<p>This lesson applies previous knowledge of angle and side measurement to the concept of proving triangle congruence. Students will:&nbsp;<br>- show congruence of triangle using the definition.<br>- show congruence of triangles using side-side-side (SSS) and introduce the concepts of side-angle-side (SAS), angle-angle-side (AAS), and angle-side-angle (ASA).</p>

<p>- What does it mean for two shapes to be congruent, and how can we quickly evaluate whether two shapes are congruent?</p>

<p>- Congruent: Having the same size and shape; congruent figures have corresponding sides and angles congruent.&nbsp;<br>- Congruent Triangles: Two or more triangles with congruent corresponding sides and angles; triangles that are identical; those with the same size and shape.<br>- Corresponding Angles: Angles in the same relative position in similar or congruent figures.<br>- Corresponding Sides: Sides in the same relative position in similar or congruent figures.<br>- Similar Triangles: Two or more triangles whose corresponding angles are congruent. Congruent triangles are also similar triangles.</p>

<p>- ribbon or string 12½ inches long, two per student<br>- standard-sized paper or larger<br>- markers and/or colored pencils<br>- protractors or templates<br>- ruler and pencil<br>- identical sets of popsicle sticks where each set has three popsicle sticks of different lengths, one or two sets per group<br>- Lesson 1 Exit Ticket and KEY (M-G-4-1_Lesson 1 Exit Ticket and KEY)</p>

<p>- When evaluating group activities, find the most specific and least necessary elements that make two triangles congruent. Ask students to identify some redundant attributes of two congruent triangles. For example, if two triangles have side-angle-side correspondence, they must also have side-side-side correspondence. Triangles will have corresponding side and corresponding angle congruence.&nbsp;<br>- Make observations during group activities and class discussions. Check specified measurements of side length and angle size. Make sure that students are comparing corresponding sides and angles, and emphasize that comparing non-corresponding sides and angles is pointless.&nbsp;<br>- Lesson 1 Exit Ticket demands specific identification of the attributes of congruent triangles, the identification of corresponding sides and angles, and the minimal requirements for congruence (M-G-4-1_Lesson 1 Exit Ticket and KEY).</p>

<p>Active Engagement, Modeling, Explicit Instruction<br>W: This lesson covers congruence, similarity, and related geometric properties. Following this lesson, students will grasp how to prove that triangles are congruent. Students study the definition of triangle congruence since it is necessary in real-world situations to identify lengths and angles in congruent objects.&nbsp;<br>H: In the first activity, students investigate the relationship between perimeter and congruence. Students use tangible objects (ribbons and rulers) to demonstrate that equal perimeters are necessary but insufficient to make for congruence.&nbsp;<br>E: In Activity 2, students measure side lengths and angles to determine to what degree side-angle-side congruence implies triangle congruence. The use of hand-drawn and hand-measured triangles fosters engagement with the requirements of triangle congruence.&nbsp;<br>R: In Activity 3, students apply their knowledge of side-angle-side congruence from Activity 2 to better grasp triangle congruence. Side-angle-side congruence occurs when the side opposite the angle between the two given sides is congruent to the corresponding side of the second triangle due to the angle congruence of the two corresponding angles.&nbsp;<br>E: Students are required to consider whether triangle congruence can be established by congruence if only one pair of corresponding sides compels them to develop one or more counterexamples to show the insufficiency of one pair of congruent corresponding sides. In this case, students may visualize for themselves the increase and decrease in the length of the third side as the two congruent corresponding sides open and close in accordance with the increase and decrease of the angle between them.&nbsp;<br>T: The unit includes small group and partner activity, such as pair-share, to foster collaboration and learning among students. Students can help each other find and understand mathematical concepts and vocabulary. Encourage students to apply their knowledge by creating, explaining, and solving real-world problems.&nbsp;<br>O: Small-group activities in Lesson 1 support varied learning styles. Visual aids and spatial relationships are demonstrated throughout the activities. The goal is to have a knowledge of the definition of congruence and apply it to triangles. First, students use a discovery approach to get an understanding of congruent triangles. Then, students devise methods to prove congruency using congruence theorems.</p>

