to travel an

additional 15 miles northeast. Jeremy’s family is driving 35 miles due south and

taking an exit to travel an additional 15 miles southwest. Use the diagram

shown to determine which family is further from the diner.

Explain your reasoning.

The Hinge Converse Theorem states: “If two sides of one triangle are congruent to two sides of another triangle

and the third side of the first triangle is longer than the third side of the second triangle, then the included angle

of the first pair of sides is larger than the included angle of the second pair of sides.”


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Example 11: Which of the following is a possible length for AH: 20 cm, 21 cm, or 24 cm?

Explain your choice.


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Additional Notes

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