<p>Students will apply their knowledge from Lesson 1 about utilizing definitions to prove congruency. Students will study the concept of Corresponding Parts of Congruent Triangles Are Congruent (CPCTC) and how to apply it in proofs. Students will:&nbsp;<br>- use triangle theorems to prove triangle congruence.<br>- use CPCTC in conjunction with triangle congruence to prove other statements about triangles.</p>

<p>- How can we use congruence to learn about the parts of congruent triangles?&nbsp;<br>- How does this relate to real-world problems?</p>

<p>-<strong> AAS (Angle-Angle-Side correspondence):</strong> If two pairs of corresponding angles have the same measure and the pair of third sides (not included) has the same length, the two triangles are congruent.&nbsp;<br>- <strong>ASA (Angle-Side-Angle correspondence):</strong> If two pairs of corresponding angles have the same measure and the pair of corresponding sides has the same length, the two triangles are congruent.<br>- <strong>Congruent:</strong> Having the same size and shape.<br>- <strong>Corresponding Angles:</strong> Angles in the same relative position in similar or congruent figures.<br>- <strong>Corresponding Sides:</strong> Sides in the same relative position in similar or congruent figures.<br>- <strong>Deductive Reasoning:</strong> A method which arrives at conclusions from accepted principles; reasoning such that the conclusion necessarily follows from a set of premises.<br>- <strong>Hypotenuse-leg:</strong> In right triangles, if the hypotenuse and one leg of one triangle are congruent to the hypotenuse and another leg of a second triangle, then the two triangles are congruent.<br>- <strong>Included Angle:</strong> An angle of a triangle whose vertex is the common endpoint of two consecutive sides of a triangle.<br>- <strong>Included Side:</strong> A side of a triangle whose endpoints are the vertices of two consecutive angles of the triangle.<br>- <strong>Inductive Reasoning:</strong> Drawing conclusions from several known cases; reasoning from the particular to the general. The premises of an inductive logical argument indicate some degree of support for the conclusion; they suggest truth, but do not ensure it.<br>- <strong>SAS (Side-Angle-Side Correspondence):</strong> If two pairs of corresponding sides have the same length and the pair of corresponding angles has the same measure, the two triangles are congruent.<br>- <strong>SSS (Side-Side-Side Correspondence):</strong> If three pairs of corresponding sides have the same length of measure, the two triangles are congruent.</p>

<p>- paper and markers or colored pencils<br>- graph paper<br>- rulers<br>- protractors<br>- Lesson 2 Exit Ticket and KEY (M-G-4-2_Lesson 2 Exit Ticket and KEY)</p>

<p>- In Activity 3, conducted in pairs, each student must evaluate the work of his or her partner. Students should assess what each partner completed correctly, which parts were incorrect, and what knowledge was required to finish the work successfully. Remind students to distinguish between errors of measurement and computation and errors of correspondence and alignment.&nbsp;<br>- Observations of student work during group activities and class discussions are most successful when particular categories of incorrect or inaccurate work are given as examples for correction and greater comprehension. Encourage students to analyze the work of other members in the group, determining what is and is not correct, why, and what modifications are necessary.&nbsp;<br>- Lesson 2 Exit Ticket summarizes the knowledge and skill required to determine the necessary conditions for congruence, as well as the limitations on the remaining sides and angles once the conditions have been established (M-G-4-2_Lesson 2 Exit Ticket and KEY).</p>

