You will use sides and angles to prove congruence. Essential Question: How can you use two sides and...
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Warm-Up Exercises • You will use sides and angles to prove congruence. • Essential Question: How can you use two sides and an angle to prove triangles congruent? 4.5 Prove Triangles Congruent by SAS and HL predict You will learn how to answer this question by using the SAS Post. and the HL Thm.
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Transcript of You will use sides and angles to prove congruence. Essential Question: How can you use two sides and...
• You will use sides and angles to prove congruence.
• Essential Question: How can you use two sides and an angle to prove triangles congruent?
4.5 Prove Triangles Congruent by SAS and HL
predict
You will learn how toanswer this question by using theSAS Post. and the HL Thm.
Warm-Up Exercises
Prove: ∆CDF ∆EDF
Given: DF bisects CE, DC DE
C
F
E
D
DF bisects CE
DC DE
CF EF
def. of bisector
given
given
DF DF Refl. Prop. of Segs
∆CDF ∆ EDF
SSS
Warm-Up ExercisesEXAMPLE 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
Warm-Up ExercisesEXAMPLE 1 Use the SAS Congruence Postulate
STATEMENTS REASONS
5. ABC CDA SAS Congruence Postulate
5.
Warm-Up ExercisesEXAMPLE 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 Is there only one way to match thevertices to get a true congruencestatement?
Warm-Up ExercisesGUIDED 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
Warm-Up ExercisesGUIDED 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
Warm-Up ExercisesEXAMPLE 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
Warm-Up Exercises
STATEMENTS REASONS
EXAMPLE 3 Use the Hypotenuse-Leg Congruence Theorem
1. WY XZ 1. Given
4. 4. Definition of a right triangle
WYZ and XZY are right triangles.
L ZY YZ5. 5. Reflexive Property of Congruence
6. WYZ XZY 6. HL Congruence Theorem
3. 3. Definition of linesZ and Y are right angles
2. 2. WZ ZY, XY ZY Given
Warm-Up ExercisesEXAMPLE 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?
Warm-Up ExercisesEXAMPLE 4 Choose a postulate or theorem
SOLUTION
RPQ and RPS are right angles, so they are congruent. So, two sides and their included angle are congruent.
You are given that PQ PS . By the Reflexive Property, RP RP . By the definition of perpendicular lines, both
You can use the SAS Congruence Postulate to conclude that .PQR PSR
ANSWER
Warm-Up ExercisesGUIDED 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.
Warm-Up ExercisesGUIDED PRACTICE for Examples 3 and 4
STATEMENTS REASONS
L BC CB5. 5. Reflexive Property of Congruence
6. ACB DBC 6. HL Congruence Theorem
Warm-Up ExercisesGUIDED 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
Warm-Up ExercisesDaily Homework Quiz
Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem.
1. ABE, CBD
ANSWER SAS Post.
Warm-Up ExercisesDaily Homework Quiz
Is there enough given information to prove the triangles congruent? If there is, state the postulate or theorem.
2. FGH, HJK
ANSWER HL Thm.
Warm-Up ExercisesDaily Homework Quiz
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.
Warm-Up ExercisesDaily Homework Quiz
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
• You will use sides and angles to prove congruence.
• Essential Question: How can you use two sides and an angle to prove triangles congruent?
• Triangles are congruent by the SAS Congruence Postulate.• Right triangles are congruent by the HL Congruence Theorem.
You can prove triangles congruent if you know that two sides and the included angle of one triangle are congruent to two sides and the included angle of the other. If the triangles are right triangles, you can prove them congruent if they have congruent hypotenuses and apair of congruent legs.
