Geometry Section 4-5 1112
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Transcript of Geometry Section 4-5 1112
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SECTION 4-5Proving Congruence: ASA, AAS
Wednesday, February 8, 2012
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ESSENTIAL QUESTIONS
How do you use the ASA Postulate to test for triangle congruence?
How do you use the AAS Postulate to test for triangle congruence?
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VOCABULARY1. Included Side:
Postulate 4.3 - Angle-Side-Angle (ASA) Congruence:
Theorem 4.5 - Angle-Angle-Side (AAS) Congruence:
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VOCABULARY1. Included Side: The side between two consecutive
angles in a triangle
Postulate 4.3 - Angle-Side-Angle (ASA) Congruence:
Theorem 4.5 - Angle-Angle-Side (AAS) Congruence:
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VOCABULARY1. Included Side: The side between two consecutive
angles in a triangle
Postulate 4.3 - Angle-Side-Angle (ASA) Congruence: If two angles and the included side of one triangle are congruent to two angles and included side of a second triangle, then the triangles are congruent
Theorem 4.5 - Angle-Angle-Side (AAS) Congruence:
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VOCABULARY1. Included Side: The side between two consecutive
angles in a triangle
Postulate 4.3 - Angle-Side-Angle (ASA) Congruence: If two angles and the included side of one triangle are congruent to two angles and included side of a second triangle, then the triangles are congruent
Theorem 4.5 - Angle-Angle-Side (AAS) Congruence: If two angles and the nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of a second triangle, then the triangles are congruent
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WHAT ABOUT SSA?
Try drawing a triangle with two sides that are 5 inches and 3 inches. Then, draw a nonincluded angle that is 30°. See
how many different triangles you can draw.
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EXAMPLE 1Prove the following.
Prove: WRL ≅EDLGiven: L is the midpoint of WE, WR ED
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EXAMPLE 1Prove the following.
Prove: WRL ≅EDLGiven: L is the midpoint of WE, WR ED
1. L is the midpoint of WE, WR ED
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EXAMPLE 1Prove the following.
Prove: WRL ≅EDL
1. Given
Given: L is the midpoint of WE, WR ED
1. L is the midpoint of WE, WR ED
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EXAMPLE 1Prove the following.
Prove: WRL ≅EDL
1. Given
Given: L is the midpoint of WE, WR ED
2. WL ≅ EL
1. L is the midpoint of WE, WR ED
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EXAMPLE 1Prove the following.
Prove: WRL ≅EDL
1. Given
Given: L is the midpoint of WE, WR ED
2. WL ≅ EL 2. Def. of midpoint
1. L is the midpoint of WE, WR ED
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EXAMPLE 1Prove the following.
Prove: WRL ≅EDL
1. Given
Given: L is the midpoint of WE, WR ED
2. WL ≅ EL 2. Def. of midpoint3. ∠WLR ≅ ∠ELD
1. L is the midpoint of WE, WR ED
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EXAMPLE 1Prove the following.
Prove: WRL ≅EDL
1. Given
Given: L is the midpoint of WE, WR ED
2. WL ≅ EL 2. Def. of midpoint3. ∠WLR ≅ ∠ELD 3. Vertical Angles
1. L is the midpoint of WE, WR ED
Wednesday, February 8, 2012
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EXAMPLE 1Prove the following.
Prove: WRL ≅EDL
1. Given
Given: L is the midpoint of WE, WR ED
2. WL ≅ EL 2. Def. of midpoint3. ∠WLR ≅ ∠ELD 3. Vertical Angles4. ∠LWR ≅ ∠LED
1. L is the midpoint of WE, WR ED
Wednesday, February 8, 2012
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EXAMPLE 1Prove the following.
Prove: WRL ≅EDL
1. Given
Given: L is the midpoint of WE, WR ED
2. WL ≅ EL 2. Def. of midpoint3. ∠WLR ≅ ∠ELD 3. Vertical Angles4. ∠LWR ≅ ∠LED 4. Alternate Interior Angles
1. L is the midpoint of WE, WR ED
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EXAMPLE 1Prove the following.
