* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download File
Survey
Document related concepts
Dessin d'enfant wikipedia , lookup
Multilateration wikipedia , lookup
Line (geometry) wikipedia , lookup
Euler angles wikipedia , lookup
Golden ratio wikipedia , lookup
Tessellation wikipedia , lookup
Apollonian network wikipedia , lookup
History of geometry wikipedia , lookup
Rational trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Reuleaux triangle wikipedia , lookup
History of trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Transcript
Geometry CP Unit 4: Congruency of Triangles Outline NOTE: Mr. Beegle reserves the right to amend, add to, or take away from assessments, assignments, and/or activities on an as-needed basis. NOTE: Expect pop quizzes, opensting tickets, and buzz tickets as well. NOTE: Dates are subject to change. DATE: Lesson: Assessment: _______ 4.1—Congruent Polygons ___________________________ _______ 4.2—Triangle Congruence ___________________________ _______ 4.3—Analyzing Triangle Congruence ___________________________ _______ 4.4—Using Triangle Congruence ___________________________ _______ Unit 4 Test ___________________________ _______ Match.com Project ___________________________ Geometry CP 4.1—Congruent Polygons (pp. 210-213) Upon completing this lesson, students will be able to: 1. Define congruent polygons. 2. Solve problems by using congruent polygons. Remember from previous lessons that is something is congruent, that it ________________________________________________________________________. Another way to look at it is that you could take any polygon and “slide” it on top of the same polygon. Naming Polygons • Here we see a triangle. • When naming a polygon, start at any point on the figure, then go ______________________________________________________________________. • Using this rule, we can name this triangle six different ways. • ________________________________________________________________________. Corresponding Sides and Angles Two polygons can be corresponding if they ______________________________________________________ _______________________________________________________ Here we have quadrilaterals ADCB and EHGF. How can we prove that they are corresponding? Notice that when we named the triangle in the earlier side, we came up with 6 different ways. Notice that when we named the quadrilateral in the last side, we came up with 8 different ways. This lets us see a pattern in that ______________________________________________ ___________________________________________________________________________________ in order to show the different ways that we can either name the polygon or show correspondence. Congruency Statements Whenever we write a congruency statement about two polygons, we have to write the letters of the vertices in the proper order so that they correspond. How would we write a congruency statement using these quadrilaterals? Polygon Congruence Postulate States that two polygons are congruent if and only if there is a correspondence between their sides and angles such that: ___________________________________________________________________________________ ___________________________________________________________________________________ So if you remember from the previous slide, the number of ways to prove congruency is twice the number of sides of the polygon. Geometry CP 4.2—Triangle Congruence (pp. 217-220) Upon completing this lesson, students will be able to: 1. Explore triangle rigidity. 2. Develop congruence postulates for triangles (SSS, SAS, and ASA) There are many different ways to determine that a triangle and congruent. Almost all of them involve either angles or sides. SSS Postulate ___________________________________________________________________________________ ___________________________________________________________________________________ ___________________________________________________________________________________ SAS Postulate ___________________________________________________________________________________ ___________________________________________________________________________________ ___________________________________________________________________________________ ASA Postulate ___________________________________________________________________________________ ___________________________________________________________________________________ ___________________________________________________________________________________ Geometry CP 4.3—Analyzing Triangle Congruence (pp. 226-229) Upon completing this lesson, students will be able to: 1. Identify and use the SSS, SAS, and ASA Congruence Postulates and the AAS and HL Congruence Theorems. 2. Use counterexamples to prove that other side and angle combinations cannot be used to prove triangle congruence. AAA Postulate? Angle-Angle-Angle? In looking at the triangles on the next slide, what can we prove? What do we need to prove congruency? So as we saw from the previous slide, we proved they were congruent through their angles, but not necessarily their sides. Thus, AAA is not a way to prove congruency. AAS Theorem ___________________________________________________________________________________ ___________________________________________________________________________________ ___________________________________________________________________________________ Difference between AAS and ASA AAS is a ___________________________, ASA is a __________________________________. With AAS, the congruent parts must correspond. HL Theorem ___________________________________________________________________________________ ___________________________________________________________________________________ ___________________________________________________________________________________ What makes the HL Theorem different from the other postulates and theorems we’ve looked at? Geometry CP 4.4—Using Triangle Congruence (pp. 245-249) Upon completing this lesson, students will be able to: 1. Use congruence of triangles to conclude congruence of corresponding parts. 2. Develop and use the Isosceles Triangle Theorem Go back to the triangles in the previous slide. How many ways can we prove the triangles congruent Name them. If you can prove that all sides of a triangle are congruent and that all angles of a triangle are congruent, you can use CPCTC ___________________________________________________________________________________ You only use CPCTC after determining ___________________________ way that two sets of triangles are congruent. In order to properly use CPCTC in context, we can use a __________________________________________________. Given: Two segments, MP and NQ, bisect each other at Point O Prove: MN ≅ PQ Isosceles Triangle Isosceles triangles have at least ______________________ congruent sides. Made up of two ______________________ and a ______________________. The angles opposite the legs are ____________________________________________. The angle opposite the base is the _________________________________________. Isosceles Triangle Theorem ___________________________________________________________________________________ ___________________________________________________________________________________ Corollary Corollaries are ______________________________________________ that can be derived from original theorems. Two common ones are: The measure of each angle of an equilateral triangle is 60° The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.