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Lesson 4.3: Congruent Triangles Page 231 in text Learning Objectives: The learners will use properties of congruent triangles The learners will prove triangles congruent by using the definition of congruence. Common Core Standards: Prove triangles congruent by using the definition of congruence G-CO.7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Continuity: Previous Lessons Yesterday we proved several angle relationships in triangles using interior and exterior angles. This Lesson Today, we will prove triangles (polygons) congruent by examining their corresponding sides and angles. Next Lesson Next, we will examine more specific triangle congruence theorems. Opening: ο· ο· Give Homework Quiz on 4.1 Go over homework from 4.2 (2-5, 6-14E, 24, 28) Launch: Warm Up Look at the figure, what do we know about βπ΄π΅πΆ and βπ·πΈπΉ? ο· ο· β π΅ β β πΈ β π΄ β β π· How do we know this? ο· Because of the arc marks. We use these to show angles are congruent. What can we conclude about β πΆ & β πΉ? ο· β πΆ β β πΉ How do we know this? ο· Third Angles Theorem Review Congruence What do we know? Congruent_________ Definition Segment Two line segments are congruent if they have the same length. Picture Symbols Μ Μ Μ Μ π΄π΅ β Μ Μ Μ Μ ππ is read as "The line segment AB is congruent to the line segment PQ". They need not lie at the same angle or position on the plane Note the single 'tic' marks on the lines. These are a graphical way to show that the two line segments are congruent. Angle Two angles are congruent if they have the same measure. β ABC β β PQR is read as "The angle ABC is congruent to the angle PQR" Congruent angles may lie in different orientations or positions. Rays and lines cannot be congruent because they do not have both end points defined, and so have no definite length. Explore: ο· Naming Congruent Corresponding Parts o How do you name a polygon? ο§ You must write the vertices in consecutive order. ο· What does consecutive mean? ο§ You can write them in counterclockwise or clockwise order ο· Using the triangles to the right we can name them: o βπ΄π΅πΆ and βπ·πΈπΉ o βπΆπ΅π΄ and βπΉπΈπ· ο§ Give a four sided polygon example to show how you can NOT label A B C D o Polygon ABCD o NOT polygon ADCB o Not consecutive o In a congruence statement, the order of the vertices indicates the corresponding parts ο§ Example: Congruent triangles have the same size and the same shape. The corresponding sides and the corresponding angles of congruent triangles are equal. Note: ο· Using Corresponding Parts of Congruent Triangles If two figures have the same shape and the same size, then they are said to be congruent figures. For example, rectangle ABCD and rectangle PQRS are congruent rectangles as they have the same shape and the same size. Side AB and side PQ are in the same relative position in each of the figures. We say that the side AB and side PQ are corresponding sides. Congruent figures are exact duplicates of each other. One could be fitted over the other so that their corresponding parts coincide. The concept of congruent figures applies to figures of any type. Just to recap the ongoing discussion: ο· ο· Angles and sides of two plane figures are said to be corresponding if they are in the same relative positions in each of the figures. If one figure coincides with another after a transformation (i.e. a translation, reflection or rotation) that moves the points on the figure but does not alter its angles or side-lengths, then the figures are said to be congruent. ο· Proving Triangles are congruent o You must show all three pairs of sides and all three pairs of angles are congruent o Geometric figures are congruent if they are the same sized and shape. o Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides. o Two polygons are congruent if and only if their corresponding sides and angles are congruent. ο§ Triangles that are the same size and shape are congruent. Reflect: Exit Slip ο· ο· What did you learn? What are you still confused about? Homework: ο· Page 234: 2-11, 17-19 Warm UP Recall Angle Measures Acute 0 < β’π΄ < 90 Right m ο A = 90 Obtuse 90 < β’π΄ < 180 Straight m ο A = 180 Classifying Triangles by Angle Measure and Side Length (By Angles): Acute 3 acute ο βs Right 1 right ο (2 acute ο βs) Obtuse 1 obtuse ο (2 acute ο βs) Isosceles ο£ 2 ο sides Equilateral 3 ο sides (By Sides): Scalene No ο sides Recall Parallel Postulate How many lines can be drawn through N parallel to Μ Μ Μ Μ Μ π΄π·? N M P Exactly 1 by the Parallel Postulate. Equiangular 3 congruent ο βs (all acute ο βs) Review Exterior and Interior Angles Triangle Students will Have students (with partners) rip Gives students a 10 Students will work with Sum Theorem: Proving Triangles Sum is 180 degrees. This is clearly something weβve known and accepted for awhile but now we are going to prove it! βdiscoverβ 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. corners of each triangle (each type of triangle by angle) and line them up on a straight line on their paper (straight angle) β have them make conjectures [the interior angles in a triangle add up to 180 degrees]. Sketch the proof for The Triangle Sum Theorem β refer them to Page 223 in their book to see the full proof. Mentions use of Auxiliary line. Mentions corollary to the Triangle Sum Theorem [in a right triangle, the acute angles are complementary] Sketches proof out loud with students. non-precise justification of the Triangle Sum Theorem. Allows them to learn in a concrete, hands-on way. Allows students to see the idea of the proof, but puts responsibility on them to look it up later. minutes partners to make conjectures. They will make the correct conjecture, or if they make an incorrect conjecture, this allows me to target misconceptions. 