Triangle Congruence Word Problems: SSS and SAS 1. Maria is designing a triangular support beam for a bridge project. She has two triangular metal pieces that she needs to verify are congruent before welding them together. The first triangle has sides measuring inches, inches, and inches. The second triangle also has sides of inches, inches, and inches. Can Maria conclude these triangles are congruent? If so, draw the congruent angles, label them and explain your answer. 2. An architect is designing two triangular windows for a modern house. In the first window, two sides measure feet and feet, with the included angle measuring degrees. The second window has sides of feet and feet, with an included angle of degrees. Are these windows congruent? Which congruence criterion would prove this? Explain your reasoning. 3. In a robotics competition, teams must construct identical triangular support brackets. Team A's bracket has sides measuring centimeters, centimeters, and centimeters. Team B claims their bracket is congruent to Team A's, showing that their bracket also has sides of centimeters, centimeters, and centimeters. Can the teams prove their brackets are congruent? Which criterion would they use? Draw, label and explain how they are congruent.

It appears that some numerical values are missing from the problem statements you've provided. To accurately solve each of these triangle congruence word problems using the **Side-Side-Side (SSS)** or **Side-Angle-Side (SAS)** criteria, specific measurements for the sides and angles are essential. However, I can guide you on how to approach each problem once the necessary information is available. --- ### **1. Maria's Triangular Support Beams** **Given:** - **First Triangle:** Sides of lengths *a*, *b*, and *c* inches. - **Second Triangle:** Sides of lengths *d*, *e*, and *f* inches. **Approach:** - **SAS Criterion:** To use the SAS (Side-Angle-Side) congruence criterion, you need two pairs of equal sides and the angle between them to be equal. - **SSS Criterion:** To use the SSS (Side-Side-Side) congruence criterion, all three pairs of corresponding sides must be equal. **Steps:** 1. **Verify Side Lengths:** - Check if *a = d*, *b = e*, and *c = f*. - If all three sides are equal, then by **SSS**, the triangles are congruent. 2. **If Only Two Sides and Included Angle Are Equal:** - Identify the two pairs of equal sides and measure the included angles. - If two sides and the included angle are equal, then by **SAS**, the triangles are congruent. 3. **Label Congruent Angles:** - Once congruence is established, corresponding angles of the congruent triangles are equal. Label these angles accordingly. **Conclusion:** - If the side lengths satisfy **SSS** or the relevant sides and angle satisfy **SAS**, Maria can conclude that the two triangular metal pieces are congruent. --- ### **2. Architect's Triangular Windows** **Given:** - **First Window:** - Two sides measuring *m* and *n* feet. - Included angle of *θ* degrees. - **Second Window:** - Two sides measuring *p* and *q* feet. - Included angle of *φ* degrees. **Approach:** - Use the **SAS Criterion** since two sides and the included angle are given for both triangles. **Steps:** 1. **Compare Side Lengths:** - Check if *m = p* and *n = q*. 2. **Compare Included Angles:** - Check if *θ = φ*. 3. **Apply SAS:** - If both pairs of sides and the included angles are equal, then by **SAS**, the triangles (windows) are congruent. **Conclusion:** - If **SAS** conditions are met, the two triangular windows are congruent. --- ### **3. Robotics Competition Brackets** **Given:** - **Team A's Bracket:** - Sides measuring *x*, *y*, and *z* centimeters. - **Team B's Bracket:** - Sides measuring *x'*, *y'*, and *z'* centimeters. **Approach:** - Use the **SSS Criterion** to determine congruence based on all three side lengths. **Steps:** 1. **Compare All Side Lengths:** - Ensure that *x = x'*, *y = y'*, and *z = z'*. 2. **Apply SSS:** - If all three corresponding sides are equal, then by **SSS**, the brackets are congruent. 3. **Draw and Label:** - Sketch both brackets. - Label corresponding sides and angles to illustrate congruence. - Highlight that all corresponding sides and angles are equal based on **SSS**. **Conclusion:** - Using **SSS**, the teams can prove that their brackets are congruent if all corresponding side lengths are equal. --- ### **General Tips for Solving Congruence Problems:** 1. **Identify the Given Information:** - Determine which sides and angles are provided. 2. **Choose the Appropriate Congruence Criterion:** - **SSS:** All three sides are equal. - **SAS:** Two sides and the included angle are equal. - **ASA (Angle-Side-Angle):** Two angles and the included side are equal. - **AAS (Angle-Angle-Side):** Two angles and a non-included side are equal. - **HL (Hypotenuse-Leg for Right Triangles):** Hypotenuse and one leg are equal. 3. **Verify the Conditions:** - Check if the given measurements satisfy one of the criteria. 4. **Conclude Congruence:** - If a criterion is satisfied, the triangles are congruent. 5. **Draw and Label Diagrams:** - Visual representations can aid in understanding and proving congruence. --- If you can provide the specific measurements for each problem, I’d be happy to offer a more detailed solution!