Formal Geometry Unit 4 – Congruent Triangles

Section 4.1 – Angles Triangles Classifying Triangles by Angles

Acute Triangle-

Obtuse Triangle-

Right Triangle-

Equiangular triangle- Classifying Triangles by Sides

Scalene Triangle-

Isosceles Triangle-

Equilateral Triangle- Examples 1-3: Classify each triangle as acute, equiangular, obtuse, or right.

1. ∆𝐴𝐵𝐷 2. ∆𝐵𝐷𝐶 3. ∆𝐴𝐵𝐶

Examples 4-5: 𝑪 is the midpoint of 𝑩𝑫̅̅̅̅̅ and point 𝑬 is the midpoint of 𝑫𝑭̅̅ ̅̅ ̅. Classify each triangle as isosceles, scalene, or equilateral.

4. ∆𝐴𝐵𝐷 5. ∆𝐴𝐵𝐶

Example 6-8: Find 𝒙 and the measures of the unkown sides of the triangle. 6.

7. ∆𝐹𝐻𝐺 is an equilateral triangle with 𝐹𝐻 = 6𝑥 + 1, 𝐹𝐺 = 3𝑥 + 10 and 𝐻𝐺 = 9𝑥 − 8

8. ∆𝑀𝑁𝑃 is an isosceles triangle with 𝑀𝑁̅̅ ̅̅ ̅ ≅ 𝑁𝑃̅̅ ̅̅ . 𝑀𝑁 is two less than five times 𝑥, 𝑁𝑃 is seven more

than two times 𝑥, and 𝑃𝑀 is two more than three times 𝑥.

Theorem List

Theorem 4.1- Definitions

Exterior angles-

Remote interior angles-

Theorem List

Theorem 4.2- Corollary-

Corollary 4.1-

Corollary 4.2- Examples 9-10: Find the measure of each numbered angle. 9. 10. Examples 11-12: Find each measure.

11. 𝑚∠1 12. 𝑚∠3

Section 4.2 – Congruent Triangles Definitions

Congruent Polygons-

Corresponding parts- Theorem List

Theorem 4.3-

Theorem 4.4-

Properties of Triangle Congruence

Example 1: Find 𝒙 and 𝒚. 1.

Example 2: Draw and label a figure to represent the congruent triangles. Then find 𝒙

and 𝒚.

2. ∆𝐽𝐾𝐿 ≅ ∆𝑀𝑁𝑃, 𝐽𝐾 = 12, 𝐿𝐽 = 5, 𝑃𝑀 = 2𝑥 − 3,𝑚∠𝐾 = 67,𝑚∠𝐿 = 𝑦 + 4, and 𝑚∠𝑃 = 2𝑦 − 15 Example 3: Write a 2 Column Proof Given: ∠𝐴 ≅ ∠𝐷

Prove: ∠𝐵 ≅ ∠𝐸

Section 4.3/4.4 Day 1– Proving Congruent Triangles Postulate List

Postulate 4.1-

Postulate 4.2-

Postulate 4.3- Theorem List

Theorem 4.5-

Proving Triangles Congruent

SSS SAS ASA AAS

Examples 1-4: Determine which postulate can be use to prove that the triangles are congruent. If it is not possible prove congruence, write not possible. 1. 2. 3. 4. Example 5: Write a congruence statement for each pair of triangles represented:

5. 𝐹𝐴̅̅ ̅̅ ≅ 𝐻𝑂̅̅ ̅̅ , 𝐴𝑇̅̅ ̅̅ ≅ 𝑂𝐺̅̅ ̅̅ ∠𝐴 ≅ ∠𝑂 Examples 6-7: Determine whether each pair of triangles is congruent. If so, write the congruence statement and why the triangles are congruent. 6. 7.

