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Chapter 4: Triangle Congruence Lesson 4.1 Classifying Triangles Learning Targets LT-1: Use triangle classification to solve problems involving angle measure and side lengths. Success Criteria Classify triangles by angle measure. Classify triangles by side length. Use triangle classification to solve problems. • • • Directions: Work with a partner to find answers to the following questions. Ex #1: Create a diagram that will help you remember the two ways to classify triangles. Be sure to include the classification names in each category. Ex #2: Classify each triangle by its angle measures. a. △ABD b. △ACD c. △BCD Ex #3: Classify each triangle by its side lengths. a. △EFH b. △FGH c. △EGH page1 Ex #4: Use triangle classification to solve problems. Find the side lengths of the triangle. a. JK b. KL c. JL Ex #5: Use triangle classification to solve probems. Draw an example of each type of triangle, or explain why it is not possible. a. isosceles right b. equiangular obtuse c. scalene right Ex #6: Use triangle classification to solve problems. An isosceles triangle has a perimeter of 34cm. The congruent sides measure (4x-1) cm. The length of the third side is x cm. What is the value of x? Ex #7: Use triangle classification to solve problems. Is every isosceles triangle equilateral? Is every equilateral triangle isosceles? Explain. page2 Chapter 4 Lesson 2: Angle Relationships in Triangles Learning Targets LT-2: Use interior and exterior angle theorems to solve problems involving triangles. Success Criteria • • • • • Use the Triangle Sum Theorem to solve problems. Find angle measures in right triangles. Apply the Exterior Angle Theorem. Apply the Third Angle Theorem. auxiliary line: Theorem Triangle Sum Theorem: C A B B 4 G1.2.1 – Proof of the Triangle Sum Theorem Statement ∆ABC and line l // to A 1 3 l 5 2 C Reason AC Alternate Interior Angle Theorem Definition of Congruent Angles Angle Addition Postulate and Defn. of Straight Angle m∠1 +m∠2+ m∠3 = 180 Ex# 1: Use the Triangle Sum Theorem to Solve Problems. After an accident, the position of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find the indicated angle measures. a. m∠XYZ b. m∠YWZ page3 • corollary: Corollary Hypothesis Conclusion The acute angles of a right triangle are complementary. The measure of each angle of an equiangular triangle is 60°. Ex #2: Find Angle Measures In Right Triangles. One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle? Theorem Exterior Angle Theorem: The measures of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. 2 1 3 4 Ex# 3: Apply The Exterior Angle Theorem. Find m∠B. Theorem Hypothesis Third Angles Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. page4 Conclusion Ex# 4: Apply The Third Angle Theorem. Find m∠K and m∠J. Chapter 4 Lesson 3: Congruent Triangles Learning Targets Success Criteria LT-3: Use provided information to prove triangles congruent and/or to solve problems involving congruent triangles. • • • • Name congruent and corresponding parts of congruent figures. Use corresponding parts of congruent figures to solve problems. Prove triangles congruent. congruent triangles: Properties of Congruent Polygons Diagram Corresponding Angles Corresponding Sides Ex #1: Name Congruent Corresponding Parts. Given: ΔPQR ≅ ΔSTW. Identify all pairs of congruent corresponding parts. Ex #2: Using Corresponding Parts of ≅ Triangles To Solve Problems. Given: ΔABC ≅ ΔDBC. A. Find the value of x. B. Find m∠DBC. page5 Ex #3: Prove Triangles Congruent Given: ∠YWX and ∠YWZ are right angles YW bisects ∠XYZ W is the midpoint of XZ XY ≅ YZ Prove: ΔXYW ≅ ΔZYW Statement Reason ∠YWX and ∠YWZ are right angles YW bisects ∠XYZ 1. W is the midpoint of XZ XY ≅ YZ 2. 3. 4. 5. 6. 