5.1 Congruent Triangles - mr. morrison's geometry...

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Geometry Chapter 4 Notes pg. 1 5.1 Congruent Triangles Two figures are congruent if they have the same __________ and the same _____________. Definition of Congruent Triangles ΔABC ≅ΔDEF if and only if Corresponding Angles are congruent: Corresponding Sides are congruent: A ____ ____ B ____ ____ C ____ ____ 1. Write a congruence statement. 2. Given: ∆ ≅ ∆. Name the 6 congruent corresponding parts. 3. Given: ∆ ≅ ∆. Find the value of x. 4. NPLM EFGH. Find the value of each variable. ∠ = 57°, ∠ = 64°, ∠ = (5 + 4)° x = _______ y = _________

Transcript of 5.1 Congruent Triangles - mr. morrison's geometry...

Page 1: 5.1 Congruent Triangles - mr. morrison's geometry …morrisongeometry.weebly.com/.../5/28450119/unit_5_notes.pdfGeometry Chapter 4 Notes pg. 2 Third Angles Theorem If two angles of

Geometry Chapter 4 Notes pg. 1

5.1 Congruent Triangles

Two figures are congruent if they have the same __________ and the same _____________.

Definition of Congruent Triangles

ΔABC ≅ΔDEF if and only if

Corresponding Angles are congruent: Corresponding Sides are congruent:

∠A ≅____ 𝐴𝐵̅̅ ̅̅ ≅____

∠B ≅____ 𝐵𝐶̅̅ ̅̅ ≅____

∠C ≅____ 𝐶𝐴̅̅ ̅̅ ≅____

1. Write a congruence statement. 2. Given: ∆𝑋𝑌𝑍 ≅ ∆𝑅𝑆𝑇. Name the 6 congruent

corresponding parts.

3. Given: ∆𝐵𝐿𝑈 ≅ ∆𝑀𝑂𝑁. Find the value of x. 4. NPLM ≅ EFGH. Find the value of each variable.

𝑚∠𝐿 = 57°, 𝑚∠𝑀 = 64°, 𝑚∠𝑈 = (5𝑥 + 4)° x = _______ y = _________

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Geometry Chapter 4 Notes pg. 2

Third Angles Theorem

If two angles of one triangle are congruent to two angles of a second triangle,

then the third angles are also congruent.

5. Solve for the value of x.

Proving Triangles are Congruent

Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ , 𝐴𝐵̅̅ ̅̅ ∥ 𝐶𝐷̅̅ ̅̅

E is the midpoint of 𝐵𝐶̅̅ ̅̅ 𝑎𝑛𝑑𝐴𝐷̅̅ ̅̅ .

Prove: ∆𝐴𝐸𝐵 ≅ ∆𝐷𝐸𝐶

Statements Reasons

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Geometry Chapter 4 Notes pg. 3

5.2 – 5.4 Proving Triangles are Congruent: SSS, SAS, ASA, AAS, and HL

Warm Up

∆DEF ≅ ∆MNO. Complete the statements.

1. m∠E = m∠ ______ 2. DF = ________

Use the given information to find the value of the variables.

3. ∆ABC ≅ ∆PQR 4. ∆JKL ≅ ∆XYZ

Side-Side-Side (SSS) Congruence Postulate

If three sides of one triangle are congruent to three sides of a second triangle,

then the two triangles are congruent.

Side-Angle-Side (SAS) Congruence Postulate

If two sides and the included angle of one triangle are congruent to two sides

and the included angle of a second triangle, then the two triangles are congruent.

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Geometry Chapter 4 Notes pg. 4

Angle-Side-Angle (ASA) Congruence Postulate

If two angles and the included side of one triangle are congruent to two

angles and the included side of a second triangle, then the two triangles are congruent.

Angle-Angle-Side (AAS) Congruence Postulate

If two angles and a NON-included side of one triangle are congruent to

two angles and the corresponding NON-included side of a second triangle,

then the two triangles are congruent.

Hypotenuse-Leg (HL) Congruence Theorem

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse

and a leg of a second right triangle, then the two triangles are congruent.

Counterexample to show that Angle-Side-Side is not a valid reason to prove triangles are congruent.

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Geometry Chapter 4 Notes pg. 5

Identify which property will prove these triangles congruent.

SSS SAS ASA AAS HL NONE

WARM UP 5.2 – 5.4 DAY 2

For each triangle, name the included angle for the sides given.

1. ∆ABC: sides 𝐴𝐵̅̅ ̅̅ 𝑎𝑛𝑑𝐴𝐶̅̅ ̅̅ 2. ∆DEF: sides 𝐷𝐹̅̅ ̅̅ 𝑎𝑛𝑑𝐸𝐷̅̅ ̅̅

State the congruence postulate or theorem that proves the triangles congruent. Then state the congruence

statement.

3. 4. 5.

6. 7. 8.

