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 ≅____ 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 = _________
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
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.
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.
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
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
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
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
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 ≅ _______________
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
Geometry Chapter 4 Notes pg. 11
4. Given: ∠M ≅ ∠N
∠OKL ≅ ∠OLK
Prove: 𝑀𝐾̅̅ ̅̅ ̅ ≅ 𝑁𝐿̅̅ ̅̅
Statements Reasons
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 __________________________.
Geometry Chapter 4 Notes pg. 13
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.
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
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