7. Ex #4: Prove Triangles Congruent Given: PR and QT bisect each other. ∠PQS ≅ ∠RTS QP ≅ RT Prove: ΔQPS ≅ ΔTRS Statement 1. Reason PR and QT bisect each other QP ≅ RT ∠PQS ≅ ∠RTS and 2. 3. 4. 5. page6 Chapter 4 Lesson 4: Triangle Congruence: SSS and SAS Learning Targets Success Criteria LT-4: Prove triangles are congruent using SSS and SAS Theorems. • • Use SSS to Prove Triangles Congruent. Use SAS to Prove Triangles Congruent. Triangles can be proved congruent without using all six pairs of corresponding and congruent parts (three pairs of sides and three pairs of angles). This lesson will show how to prove triangles congruent using just three pairs of congruent corresponding parts. Side-Side-Side (SSS) Congruence Postulate Hypothesis Conclusion If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Side-Angle-Side (SAS) Congruence Postulate Hypothesis If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Ex #1: Use SSS To Prove Triangles Congruent. Use SSS to explain why ΔABC ≅ ΔDBC. Ex #2: Use SAS To Prove Triangles Congruent. The diagram shows part of the support structure for a tower. Use SAS to explain why ΔXYZ ≅ ΔVWZ. page7 Conclusion Ex #3: Use SSS and SAS To Prove Triangles Congruent. Use the given information to show that the triangles are congruent for the given values of the variable. Give the name of the theorem that supports your work. A: ΔMNO ≅ ΔPQR, when x = 5. B: ΔSTU ≅ ΔVWX, when y = 4. Ex #4: Prove Triangles Congruent Given: BC // AD , BC ≅ AD Prove: ΔABD ≅ ΔCDB Statement BC 1. Reason // AD , BC ≅ AD 2. 3. 4. Chapter 4 Lesson 5: Triangle Congruence: ASA, AAS, and HL Learning Targets Success Criteria LT-5: Prove triangles congruent by using ASA, AAS, and HL Theorems. • • • • included side: page8 Use ASA to prove triangles congruent. Use AAS to prove triangles congruent. Use HL to prove triangles congruent. Angle-Side-Angle (ASA) Congruence Postulate Hypothesis Conclusion If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Ex #1: Use ASA to Prove Triangles Congruent. Determine if you can use ASA to prove the triangles congruent. Explain. Angle-Angle-Side (AAS) Congruence Theorem Hypothesis If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent. Proof of Angle-Angle-Side Congruence Given: ∠G ≅ ∠K, ∠J ≅ ∠M, HJ ≅ LM Prove: ΔGHJ ≅ ΔKLM Statement 1. ∠G ≅ ∠K, ∠J ≅ ∠M, Reason HJ ≅ LM 2. 3. 4. page9 Conclusion Ex #2: Use AAS to prove the triangles congruent. Given: AB // ED , BC ≅ DC Prove: ΔABC ≅ ΔEDC Statement 1. AB Reason // ED , BC ≅ DC 2. 3. 4. Hypotenuse-Leg (HL) Congruence Theorem Hypothesis Conclusion If the hypotenuse and a leg of one triangle are congruent to the hypotenuse and a leg of another right triangle, then the triagles are congruent. Ex #3 Use HL to Prove Triangles Congruent. Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. A. B. page10 Chapter 4 Lesson 6: Triangle Congruence: CPCTC Learning Targets Success Criteria LT-6: Use Corresponding Parts of Congruent Triangles are Congruent(CPCTC) to prove statements about triangles and/or solve problems involving triangles. • • Use CPCTC to find angle measures and side lengths. Use CPCTC in a proof. Ex #1: Use CPCTC to Find Angle Measures and Side Lengths. A and B are on the edges of a ravine. What is AB? Explain. Ex2: Use CPCTC in a Proof. Given: YW bisects XZ and XY ≅ YZ Prove: ∠XYW ≅ ∠ZYW Statement 1. Reason YW bisects XZ XY ≅ YZ 2. 3. 4. 5. page11 Ex #3: Using CPCTC in a Proof. Given: NO // MP , ∠N ≅ ∠P Prove: MN // OP Statement 1. NO // Reason MP , ∠N ≅ ∠P 2. 3. 4. 5. 