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Lesson 4.2: Angle Relationships in Triangles Page 223 in text Learning Objectives: The learners will be able to measures of interior and exterior angles of triangles The learners will be able to apply theorems about the interior and exterior angles of triangles. Common Core Standards: Prove geometric theorems G-CO.10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Continuity: Previous Lessons Yesterday we learned how to classify triangles according to side length and angle measure. This Lesson Today, we will prove that the angle measures of a triangle add up to 180 degrees, and we will prove that the measure of an exterior angle of a triangle is equal to the sum of the two nonadjacent angles of the triangle Next Lesson Next, we’ll add to our knowledge by exploring congruence in the context of triangles. Lesson Overview Opening: Hand back tests from Chapter 3 Launch: Warm Up Review Angle Measures Review Triangle Classifications Review Interior and Exterior Angles Explore: Triangle Sum Activity o Tearing triangles angles to show they add up to 180o o Folding triangle into a rectangle to show angle sum is 180o Triangle Sum Proof o Corollaries What is a corollary anyways? Exterior Angle Proof o Why does this make sense? o Reflect back to Triangle Sum Activity Third Angles Theorem Reflect: Exit Slip How are the three main theorems we learned today connected? What did you learn? What are you still confused about? Homework: 2-5, 6-14E, 24, 28, Detailed Outline Warm UP Objective Activity Recall Angle Acute Right Obtuse Measures to connect 0 < ∢𝐴 < 90 m A = 90 90 < ∢𝐴 < 180 what learners know about angles to what they will learn about classifying triangles by angle measure. Classify Triangles by (By Angles): Angle Measure and Acute Right Obtuse Side Length by 3 acute ’s 1 right 1 obtuse identifying (2 acute ’s) (2 acute ’s) properties and attributes of the figures. (By Sides): Scalene No sides Isosceles 2 sides Equilateral 3 sides Straight m A = 180 Equiangular 3 congruent ’s (all acute ’s) Teacher Use Effective Questioning from PBS sheet. Timing 2 min Draw visual on board that looks like picture for learners who need organization and visual. 2 min Recall Parallel Postulate How many lines can be drawn through N parallel to ̅̅̅̅̅ 𝑀𝑃 ? N M P Answer: Exactly 1 by the Parallel Postulate. Review Exterior and Interior Angles Introduce auxiliary line. If learners cannot recall have them flip back in their textbook. Have students label all angle relationships. 3 min 3 min Dialogue for Intro. Proving Triangles Sum is 180 degrees. This is clearly something we’ve known and accepted for awhile but now we are going to prove it! Objective of Activity Students will “discover” or convince themselves that the Triangle Sum Theorem should in fact be true. Students will help with the proof of the Triangle Sum Theorem (sketch of the proof), thus working on their proof skills. Explore Part 1: Triangle Sum Theorem Outline of Activity Teacher Have students (with partners) rip corners of each triangle (each type of triangle by angle) and line them up on a straight line on their paper (straight angle) – have them make conjectures [the interior angles in a triangle add up to 180 degrees]. Sketch the proof for The Triangle Sum Theorem – refer them to Page 223 in their book to see the full proof. Mentions use of Auxiliary line. ***(SEE SKETCH BELOW) Mentions corollary to the Triangle Sum Theorem [in a right triangle, the acute angles are complementary] CHECK 2 in text Sketches proof out loud with students. Timing Gives students a 15 non-precise minutes justification of the Triangle Sum Theorem. Allows them to learn in a concrete, handson way. Allows students to see the idea of the proof, but puts responsibility on them to discover it. Once they understand the purpose, they can delve into the formal proof and really make sense of the idea being discussed. Assessment Students will work with partners to make conjectures. They will make the correct conjecture, or if they make an incorrect conjecture, this allows me to target misconceptions. Sketch of Proof (with guided questing from teacher) Triangle Sum Proof: With what we know about parallel lines and alternate interior angles, it's pretty straight forward: Construct Auxiliary Line: a line that is added to a figure to aid in a proof. How can we justify the auxiliary lines existence? That is, why are we allowed to construct this line? Through any two points there is exactly one line. ̅̅̅̅? But how can we make our auxiliary parallel to AC By the parallel postulate!! What kind of Angles did we create? Label Interior and Exterior What would the transversal be in each