Chapter 4 4-5 congruent triangle : SSS and SAS

SAT Problem of the day

ObjectivesApply 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.

SAS Congruence

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

Proofs Given: BC ║ AD, BC AD Prove: ∆ABD ∆CDB

ReasonsStatements

5. SAS Steps 3, 2, 45. ∆ABD ∆ CDB

4. Reflex. Prop. of

3. Given

2. Alt. Int. s Thm.2. CBD ABD

1. Given1. BC || AD

3. BC AD

4. BD BD

Proofs Given: QP bisects RQS. QR QS Prove: ∆RQP ∆SQP

ReasonsStatements

5. SAS Steps 1, 3, 45. ∆RQP ∆SQP

4. Reflex. Prop. of

1. Given

3. Def. of bisector3. RQP SQP

2. Given2. QP bisects RQS

1. QR QS

4. QP QP

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

Have a great day