Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Multilateration wikipedia , lookup
Line (geometry) wikipedia , lookup
Rational trigonometry wikipedia , lookup
History of geometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
History of trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Euler angles wikipedia , lookup
Lesson 23 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Lesson 23: Base Angles of Isosceles Triangles Classwork Opening Exercise Describe the additional piece of information needed for each pair of triangles to satisfy the SAS triangle congruence criteria. 1. 2. Given: π΄π΅ = π·πΆ Prove: β³ π΄π΅πΆ β β³ π·πΆπ΅ Given: π΄π΅ = π π Μ Μ Μ Μ π΄π΅ β₯ Μ Μ Μ Μ π π Prove: β³ π΄π΅πΆ β β³ π ππ Exploratory Challenge Today we examine a geometry fact that we already accept to be true. We are going to prove this known fact in two ways: (1) by using transformations and (2) by using SAS triangle congruence criteria. Here is isosceles triangle π΄π΅πΆ. We accept that an isosceles triangle, which has (at least) two congruent sides, also has congruent base angles. Label the congruent angles in the figure. Now we will prove that the base angles of an isosceles triangle are always congruent. Lesson 23: Date: Base Angles of Isosceles Triangles 5/3/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.125 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 23 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Prove Base Angles of an Isosceles are Congruent: Transformations Given: Isosceles β³ π΄π΅πΆ, with π΄π΅ = π΄πΆ Prove: mβ π΅ = mβ πΆ Construction: Draw the angle bisector βββββ π΄π· of β π΄, where π· is the intersection of the Μ Μ Μ Μ . We need to show that rigid motions will map point π΅ to point πΆ and bisector and π΅πΆ point πΆ to point π΅. β‘ββββ . Through the reflection, we want to demonstrate two Let π be the reflection through π΄π· βββββ , and (2) π΄π΅ = π΄πΆ. pieces of information that map π΅ to point πΆ and vice versa: (1) βββββ π΄π΅ maps to π΄πΆ β‘ββββ , π(π΄) = π΄. Reflections preserve angle measures, so the measure of the reflected Since π΄ is on the line of reflection, π΄π· βββββ ) = βββββ angle π(β π΅π΄π·) equals the measure of β πΆπ΄π·; therefore, π(π΄π΅ π΄πΆ . Reflections also preserve lengths of segments; therefore, the reflection of Μ Μ Μ Μ π΄π΅ will still have the same length as Μ Μ Μ Μ π΄π΅ . By hypothesis, π΄π΅ = π΄πΆ, so the length of the reflection will also be equal to π΄πΆ. Then π(π΅) = πΆ. Using similar reasoning, we can show that π(πΆ) = π΅. βββββ ) = βββββ βββββ ) = βββββ Reflections map rays to rays, so π(π΅π΄ πΆπ΄ and π(π΅πΆ πΆπ΅ . Again, since reflections preserve angle measures, the measure of π(β π΄π΅πΆ) is equal to the measure of β π΄πΆπ΅. We conclude that mβ π΅ = mβ πΆ. Equivalently, we can state that β π΅ β β πΆ. In proofs, we can state that βbase angles of an isosceles triangle are equal in measureβ or that βbase angles of an isosceles triangle are congruent.β Prove Base Angles of an Isosceles are Congruent: SAS Given: Isosceles β³ π΄π΅πΆ, with π΄π΅ = π΄πΆ Prove: β π΅ β β πΆ βββββ of β π΄, where π· is the intersection of the Construction: Draw the angle bisector π΄π· Μ Μ Μ Μ Μ bisector and π΅πΆ. We are going to use this auxiliary line towards our SAS criteria. Lesson 23: Date: Base Angles of Isosceles Triangles 5/3/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.126 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 23 M1 GEOMETRY Exercises 1. Given: Prove: 2. 3. Μ Μ Μ bisects πΎπΏ Μ Μ Μ Μ π½πΎ = π½πΏ; π½π Μ Μ Μ Μ Μ Μ Μ π½π β₯ πΎπΏ Prove: π΄π΅ = π΄πΆ, ππ΅ = ππΆ Μ Μ Μ Μ π΄π bisects β π΅π΄πΆ Given: π½π = π½π, πΎπ = πΏπ Prove: β³ π½πΎπΏ is isosceles Given: Lesson 23: Date: Base Angles of Isosceles Triangles 5/3/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.127 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 23 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 4. Given: β³ π΄π΅πΆ, with mβ πΆπ΅π΄ = mβ π΅πΆπ΄ Prove: π΅π΄ = πΆπ΄ (Converse of base angles of isosceles triangle) Hint: Use a transformation. 5. Given: Μ Μ Μ Μ β₯ ππ Μ Μ Μ Μ Μ is the angle bisector of β π΅ππ΄, and π΅πΆ Μ Μ Μ Μ β³ π΄π΅πΆ, with ππ Prove: ππ΅ = ππΆ Lesson 23: Date: Base Angles of Isosceles Triangles 5/3/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.128 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 23 M1 GEOMETRY Problem Set 1. 2. 3. Given: π΄π΅ = π΅πΆ, π΄π· = π·πΆ Prove: β³ π΄π·π΅ and β³ πΆπ·π΅ are right triangles Given: Μ Μ Μ Μ π΄πΆ = π΄πΈ and Μ Μ Μ Μ π΅πΉ βπΆπΈ Prove: π΄π΅ = π΄πΉ In the diagram, β³ π΄π΅πΆ is isosceles with Μ Μ Μ Μ π΄πΆ β Μ Μ Μ Μ π΄π΅ . In your own words, describe how transformations and the properties of rigid motions can be used to show that β πΆ β β π΅. Lesson 23: Date: Base Angles of Isosceles Triangles 5/3/17 © 2014 Common Core, Inc. Some rights reserved. commoncore.org S.129 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.