<p>Active Engagement, Modeling, Explicit Instruction<br>W: This lesson teaches students how to use congruence theorems (SSS, SAS, AAS, ASA) to show that other corresponding parts of congruent triangles are also congruent. Students learn how to prove the congruence of geometric shapes since these concepts are used in real-world situations. Students can use deductive and inductive reasoning to hypotheses about two-dimensional objects, thereby developing real-world reasoning skills. &nbsp;<br>H: Activity 1 begins with a measured drawing of a triangle built, successively, from side length 7 centimeters, angle measure 40 degrees, side length 8 centimeters. From these three triangle parts only, students can visually express the remaining possibilities for the third side. The third side can be defined by its length, the angle measure at its left end point, or the angle at its right end point. &nbsp;<br>E: In Activity 2, the triangle construction follows the same steps as in Activity 1. The difference is that the angle between the first two drawn sides is not stated; at this point, the only correspondence between the various triangles is between two adjacent sides. &nbsp;<br>R: In Activity 3, the triangle's side and angle measurements are sufficient to constrain the missing measurements. The activity initially encourages a reconsideration of the possibilities of constructing more than one alternate triangle after establishing three parts of a triangle that force congruence.&nbsp;<br>E: Each pair of students considers possible triangles based on their initial three measurements.: three sides; one side, one included angle, one side; one angle, one included side, one included angle; and two angles, one non-included side. After all three triangle parts have been defined and drawn, the congruence relationship is visually and logically demonstrated, as no other possible sides or angles can be created to complete the triangle. &nbsp;<br>T: By having students work in pairs on a regular basis, students learn and support one another. Working in pairs on a frequent basis teaches students how to collaborate well with specific students and develop strategies to assist one another. Students who are mathematically proficient, for example, learn to assist other students who may struggle to follow specific instructions or complete activities. &nbsp;<br>O: During the first two activities, students work with partners while you provide overall instruction. The partner activities give each student time to focus carefully on the new topics while simultaneously providing a resource (the other student) to help answer exploratory questions about the new material. While the third activity is still designed to be completed with a partner, students work on their own initially to form their triangle. This activity leads into the repetition of Activity 3 (using different triangle congruence theorems) and, finally, the Exit Ticket for Lesson 2, which should be completed individually (M-G-4-2_Lesson 2 Exit Ticket and KEY).</p>