case? Does this make sense in terms of the activity we did when we tore angles from the triangles? COROLLARIES for triangle sum theorem: What is a corollary anyways? A theorem whose proof follows directly from another theorem. So basically, we get these for free, well almost free. CHECK 2 in text Part 2: Exterior Angles Theorem Dialogue for Intro. Part 2: Exterior Angles. In the first activity we were able to create and examine the features of a triangle. In doing this, we learned how to use deductive reasoning to formally prove the sum of the measures of the interior angles is equal to the measure of a straight angle (180o) of a line drawn through one of the vertices. We will now use that information to examine the exterior angles of a triangle. Objective of Activity Students will use what they know to help them sketch the Exterior Angle theorem. Students will see why this theorem seems logical and then they will formulate a proof. Outline of Activity Recall what we know about triangles Define terms: remote interior angles, exterior angles, and interior angles.***See Definition Whiteboard Activity BELOW Read the Exterior Angles Theorem from book on page 225. Do CHECK 3 in book with partners. Teacher uses GEOGEBRA to “convince them” in a non-precise sense that the theorem is true. Then sketch the proof for The Exterior Angle Theorem on the board– refer them to Page 225 in their book to examples of the Theorem. ***See Geogebra sketch BELOW Teacher Allows students to see visual representation of the material and build off prior knowledge by connecting familiar concepts with the new ideas. Includes technology in the classroom for the purpose of making a conjecture. Timing 20 min Assessment Students will be enthusiastic, or at least engaged in watching Geogebra. They will have ideas of how to sketch the proof of this theorem and will believe it to be true. If the students aren’t asking the right kinds of questions or are applying the wrong vocabulary, I can stop the demonstration and review the main ideas we are using to construct the proof. Interior and Exterior Angles in Triangle Whiteboard Organization An interior angle is formed by two sides of a triangle.( inside the figure) In figure: ∠1, ∠2, ∠3 An exterior angle is formed by one side of the triangle and the extension of the adjacent side. (outside the figure) CHECK 3 IN figure: ∠4 Each exterior angle has two remote interior angles. In figure: ∠1 𝑎𝑛𝑑 ∠2 A remote interior angle is an interior angle that is not adjacent to the exterior angle. (Interior and away from exterior) Exterior Angle: A better visual of why the proof make sense Geometers Sketchpad Proof (tying everything back together) Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. Using Geogebra, try 2 examples to see if the theorem is true. In the figures above the exterior angle ∠ABP is equal to the sum of the remote interior angles ∠BAC and ∠ACB. Now that the conjecture is believed to be true, work through a formal proof with students. Dialogue for Intro. Next we are going to look at the Third Angles Theorem. We will discuss why it makes sense and how we can use it. Reflection Objective Summarize lesson. Complete Exit Slip Apply understanding to homework. Objective of Activity The students will understand how to compare angles amongst two triangles and will understand how to find the third angle of a triangle by applying the previous theorems. PART 3: Third Angles Theorem Outline of Activity Teacher Read the Third Angles Theorem Aloud to the class. Let them work on CHECK 4 in the text in partners. As a class discuss our findings and connections. Timing Gives the 10 min students freedom to explore the third theorem, which is fairly straightforward, with one another. Asks questions that force the learners to use the exterior angles theorem and triangle sum theorem. Activity Today we learned how to classify triangles, we learned that the angle measures of a triangle add up to 180 degrees, and we learned that the measure of an exterior angle of a triangle is equal to the sum of the two nonadjacent angles of the triangle. Now, it’s time to try some problems that apply what you know from today’s lesson and from your previous experience with angles, side lengths, and triangles, to some homework problems. Tomorrow we’ll add to our knowledge of triangles by exploring congruence in the context of triangles. Assessment Students will be given the freedom to explore examples that force them to apply all of the theorems they learned to find the third angle. The teacher can walk around and ask questions to ensure learners understand the material. Teacher Wraps up what was covered. Assesses student learning. Timing 10 min ……… .