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Transcript
Lesson 4.1 Classifying Triangles Learning Target: • Use triangle classification to solve problems involving angle measure and side lengths. Directions: Work with a partner to find answers to the following questions. 1. Create a diagram that will help you remember the two ways to classify triangles. Be sure to include the classifications in each category. 2. Classify each triangle by its angle measures. a. △ABD b. △ACD c. △BCD 3. Classify each triangle by its side lengths. a. △EFH b. △FGH c. △EGH page1 4. Find the side lengths of the triangle. a. JK b. KL c. JL 5. Draw an example of each type of triangle, or explain why it is not possible. a. isosceles right b. equiangular obtuse c. scalene right 6. 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? 7. Is every isosceles triangle equilateral? Is every equilateral triangle isosceles? Explain. Geometry Chapter 4 Lesson 2: Angle Relationships in Triangles Learning Target: • Use interior and exterior angle theorems to solve problems involving triangles. • auxiliary line: Theorem Triangle Sum Theorem: C A page2 B B 4 1 A G1.2.1 Statement 3 l 5 2 C Reason ∆ABC and line l // to AC Alternate Interior Angle Theorem Definition of Congruent Angles Angle Addition Postulate and Defn. of Straight Angle m∠1 +m∠2+ m∠3 = 180 Ex1: 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. Y 12° a. m∠XYZ b. m∠YWZ W • 62° 40° X corollary: Corollary Hypothesis The acute angles of a right triangle are complementary. The measure of each angle of an equiangular triangle is 60°. page3 Conclusion Z Ex2: 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 Ex3: Find m∠B. Theorem Hypothesis Conclusion 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. Ex4: Find m∠K and m∠J. ( 4y 2 K J ( 6y 2 - 40)° )° Geometry Chapter 4 Lesson3: Congruent Triangles Learning Target: • Use provided information to prove triangles congruent and/or to solve problems involving congruent triangles. • congruent triangles: page4 Properties of Congruent Polygons Diagram Corresponding Angles Naming Congruent Corresponding Parts Ex1: Given: ΔPQR ≅ ΔSTW. Identify all pairs of congruent corresponding parts. Corresponding Sides Using Corresponding Parts of ≅ Triangles B Ex2: Given: ΔABC ≅ ΔDBC. A. Find the value of x. B. Find m∠DBC. D 49.3 C ° (2x - 16)° page5 A Y Proving Triangles Congruent Ex3: Given: ∠YWX and ∠YWZ are right angles. YW bisects ∠XYZ. W is the midpoint of XZ . XY ≅ YZ Z X W Prove: ΔXYW ≅ ΔZYW Statement Reason 1. 2. 3. 4. 5. 6. 7. Ex4: The diagonal bars across a gate give it support. Since the angle measures and the lengths of the corresponding sides are the same, the triangles are congruent. R Given: PR and QT bisect each other. Q ∠PQS ≅ ∠RTS S QP ≅ Prove: RT ΔQPS ≅ ΔTRS Statement P T Reason 1. 2. 3. 4. 5. page6 Geometry Chapter 4 Lesson 4: Triangle Congruence: SSS and SAS Learning Target: • Prove triangles are congruent using SSS and SAS theorems. 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 Conclusion 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. Ex1: Use SSS to explain why ΔABC ≅ ΔDBC. B A C Ex2: The diagram shows part X of the support structure for a tower. Use SAS to explain why ΔXYZ ≅ ΔVWZ. D W page7 Y Z V Ex3: 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. 6 M N Q 5 B: ΔSTU ≅ ΔVWX, when y = 4. U 3x - 9 x P x+2 7 O R X (20y + 12)° y+3 2y + 3 T S 92° 11 7 W V B Ex4: Proving Triangles Congruent Given: BC // AD , BC ≅ AD A Prove: ΔABD ≅ ΔCDB Statement C D Reason 1. 2. 3. 4. Geometry Chapter 4 Lesson 5: Triangle Congruence: ASA, AAS, and HL Learning Target: • Prove triangles congruent by using ASA, AAS, and HL theorems. • included side: Angle-Side-Angle (ASA) Congruence Postulate Hypothesis 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. page8 Conclusion Ex1: A mailman has to collect mail from mailboxes at A and B and drop it off at the post office at C. Does the table give enough information to determine the location of the mailboxes and the post office? Bearing Distance A to B N 65° E 8 mi B to C N 24° W C to A S 20° W Ex2: Determine if you can use ASA to prove the triangles congruent. Explain. 1 2 3 4 m∠1 = m∠2 m∠3 = m∠4 Angle-Angle-Side (AAS) Congruence Theorem Hypothesis Conclusion 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 ≅ Prove: ΔGHJ ≅ ΔKLM Statement L H LM J G Reason 1. 2. 3. 4. page9 K M Using AAS to Prove Triangles Congruent Ex3: Use AAS to prove the triangles congruent Given: AB // ED , BC ≅ DC Prove: ΔABC ≅ ΔEDC Statement Reason 1. 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. Ex4: 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 Geometry Chapter 4 Lesson 6: Triangle Congruence: CPCTC Learning Target: Use corresponding parts of congruent triangles to prove statements about triangles and/or solve problems involving triangles. You can use congruent triangles to estimate distances. Ex1: A and B are on the edges of a ravine. What is AB? Y Ex2: Proving Corresponding Parts Congruent Given: YW bisects XZ and XY ≅ YZ Prove: ∠XYZ ≅ ∠ZYW X Statement Reason 1. 2. 3. 4. 5. page11 W Z Ex3: Using CPCTC in a Proof Given: NO // MP , ∠N ≅ ∠P Prove: MN // OP M Statement O N P Reason 1. 2. 3. 4. 5. 6. Ex4: Using CPCTC in the Coordinate Plane 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 • Appropriately place geometric figures in the coordinate plane to prove statements and/or solve problems about these figures. 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. page12 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 Ex1: Position a square with a side length of 6 units in the coordinate plane. Ex2: Writing 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. 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. page13 Assigning Coordinates to Vertices Ex3: Position each figure in the coordinate plane and give the coordinates of each vertex. A. rectangle with width m and length twice the width B. right triangle with legs of lengths s and r Ex4: Writing a Coordinate Proof Given: Rectangle PQRS with P(0, b), Q(a, b) R(a, 0), and S(0, 0). Prove: The diagonals are congruent. Geometry Chapter 4 Lesson 8: Isosceles and Equilateral Triangles Learning Target: • 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 page14 Isosceles Triangle Theorem Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Hypothesis Conclusion A B Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. C A B C B Proof of Isosceles Triangle Theorem Given: AB ≅ AC Prove: ∠B ≅ ∠C A X C Statement Reason 1 2 3 4 5 6 7 Ex1: The length of YX is 20 ft. Explain why the length of YZ is the same. Ex2: Find each angle measure. a. m∠F b. m∠G page15 Equilateral Triangle Corollary Hypothesis Conclusion If a triangle is equilateral, then it is equiangular. If a triangle is equiangular, then it is equilateral. Ex3: Find each value. a. x b. y Ex4: 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 AB , and y is the midpoint of AC . 1 AC Prove: XY = 2 page16