<p>Provide students with the following notes to read and/or copy.</p><p><strong>Definition of congruent triangles: </strong>Congruent triangles are triangles that have the same size and shape. They have corresponding angles with the same measurement and corresponding sides with the same length.</p><p>Use the following to determine the congruency of two triangles:</p><p><strong>SSS:</strong> If three pairs of corresponding sides have the same length, then the two triangles are congruent.</p><p><strong>SAS:</strong> If two pairs of corresponding sides have the same length and the included corresponding angles have the same measure, the two triangles are congruent.</p><p><strong>ASA:</strong> If two pairs of corresponding angles have the same measure and the pair of included corresponding sides have the same length, the two triangles are congruent.</p><p><strong>AAS:</strong> If two pairs of corresponding angles have the same measure and the third sides (not included) are the same length, the two triangles are congruent.</p><p>Hypotenuse-Leg: In right triangles, the hypotenuse and one leg of one triangle are congruent to the hypotenuse and another leg of another triangle, implying that the two triangles are congruent.</p><p><strong>CPCTC: </strong>Corresponding parts of congruent triangles are congruent.</p><figure class="image"><img style="aspect-ratio:643/477;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_22.png" width="643" height="477"></figure><p><strong>Activity 1: Building of Triangles with Specific Given Information</strong></p><p>Divide the students into groups of three or four. Let each group draw two congruent triangles. Start each group by drawing a side with length 7 cm and marking one endpoint <i>A</i> and the other <i>B</i>. Next, each group should create ∠<i>A</i> with a measure of 40 degrees and utilize it to draw \(\overline{AC}\) with a length of 8 cm. At this point, ask students if all of their pictures should be identical. Now have them sketch the third side (by connecting <i>B</i> and <i>C</i>). Again, ask students if their pictures are identical. Remind students that all triangles have \(\overline{AB}\) and \(\overline{AC}\) of 7 and 8 cm, respectively, and ∠<i>A</i> of 40 degrees. Ask students, <strong>"How do we know that the triangles are congruent?"</strong> (<i>SAS</i>)</p><p>Now, ask students to measure \(\overline{BC}\), round to the nearest cm, and share their measurements. Students should discover that \(\overline{BC}\) has the same measure in each triangle. Use the following questions to introduce CPCTC:</p><figure class="image"><img style="aspect-ratio:651/284;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_23.png" width="651" height="284"></figure><p><strong>"Did you know that \(\overline{BC}\) would be the same in each triangle?"</strong> (<i>yes</i>)<br><strong>"How did \(\overline{BC}\) end up being the same when all students drew their triangles on their own?"</strong> (<i>the triangles were congruent by ASA</i>)&nbsp;<br><strong>"What does the definition of congruence mean about the corresponding parts of the triangles?"</strong> (<i>the corresponding parts are congruent</i>)<br>Following the discussion, write the following notes:</p><p><strong>CPCTC:</strong> <strong>C</strong>orresponding <strong>P</strong>arts of <strong>C</strong>ongruent <strong>T</strong>riangles are <strong>C</strong>ongruent</p><p><strong>Activity 2: Construction of Triangles with Limited Given Information</strong></p><p>Have one member from each group build a triangle. \(\overline{AB}\) and \(\overline{BC}\) should be 5 and 7 cm, respectively. Angle <i>A</i> (the angle between \(\overline{AB}\) and \(\overline{AC}\)) can be any angle the student chooses, and it should be constructed by connecting the endpoints of the 5- and 7-cm sides.</p><p>Once each student has constructed a triangle, ask him or her to measure \(\overline{BC}\) and record the measure. Encourage students to compare the length of \(\overline{BC}\) with their partner. Is the length of \(\overline{BC}\) the same? (Students can make comparisons with different groups.) Use the following questions to emphasize that you cannot use CPCTC until you have proven that two triangles are congruent:</p><p><strong>"Is the third side (\(\overline{BC}\)) of your triangle congruent to your partner's?"</strong> (<i>no</i>)&nbsp;<br><strong>"How did the construction of this triangle differ from the construction of the last triangle, in which everyone's \(\overline{BC}\) was congruent?"</strong> (<i>we could chose whatever angle we wanted</i>)&nbsp;<br><strong>"What did we know about the triangles in the first example?"</strong> (<i>they were congruent</i>)<br><strong>"Do we know that the triangles in this example are congruent?" </strong>(<i>no</i>)&nbsp;<br>Reinforce the idea that before students can apply CPCTC, they must first understand that the two triangles are congruent. However, after students have demonstrated that two triangles are congruent (preferably by applying one of the short-cut theorems), they will know that all pairs of corresponding parts of the congruent triangles are congruent.</p><p><strong>Activity 3: Construction of Triangles with Congruence Theorems</strong></p><p>Divide students into pairs. Each partner should choose a single congruence theorem (SSS, SAS, AAS, or ASA) and decide on the lengths of each side and the measure of each angle. (For example, if they chose AAS, they must determine the measurements of the two angles and the length of the third side.) Then, have students independently design a triangle using the information they chose and fill in the remaining information as they wish. When students have completed their triangles individually, ask if their triangle is congruent to their partner's triangle.</p><p>Follow up by questioning students, since they believe that their triangle will be congruent to their partner's,<strong> "What will be true about the sides and angles that you did not agree on together? Will the measurements be the same? How do you know?"</strong></p><p>Then, have students measure each part of their triangle (all three sides and angles) and compare the results to the measurements of their partner's triangle. Students should ensure that all of the measurements for the the corresponding parts are congruent.</p><p>Ask them to list the theorems they used to prove their triangles were congruent, as well as the theorems that prove the other pairs of corresponding parts of their triangles are congruent.</p><p>Each pair of students should choose two of the remaining triangle congruence theorems (SSS, SAS, ASA, or AAS) and repeat the previous activity, building congruent triangles and proving that the corresponding parts of the congruent triangle are congruent.</p><p><strong>Extension:</strong></p><p>Given two triangles, ∆<i>UVW</i> and ∆<i>XYZ</i>, without knowing any other facts. what is the least information requires to prove that ∆<i>UVW</i> is <strong>not</strong> congruent to ∆<i>XYZ</i>?&nbsp;<br>Answer: Any ONE of the following:</p><p>o &nbsp; &nbsp; &nbsp; &nbsp;\(\overline{UV}\) is not congruent to \(\overline{XY}\)<br>o &nbsp; &nbsp; &nbsp; &nbsp;\(\overline{VW}\) is not congruent to \(\overline{YZ}\)<br>o &nbsp; &nbsp; &nbsp; &nbsp;\(\overline{UW}\) is not congruent to \(\overline{XZ}\)<br>o &nbsp; &nbsp; &nbsp; &nbsp;∠<i>UVW</i> is not congruent to ∠<i>XYZ</i><br>o &nbsp; &nbsp; &nbsp; &nbsp;∠<i>VWU</i> is not congruent to ∠<i>YZX</i><br>o &nbsp; &nbsp; &nbsp; &nbsp;∠<i>WUV</i> is not congruent to ∠<i>ZXY</i></p>