<p><strong>Activity 1: Exploring Perimeter and its Relation to Congruence</strong></p><p><strong>Exploration: </strong>Students will learn that two triangles with equal perimeters do not always mean that the triangles are congruent.</p><p>Divide students into groups of three or four. Each group will create triangles from two similar lengths of string or ribbon.</p><p>Create two triangles with the same perimeter. Shift the vertex of one triangle to modify the position of the vertex without affecting the perimeter. Straws with flexible joints or chenille sticks are ideal for this. Then draw a triangle. On tracing paper, draw a second triangle that has the same perimeter but different side lengths. Overlay one triangle on the other and try to get an exact fit.</p><p>Each small group should be given two similar pieces of string or ribbon with their ends stapled together, as well as several pieces of paper large enough to make triangles based on the ribbon size. (Using 12½ inches of string or ribbon allows for a ¼-inch overlap to staple together; if this length is used, 8½ × 11 inch paper is sufficient.) Have three students hold one vertex of a triangle created by the ribbon, and the fourth student trace it on paper. Cut out each group's triangles and compare them to the other groups' to check if they are congruent.</p><p>Allow students to record their discoveries regarding triangles. Students should note that, while the perimeters of the triangles are the same, the lengths of particular sides and measure of angles may differ.</p><p>Ask students what is significant about the perimeter being the same when the sides and angles are frequently different. Students should understand that just because the perimeters are the same does not mean they are congruent triangles. Then ask the students what it would mean if the sides and angles were the same. Students should note that triangles with the same sides and angles are congruent.</p><p><strong>Activity 2: Exploring Congruence by Definition (and Implying SAS)</strong></p><p>To assist students with drawing congruent triangles on paper, present the following information on the board using an overhead projector:</p><p>Angle <i>A</i> measures 90 degrees.</p><p>Side <i>AB</i> should be 9 cm.</p><p>Side <i>AC</i> should be 12 cm.</p><p>Then, connect<i> B</i> and <i>C </i>to create \(\overline{BC}\).</p><p>Students should create a triangle similar to <i>ΔABC</i>, but label it <i>DEF</i>.</p><p>Steps for proving congruence:</p><p>1. Compare the angle measurements for the corresponding angles in <i>ΔABC</i> and <i>ΔDEF</i> and ensure that they are identical.</p><p>2. Check that the lengths of the corresponding sides in <i>ΔABC </i>and <i>ΔDEF</i> and ensure that they are identical.</p><p>3. Using the definition of congruent triangles, show that <i>ΔABC</i> is congruent to (≅) <i>ΔDEF</i>.</p><p>Encourage students to discuss their observations regarding the two triangles. Students should respond by stating that the triangles have 3 pairs of congruent corresponding sides and 3 pairs of congruent corresponding angles.</p><p>Ask students to identify which angle in <i>ΔDEF ≅ ∠A</i> in <i>ΔABC</i>. Introduce the term <i>corresponding</i> for <i>∠A</i> and <i>∠D</i>. Ask students which side in <i>ΔDEF</i> is congruent to \(\overline{BC}\) in <i>ΔABC</i>. Use the term <i>corresponding</i> to refer to \(\overline{BC}\) and \(\overline{EF}\). Write an example congruence statement for the triangles: <i>ΔABC ≅ ΔDEF</i>. Consider how the statement changes if you referred to the first triangle as <i>ΔCBA</i>.</p><p>After explaining the importance of corresponding parts in triangle congruence statements, have students write their own congruence statement for the two triangles and then exchange with a partner to confirm their statement.</p><p>Remind students that they demonstrated the two triangles were congruent by observing that each pair of corresponding angles and each pair of corresponding sides had the same measurement. Affirm the concept of corresponding in terms of the order in which the triangle congruence statements are written.</p><p><strong>Activity 3: Discovering SSS</strong></p><p>Give each group one or two pairs of popsicle sticks. You can also use straws with flexible joints, chenille sticks, and wooden skewers. Students should create a triangle out of each set of popsicle sticks, with the sticks meeting at the ends but not overlapping. Have students trace the inside edge of their triangles on paper and then cut them out. Students should next compare their triangles to see if they are congruent by stacking them on top of each other.</p><p>Ask students to raise their hands if they discovered someone with a triangle similar to the one they made. Students should be aware that they did not need to measure any angles for this activity and still ended up with congruent triangles. Provide the following information for students to transcribe into their notes:</p><p><strong>SSS: </strong>If two triangles have three pairs of congruent corresponding sides, then the two triangles must be congruent.</p><p>Tell students that this is a shortcut. Remind them of the definition of congruence and explain that only showing that three corresponding pairs of sides are congruent is enough to show that two triangles are congruent.</p><p>Ask students whether knowing that one pair of corresponding sides is congruent is sufficient to determine that the triangles are congruent. Continue by asking about one pair of corresponding angles. Use their findings to demonstrate that in order to show congruence, we must first know that the corresponding parts of two triangles are congruent, and that SSS is just one of many congruence theorems.</p><p>Point out that SSS is a true shortcut-we don't need to verify that everything in the two triangles is congruent. Students should use shortcuts wherever possible-they are an important aspect of geometric proofs.</p><p><strong>Extension:</strong></p><p>Given two right triangles, <i>ΔQRS</i> and <i>ΔVRT</i>, where <i>∠QRS </i>and <i>∠VRT</i> are both right angles, what further side or angle congruences are required to prove that <i>ΔQRS ≅ ΔVRT</i>?&nbsp;<br>Answer: <i>QR ≅ VR</i> and <i>RS ≅ RT (SAS)</i>. Students should also be aware that for right triangles, there is also a congruence condition known as the <i>SSA </i>(side-side-angle), which is <i>only</i> applicable to right triangles. We can show by counterexample that for non-right triangles, <i>SSA</i> congruence may not be sufficient for triangle congruence. The Hypotenuse Leg Congruence Theorem applies to right triangles: If the hypotenuse and one leg of one right triangle match the hypotenuse and one leg of the other, the two triangles are congruent.</p>