T

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Geometry Chapter 4 Notes pg. 6

5.2 – 5.4 Proving Triangles are Congruent: SSS, SAS, ASA, AAS, and HL

Day 2: Proofs

1. Given: 𝐴𝐶̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅ , 𝐶𝐷̅̅ ̅̅ ⊥ 𝐴𝐵̅̅ ̅̅

Prove: ∆𝐴𝐷𝐶 ≅ ∆𝐵𝐷𝐶

Statements Reasons

2. Given: 𝐴𝐷̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅ , 𝐴𝐷̅̅ ̅̅ ∥ 𝐵𝐶̅̅ ̅̅

Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐶𝐷𝐴

Statements Reasons

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Geometry Chapter 4 Notes pg. 7

3. Given: 𝑃𝑄̅̅ ̅̅ bisects ∠SPT

𝑆𝑃̅̅̅̅ ≅ 𝑃𝑇̅̅̅̅

Prove: ∆𝑆𝑃𝑄 ≅ ∆𝑇𝑃𝑄

Statements Reasons

4. Given: 𝐴𝐵̅̅ ̅̅ ⊥ 𝐴𝐷̅̅ ̅̅ , 𝐷𝐸̅̅ ̅̅ ⊥ 𝐴𝐷̅̅ ̅̅

C is the midpoint of 𝐵𝐸̅̅ ̅̅

Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐶

Statements Reasons

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Geometry Chapter 4 Notes pg. 8

5. Given: 𝐴𝐷̅̅ ̅̅ ∥ 𝐶𝐸̅̅ ̅̅ , 𝐵𝐷̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅

Prove: ∆𝐴𝐵𝐷 ≅ ∆𝐸𝐵𝐶

Statements Reasons

6. Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ , 𝐴𝐵̅̅ ̅̅ ∥ 𝐶𝐷̅̅ ̅̅

Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐶𝐷𝐴

Statements Reasons

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Geometry Chapter 4 Notes pg. 9

5.5 Using CPCTC in Triangles

How many triangles can you count Five ways to prove triangles congruent:

in the diagram?? ______________________

______________________

______________________

______________________

______________________

Is it possible to prove that the triangles are congruent? If so, state Given: ∆𝐴𝐵𝐶 ≅ ∆𝑀𝑁𝑃

the postulate or theorem you would use. Explain your reasoning.

Once you have determined that two triangles are congruent,

now you can say that all of the other corresponding parts are also congruent.

1. Given: 𝐴𝐵̅̅ ̅̅ ∥ 𝐶𝐷̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅ ∥ 𝐷𝐴̅̅ ̅̅

Prove: 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅

Statements Reasons

3. ∠𝐵 ≅ _______________

4. 𝑀𝑃̅̅̅̅̅ ≅ _______________

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Geometry Chapter 4 Notes pg. 10

2. Given: A is the midpoint of 𝑀𝑇̅̅̅̅̅

A is the midpoint of 𝑆𝑅̅̅̅̅

Prove: 𝑀𝑆̅̅ ̅̅ ≅ 𝑇𝑅̅̅ ̅̅

Statements Reasons

3. Given: ∠1 ≅ ∠2

∠3 ≅ ∠4

Prove: 𝐵𝐶̅̅ ̅̅ ≅ 𝐷𝐶̅̅ ̅̅

Statements Reasons

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Geometry Chapter 4 Notes pg. 11

4. Given: ∠M ≅ ∠N

∠OKL ≅ ∠OLK

Prove: 𝑀𝐾̅̅ ̅̅ ̅ ≅ 𝑁𝐿̅̅ ̅̅

Statements Reasons

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Geometry Chapter 4 Notes pg. 12

5.5 Day 2: Proofs with Isosceles Triangles & Equilateral Triangles

Warm Up

State which postulate or theorem you can use to prove that the triangles are congruent. Then explain how proving

that the triangles are congruent proves the given statement.

∆𝐿𝑀𝐾 ≅ ∆𝑁𝑀𝐾 because _________________

𝐿𝐾̅̅ ̅̅ ≅ 𝑁𝐾̅̅̅̅̅ because _________________

How many triangles are in the figure??

If an ISOSCELES triangle has exactly two congruent sides, then the congruent sides are the ______________ of

the triangle and the noncongruent side is the _________________________.

The two angles adjacent to the base are the _____________________.

The angle opposite the base is the __________________________.

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Geometry Chapter 4 Notes pg. 13

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Geometry Chapter 4 Notes pg. 14

Given: ∆ABC is an isosceles triangle

𝐴𝐵̅̅ ̅̅ ≅ 𝐴𝐶̅̅ ̅̅ , 𝐴𝐷̅̅ ̅̅ bisects ∠CAB

Prove: ∠B ≅ ∠C

Statements Reasons

Base Angles Theorem

Two sides of a triangle are congruent if and only if

the angles opposite them are congruent.

A triangle is equilateral if and only if it is equiangular.

1. Find the value of x and y. 2. Find the values of x and y.

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Geometry Chapter 4 Notes pg. 15

Determine the values of x, y and z.

3. 4.

GUIDED PRACTICE

Solve for x and y.

1. 2.

3. Given: 𝑅𝑉̅̅ ̅̅ ≅ 𝑆𝑇̅̅̅̅ , ∠RTV and ∠SVT are right angles

Prove: ΔRTV ≅ ΔSVT

Statements Reasons