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In this lesson, we explore the principles of SSS (Side-Side-Side) and SAS (Side-Angle-Side) congruence criteria for triangles. We'll prove triangles are congruent by using these methods and apply them to solve related problems. Through examples, such as explaining congruence between triangles through shared sides and angles, students will learn how triangle rigidity ensures that a given set of side lengths leads to a unique triangle. Guided practice and homework will reinforce these concepts, enhancing problem-solving skills in geometry.
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Chapter 4 4-5 congruent triangle : SSS and SAS
Objectives Apply SSS and SAS to construct triangles and solve problems. Prove triangles congruent by using SSS and SAS.
Congruent triangles • In Lessons 4-3 and 4-4, you proved triangles congruent by showing that all six pairs of corresponding parts were congruent.
Triangle Rigidity • The property of triangle rigidity gives you a shortcut for proving two triangles congruent. It states that if the side lengths of a triangle are given, the triangle can have only one shape.
SSS congruence • For example, you only need to know that two triangles have three pairs of congruent corresponding sides. This can be expressed as the following postulate.
Example#1 • Use SSS to explain why ∆ABC ∆DBC. • Solution: It is given that AC DC and that AB DB. By the Reflexive Property of Congruence, BC BC. Therefore ∆ABC ∆DBC by SSS.
Example#2 • Use SSS to explain why • ∆ABC ∆CDA. • Solution: It is given that AB CD and BC DA. • By the Reflexive Property of Congruence, AC CA. • So ∆ABC ∆CDA by SSS.
Student guided practice • Do problems 2 and 3 in your book page 253.
Included Angle An included angle is an angle formed by two adjacent sides of a polygon. B is the included angle between sides AB and BC.
SAS Congruence • It can also be shown that only two pairs of congruent corresponding sides are needed to prove the congruence of two triangles if the included angles are also congruent.
Example#3 • The diagram shows part of the support structure for a tower. Use SAS to explain why ∆XYZ ∆VWZ. • Solution: • It is given that XZ VZ and that YZ WZ. By the Vertical s Theorem. XZY VZW. Therefore ∆XYZ ∆VWZ by SAS.
Example#4 • Use SAS to explain why ∆ABC ∆DBC. • Solution: It is given that BA BD and ABC DBC. By the Reflexive Property of , BC BC. So ∆ABC ∆DBC by SAS.
Student guided practice • Do problem 4 in your book page 253
Example#5 • Show that the triangles are congruent for the given value of the variable. • ∆MNO ∆PQR, when x = 5. • ∆MNO ∆PQR by SSS.
Example#6 • Show that the triangles are congruent for the given value of the variable. • ∆STU ∆VWX, when y = 4. ∆STU ∆VWX by SAS.
Student guided practice • Do problems 5 and 6 in your book page 253
1.BC || AD 3. BC AD 4. BD BD Proofs • Given: BC║ AD, BC AD • Prove: ∆ABD ∆CDB Statements Reasons 1. Given 2. CBD ABD 2. Alt. Int. s Thm. 3. Given 4. Reflex. Prop. of 5.∆ABD ∆CDB 5. SAS Steps 3, 2, 4
2.QP bisects RQS 1. QR QS 4. QP QP Proofs • Given: QP bisects RQS. QR QS • Prove: ∆RQP ∆SQP Statements Reasons 1. Given 2. Given 3. RQP SQP 3. Def. of bisector 4. Reflex. Prop. of 5.∆RQP ∆SQP 5. SAS Steps 1, 3, 4
Student guided practice • Do problem 7 in your book page 253
Homework • Do problems 8 to 13 in your book page 254
Closure • Today we learned about triangle congruence by SSS and SAS. • Next class we are going to continue learning about triangle congruence
