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4.4 Prove Triangles Congruent by SAS and HL
Side-Angle-Side (SAS) Congruence Postulate
• If two sides and the included angle are congruent to the corresponding sides and angles on another triangle, then the triangles are congruent.
EXAMPLE 1 Use the SAS Congruence Postulate
Write a proof.
GIVEN
PROVE
STATEMENTS REASONS
BC DA, BC AD
ABC CDA
1. Given1. BC DAS
Given2. 2. BC AD
3. BCA DAC 3. Alternate Interior Angles Theorem
A
4. 4. AC CA Reflexive Property of Congruence
S
Extra Example 1
EXAMPLE 2 Use SAS and properties of shapes
In the diagram, QS and RP pass through the center M of the circle. What can you conclude about MRS and MPQ?
SOLUTION
Because they are vertical angles, PMQ RMS. All points on a circle are the same distance from the center, so MP, MQ, MR, and MS are all equal.
MRS and MPQ are congruent by the SAS Congruence Postulate.
ANSWER
GUIDED PRACTICE for Examples 1 and 2
In the diagram, ABCD is a square with four congruent sides and four right angles. R, S, T, and U are the midpoints of the sides of ABCD. Also, RT SU and .SU VU
1. Prove that SVR UVR
STATEMENTS REASONS
1. SV VU 1. Given
3. 3. RV VR Reflexive Property of Congruence
2. 2. SVR RVU Definition of line
4. 4. SVR UVR SAS Congruence Postulate
GUIDED PRACTICE for Examples 1 and 2
2. Prove that BSR DUT
STATEMENTS REASONS
1. 1. GivenBS DU
2. 2. RBS TDU Definition of line
3. 3. RS UT Given
4. 4. BSR DUT SAS Congruence Postulate
Key Ideas for Proving SAS
• Use the given- often sides are already given in a direct (BC is congruent to DA) or in an indirect way (ABCD is a square)
• Think about what the given means- midpoint- divides a segment into two congruent partsParallel Lines- Look for alternate interior or other
relationships we’ve discussedPerpendicular- Automatically have right angles
HL
• All right triangles have two legs and one hypotenuse.
• To prove these triangles congruent- the hypotenuse and a leg for two different triangles have to be congruent to each other.
EXAMPLE 3 Use the Hypotenuse-Leg Congruence Theorem
Write a proof.
SOLUTION
Redraw the triangles so they are side by side with corresponding parts in the same position. Mark the given information in the diagram.
GIVEN WY XZ, WZ ZY, XY ZY
PROVE WYZ XZY
GUIDED PRACTICE for Examples 3 and 4
Use the diagram at the right.
3. Redraw ACB and DBC side by side with corresponding parts in the same position.
GUIDED PRACTICE for Examples 3 and 4
4.
Use the diagram at the right.
Use the information in the diagram to prove that ACB DBC
STATEMENTS REASONS
1. AC DB 1. Given
2. 2. AB BC, CD BC Given
4. 4. Definition of a right triangle
ACB and DBC are right triangles.
3. 3. Definition of linesC B
Questions to ask When Deciding Which Postulate to Use
• Can I see that all sides are going to be congruent? SSS
• Do I have congruent hypotenuses- have to have a right angle to have a hypotenuse- can I show the legs congruent? HL
• Do I have an angle congruent in both? Is it in between two sides that are congruent or I can show they’re congruent? SAS
EXAMPLE 4 Choose a postulate or theorem
Sign Making
You are making a canvas sign to hang on the triangular wall over the door to the barn shown in the picture. You think you can use two identical triangular sheets of canvas. You know that RP QS and PQ PS . What postulate or theorem can you use to conclude that PQR PSR?
Daily Homework Quiz
For use after Lesson 4.4
Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem.
1. ABE, CBD
ANSWER SAS Post.
Daily Homework Quiz
For use after Lesson 4.4
Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem.
2. FGH, HJK
ANSWER HL Thm.
Daily Homework Quiz
For use after Lesson 4.4
State a third congruence that would allow you to prove RST XYZ by the SAS Congruence postulate.
3. ST YZ, RS XY
ANSWER S Y.
Daily Homework Quiz
For use after Lesson 4.4
State a third congruence that would allow you to prove RST XYZ by the SAS Congruence postulate.
ANSWER ST YZ .
4. T Z, RT XZ
Homework
• 4.4: 1, 2-18ev, 20 – 23, 25 – 29, 34 – 38