Examples 8-9: State the 3rd congruence that must be given to prove that the Δ' ares ,

using the indicated method. (what other corresponding parts are needed) 8. Method: SAS 9. Given: 𝐶𝑇̅̅̅̅ ≅ 𝐷𝐺̅̅ ̅̅ 𝐶𝐴̅̅ ̅̅ ≅ 𝐷𝑂̅̅ ̅̅

Method: SAS

Section 4.3/4.4 Day 2– Proving Congruent Triangles 1. Given: 𝑅𝑂̅̅ ̅̅ ⊥ 𝑀𝑃̅̅̅̅̅ 𝑀𝑂̅̅ ̅̅ ̅ ≅ 𝑂𝑃̅̅ ̅̅

Prove: ∆𝑀𝑅𝑂 ≅ ∆𝑃𝑅𝑂

2. Given: 𝑆𝑉⃗⃗⃗⃗ ⃗𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝑇𝑆𝐵

𝑉𝑆⃗⃗⃗⃗ ⃗ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝑇𝑉𝐵

Prove: ∆𝑇𝑆𝑉 ≅ ∆𝐵𝑆𝑉

3. Given: 𝐸𝐴̅̅ ̅̅ ∥ 𝐷𝐵̅̅ ̅̅ 𝐸𝐴̅̅ ̅̅ ≅ 𝐷𝐵̅̅ ̅̅

𝐵 is the midpoint of 𝐴𝐶̅̅ ̅̅ Prove: ∆𝐸𝐴𝐵 ≅ ∆𝐷𝐵𝐶

Section 4.3/4.5 Day 3– No Notes

Section 4.4/4.5 Day 4– CPCTC CPCTC-

1. Given: ∠𝐻𝐺𝐽 ≅ ∠𝐾𝐽𝐺 ∠𝐾𝐺𝐽 ≅ ∠𝐻𝐽𝐺

Prove: 𝐻𝐺̅̅ ̅̅ ≅ 𝐾𝐽̅̅ ̅

2. Given: ∆𝑇𝑃𝑄 ≅ ∆𝑆𝑃𝑅 ∠𝑇𝑄𝑅 ≅ ∠𝑆𝑅𝑄

Prove: ∆𝑇𝑄𝑅 ≅ ∆𝑆𝑅𝑄

Section 4.5 Extension-HL Theorem Review In a right triangle the sides are called: Hypotenuse- Legs-

Theorem List

Theorem 4.9-

1. Given: ∠𝐴𝐵𝐷 and ∠𝐶𝐵𝐷 are right ∠′𝑠 𝐴𝐷̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅

Prove: ∆𝐴𝐵𝐷 ≅ ∆ 𝐶𝐵𝐷 2. Given: ∠𝐹𝐺𝐻 is a right ∠

∠𝐽𝐻𝐺 is a right ∠ 𝐹𝐺̅̅ ̅̅ ≅ 𝐽𝐻̅̅̅̅

Prove: ∆𝐹𝐺𝐻 ≅ ∆𝐽𝐻𝐺

3. Given: 𝐴𝐵̅̅ ̅̅ ⊥ 𝐵𝐶̅̅ ̅̅ 𝐷𝐶̅̅ ̅̅ ⊥ 𝐵𝐶̅̅ ̅̅ 𝐴𝐶̅̅ ̅̅ ≅ 𝐵𝐷̅̅ ̅̅

Prove: 𝐴𝐵̅̅ ̅̅ ≅ 𝐷𝐶̅̅ ̅̅

Section 4.6 Isosceles and Equilateral Triangles Review: Isosceles triangle-

Legs-

Base-

Vertex angle-

Base angles- Theorem List

Theorem 4.10-

Theorem 4.11-

Corollaries:

Corollary 4.3-

Corollary 4.4- Examples 1-2: Find each measure. 1. 𝐹𝐻 2. 𝑚∠𝑀𝑅𝑃 Examples 3-4: Find the value of each variable. 3. 4.

5. Given: ∆𝐴𝐵𝐶 is isosceles w/base 𝐵𝐶̅̅ ̅̅

𝐷 is the midpoint of 𝐵𝐶̅̅ ̅̅

Prove: ∆𝐴𝐵𝐷 ≅ ∆𝐴𝐶𝐷 6. Find the value of the variable.