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. ο· How can we justify the auxiliary lines existence? That is, why are we allowed to construct this line? ο· Through any two points there is exactly one line. ο· But how can we make our auxiliary parallel to Μ Μ Μ Μ ππ? ο· By the parallel postulate!! What kind of Angles did we create? Label Interior and Exterior ο· What would the transversal be in each case? ο· Does this make sense in terms of the activity we did when we tore angles from the triangles? CHECK 2 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. Interior and Exterior Angles in Triangle CHECK 3 An interior angle is formed by two sides of a triangle.( inside the figure) ο· In figure: β π, β π, β π An exterior angle is formed by one side of the triangle and the extrension of the adjacent side. (outside the figure) ο· IN figure: β π ο· Each exterior angle has two remote interior angles. ο· In figure: β π πππ β π ο· A remote interior angle is an interior angle that is not adjacent to the exterior angle. (Interior and away from exterior) Exterior Angle Theorem Exterior Angle: A better visual of why the proof make sense Exterior angle β an angle created by extending one side of a figure CHECK 3 Third Angles Theorem CHECK 4 Explore Check 4 Must use third angles theorem How is the third angles theorem related to the Triangle Sum Theorem? Exterior angle β an angle created by extending one side of a figure Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. β¦.. In the figures above the exterior angle β ABP is equal to the sum of the remote interior angles β BAC and β ACB. . Proof: β¦β¦β¦ . . If the equivalent angle is taken at each vertex, the exterior angles always add to 360° Since the measure of an exterior angle equals the sum of its two remote interior angles, the exterior angle is greater than the measure of either individual remote interior angle. 1 Hands βon Classification based on angles Students will recall angle types and will use the cut-out triangles to classify them by angles and by side lengths with a partner. Gives student pairs several different numbered triangles. Asks students to remember what acute, right and obtuse angles are. Writes on Board: (see Board Content; attached) Talks about classifications of triangles based on types of angles. (See board content; attached) Allows students to see visual representation of the material and also to manipulate the triangles and learn in a kinesthetic and hands-on way. The partner practice allows them to classify together. 15 minutes Students will correctly identify triangles according to angle. If they incorrectly identify triangles, I have a greater sense of their misconceptions, and so can target them right away or later on in the lesson. Says that these are the interior angles of the triangle. Identifies what are the exterior angles of triangles (see Board Content; attached) 5 Triangle Sum Students will Theorem βdiscoverβ or convince Instructs students to classify the angles they have (construction paper) with their partners (based on angles.) Confirms which triangles are which types, using the number on the back of each triangle for identification purposes. Have students (with partners) rip Gives students a 10 non-precise minutes corners of each triangle (each Students will work with partners to make 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. type of triangle by angle) and line them up on a straight line on their paper (straight angle) β have them make conjectures [the interior angles in a triangle add up to 180 degrees]. Sketch (in a cloud) the proof for The Triangle Sum Theorem β refer them to Page 219 in their book to see the full proof. Mentions use of Auxiliary line. justification of the Triangle Sum Theorem. Allows them to learn in a concrete, hands-on way. Allows students to see the idea of the proof, but puts responsibility on them to look it up later. conjectures. They will make the correct conjecture, or if they make an incorrect conjecture, this allows me to target misconceptions. Mentions corollary to the Triangle Sum Theorem [in a right triangle, the acute angles are complementary] Sketches proof out loud with students. 6 Exterior Angle Theorem Students will help sketch the Exterior angle theorem. Students will see why this theorem seems logical (Geogebra). Sketches the Exterior Angle Theorem-- tells them this is part of their homework. Uses GEOGEBRA to βconvince themβ in a non-precise sense that Includes technology in the classroom for the purpose of making a conjecture. 10 minutes 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 7 Closing Statements this is true (see attached) Conclusion Statement (see above) Wraps up what Exit Ticket (see attached) was covered. Homework Worksheet Assesses student learning. 10 minutes true. Students will complete the Exit Ticket and will work on homework.