6. Ex #4: Use CPCTC in a Coordinate Proof. Given: D(-5,-5), E(-3, -1), F(-2, -3), G(-2, 1), H(0, 5), and I(1, 3) Prove: ∠DEF ≅ ∠GHI Geometry Chapter 4 Lesson 7: Introduction to Coordinate Proofs Learning Target (LT-7) • Appropriately place geometric figures in the coordinate plane to prove statements and/or solve problems about these figures. page12 Learning Targets Success Criteria LT-7: Appropriately place geometric figures in the coordinate plane to prove statements and/or solve problems about these figures. • • • Position a figure in the coordinate plane. Write a proof using coordinate geometry. Assign coordinates to vertices. You have used coordinate geometry to find the midpoint of a line segment and to find the distance between two points. Coordinate geometry can also be used to prove conjectures. A coordinate proof is a style of proof that uses coordinate geometry and algebra. The first step is to position the given figure in the plane. You can use any position, but some strategies can make the steps of the proof simpler. Strategies for Positioning Figures in the Coordinate Plane ● ● ● ● Use the origin as a vertex, keeping the figure in Quadrant 1 Center the figure at the origin Center a side of the figure at the origin Use one or both axes as a side Ex #1: Position a Figure in the Coordinate Plane. A. Position a square with a side length of 6 units in the coordinate plane. Ex #2: Write a Proof Using Coordinate Geometry. Given: Rectangle ABCD with A(0, 0), B(4, 0), C(4, 10), and D(0, 10). Prove: The diagonals bisect each other. page13 B. Position a right triangle with legs of length 3 and 6 in the coordinate plane. A coordinate proof can also be used to prove that a certain relationship is always true. You can prove that a statement is true for all right triangles without knowing the side lengths. To do this, assign variables as the coordinates of the vertices. Ex #3: Assign Coordinates to Vertices. Position each figure in the coordinate plane and give the coordinates of each vertex A. A rectangle with width m and length twice the B. A right triangle with legs of lengths s and r. width. Ex #4: Write a Proof Using Coordinate Geometry. Given: Rectangle PQRS with P(0, b), Q(a, b) R(a, 0), and S(0, 0). Prove: The diagonals are congruent. page14 Chapter 4 Lesson 8: Isosceles and Equilateral Triangles Learning Targets Success Criteria LT-8: Use the properties of isosceles and equilateral triangles to prove statements and/or solve problems about them. • isosceles triangle • leg • vertex angle • base • base angle • • Apply properties of isosceles and equilateral triangles to solve problems. Write proofs using properties of isosceles and equilateral triangles. Isosceles Triangles Theorem Hypothesis Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. page15 Conclusion Proof of Isosceles Triangle Theorem Given: AB ≅ AC Prove: ∠B ≅ ∠C Statement 1 AB ≅ Reason AC 2 3 4 5 6 7 Ex #1: Use Properties of Isosceles and Equilateral Triangles to Solve Problems. The length of YX is 20 ft. Explain why the length of YZ is the same. Ex #2: Use Properties of Isosceles and Equilateral Triangles to Solve Problems. Find each angle measure. A. m∠F B. m∠G Equilateral Triangle Corollary Hypothesis If a triangle is equilateral, then it is equiangular. If a triangle is equiangular, then it is equilateral. page16 Conclusion Ex #3: Use Properties of Isosceles and Equilateral Triangles to Solve Problems. Find each value. A. x B. y Ex #4: Write proofs using properties of isosceles and equilateral triangles. Prove that the segment joining the midpoint of two sides of an isosceles triangle is half the base. Given: In isosceles ΔABC, X is the midpoint of 1 AC Prove: XY = 2 AB , and y is the midpoint of Lesson Problems 4.1 p. 219 #12-19, 21-37, 39, 41-43, 52, 56 4.2 p. 228 #15-24, 26, 28-33, 37, 41-43, 46 4.3 p. 234 #11, 13-25, 28, 30, 31-34 4.4 p. 246 #8-22, 24, 27-31, 42-44 4.5 p. 257 #11-17, 20, 22, 24, 26-29, 30 4.6 p. 263 #7-9, 11-12, 17, 18, 22, 24-28, 37 page17 AC . 4.7 p. 270 #8-13, 15-20, 22, 25, 27-30 4.8 p. 277 #12-26, 28, 29, 33, 44 page18