24
4-2 Triangle Congruence by SSS and SAS Do Now Lesson Presentation Exit Ticket
4-2 Triangle Congruence by SSS and SAS
Do Now
Lesson Presentation
Exit Ticket
4-2 Triangle Congruence by SSS and SAS
Warm Up #1
1. Name the angle formed by AB and AC.
2. Name the three sides of ABC.
3. ∆QRS ∆LMN. Name all pairs of congruent corresponding parts.
QR LM, RS MN, QS LN, Q L, R M, S N
AB, AC, BC
Possible answer: A, BAC, CAB
4-2 Triangle Congruence by SSS and SAS
Know: Solve It !
y◦
𝐿𝑀 ≅ 𝐿𝑃,𝑀𝑁 ≅ 𝑃𝑁, 𝐿𝑁 ≅ 𝐿𝑁; M ≅P, MLN ≅PNL, MNL ≅PLN
Given: ∆LMN ∆LPN
What can you conclude about the corresponding sides and angles?
4-2 Triangle Congruence by SSS and SAS
Connect Mathematical Ideas (1)(F)
How does this problem relate to a problem you have seen before ?
Communicate
4-2 Triangle Congruence by SSS and SAS
Connect to Math
SWBAT
1. Apply SSS and SAS to construct triangles and solve problems.
2. Prove triangles congruent by using SSS and SAS.
By the end of today’s lesson,
4-2 Triangle Congruence by SSS and SAS
triangle rigidity
included angle
Vocabulary
4-2 Triangle Congruence by SSS and SAS
In Lesson 4-1 Congruent Figures, you proved triangles congruent by showing that all six pairs of corresponding parts were congruent.
4-2 Triangle Congruence by SSS and SAS
Building Triangles
A.Compare your triangle with your classmates.
• The straw’s lengths of 3 in., 5 in., and 6 in.
• Thread a string through the three pieces of straw, in any order, as shown.
• Bring the ends of the string together and tie them to hold your triangle in place.
4-2 Triangle Congruence by SSS and SAS
Building Triangles
B.Make a conjecture about two triangles in which three sides of one triangle are congruent to three sides of the other triangle.
• Both triangles have the same size and shape.
• Each triangle fits exactly on top of the other triangle.
Conjecture: Triangles with congruent corresponding sides are congruent.
4-2 Triangle Congruence by SSS and SAS
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.
4-2 Triangle Congruence by SSS and SAS
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.
4-2 Triangle Congruence by SSS and SAS
Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts.
Remember!
4-2 Triangle Congruence by SSS and SAS
Example 1: Using SSS to Prove Triangle Congruence
It is given that 𝐴𝐶 𝐷𝐶 and that 𝐴𝐵 𝐷𝐵.
By the Reflexive Property of Congruence, 𝐵𝐶 𝐵𝐶.
Therefore, ∆ABC ∆DBC by SSS.
Use SSS to explain why ∆ABC ∆DBC.
4-2 Triangle Congruence by SSS and SAS
Example 2
Use SSS to explain why ∆ABC ∆CDA.
It is given that 𝐴𝐵 𝐶𝐷 and 𝐵𝐶 𝐷𝐴.
By the Reflexive Property of Congruence, 𝐴𝐶 𝐶𝐴.
So ∆ABC ∆CDA by SSS.
4-2 Triangle Congruence by SSS and SAS
An included angle is an angle formed by two adjacent sides of a polygon.
B is the included angle between sides 𝐴𝐵 and 𝐵𝐶.
4-2 Triangle Congruence by SSS and SAS
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.
4-2 Triangle Congruence by SSS and SAS
4-2 Triangle Congruence by SSS and SAS
The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides.
Caution
4-2 Triangle Congruence by SSS and SAS
Example 3: Engineering Application
The diagram shows part of the support structure for a tower. Use SAS to explain why ∆XYZ ∆VWZ.
It is given that 𝑋𝑍 𝑉𝑍 and that 𝑌𝑍 𝑊𝑍.
By the Vertical s Theorem, XZY VZW.
Therefore, ∆XYZ ∆VWZ by SAS.
4-2 Triangle Congruence by SSS and SAS
Example 4
Use SAS to explain why ∆ABC ∆DBC.
It is given that 𝐵𝐴 𝐵𝐷 and ABC DBC.
By the Reflexive Property of , 𝐵𝐶 𝐵𝐶.
So ∆ABC ∆DBC by SAS.
4-2 Triangle Congruence by SSS and SAS
The SAS Postulate guarantees that if you are given the lengths of two sides and the measure of the included angles, you can construct one and only one triangle.
4-2 Triangle Congruence by SSS and SAS
Given: BC║ AD, 𝑩𝑪 𝑨𝑫
Prove: ∆ABD ∆CDB
Example 5: Proving Triangles Congruent
Statements Reasons
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
4-2 Triangle Congruence by SSS and SAS
Example 6
Given: 𝑄𝑃 bisects RQS. 𝑄𝑅 𝑄𝑆
Prove: ∆RQP ∆SQPStatements Reasons
5. SAS Steps 1, 3, 45. ∆RQP ∆SQP4. Reflex. Prop. of
1. Given
3. Def. of bisector3. RQP SQP
2. Given2. 𝑄𝑃 bisects RQS
1. 𝑄𝑅 𝑄𝑆
4. 𝑄𝑃 𝑄𝑃
4-2 Triangle Congruence by SSS and SAS
Exit Ticket:
Which postulate, if any, can be used to prove the triangles congruent?
1. 2.
3. Given: 𝑃𝑁 bisects 𝑀𝑂, 𝑃𝑁 𝑀𝑂
Prove: ∆MNP ∆ONP