Prove: WRL ≅EDL
1. Given
Given: L is the midpoint of WE, WR ED
2. WL ≅ EL 2. Def. of midpoint3. ∠WLR ≅ ∠ELD 3. Vertical Angles4. ∠LWR ≅ ∠LED 4. Alternate Interior Angles5. WRL ≅EDL
1. L is the midpoint of WE, WR ED
Wednesday, February 8, 2012
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EXAMPLE 1Prove the following.
Prove: WRL ≅EDL
1. Given
Given: L is the midpoint of WE, WR ED
2. WL ≅ EL 2. Def. of midpoint3. ∠WLR ≅ ∠ELD 3. Vertical Angles4. ∠LWR ≅ ∠LED 4. Alternate Interior Angles5. WRL ≅EDL 5. ASA
1. L is the midpoint of WE, WR ED
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EXAMPLE 2Prove the following.
Prove: LN ≅ MNGiven: ∠NKL ≅ ∠NJM, KL ≅ JM
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EXAMPLE 2Prove the following.
Prove: LN ≅ MNGiven: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM
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EXAMPLE 2Prove the following.
Prove: LN ≅ MNGiven: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
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EXAMPLE 2Prove the following.
Prove: LN ≅ MNGiven: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
2. ∠JNK ≅ ∠KNJ
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EXAMPLE 2Prove the following.
Prove: LN ≅ MNGiven: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
2. ∠JNK ≅ ∠KNJ 2. Reflexive
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EXAMPLE 2Prove the following.
Prove: LN ≅ MNGiven: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
2. ∠JNK ≅ ∠KNJ 2. Reflexive
3. JNM ≅KNL
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EXAMPLE 2Prove the following.
Prove: LN ≅ MNGiven: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
2. ∠JNK ≅ ∠KNJ 2. Reflexive
3. JNM ≅KNL 3. AAS
Wednesday, February 8, 2012
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EXAMPLE 2Prove the following.
Prove: LN ≅ MNGiven: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
2. ∠JNK ≅ ∠KNJ 2. Reflexive
3. JNM ≅KNL 3. AAS
4. LN ≅ MN
Wednesday, February 8, 2012
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EXAMPLE 2Prove the following.
Prove: LN ≅ MNGiven: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
2. ∠JNK ≅ ∠KNJ 2. Reflexive
3. JNM ≅KNL 3. AAS
4. LN ≅ MN 4. CPCTC
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EXAMPLE 3On a template design for a certain envelope, the top
and bottom flaps are isosceles triangles with congruent bases and base angles. If EV = 8 cm and the height of
the isosceles triangle is 3 cm, find PO.
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EXAMPLE 3On a template design for a certain envelope, the top
and bottom flaps are isosceles triangles with congruent bases and base angles. If EV = 8 cm and the height of
the isosceles triangle is 3 cm, find PO.
EV ≅ PL, so each segment has a measure of 8 cm. If an auxiliary line is drawn
from point O perpendicular to PL, you will have a right triangle formed.
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EXAMPLE 3On a template design for a certain envelope, the top
and bottom flaps are isosceles triangles with congruent bases and base angles. If EV = 8 cm and the height of
the isosceles triangle is 3 cm, find PO.
EV ≅ PL, so each segment has a measure of 8 cm. If an auxiliary line is drawn
from point O perpendicular to PL, you will have a right triangle formed.
In the right triangle, we have one leg (the height) of 3 cm. The auxiliary line will bisect PL, as point O is equidistant
from P and L.
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EXAMPLE 3
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EXAMPLE 3
a2 + b2 = c2
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EXAMPLE 3
a2 + b2 = c2
42 + 32 = (PO)2
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EXAMPLE 3
a2 + b2 = c2
42 + 32 = (PO)2
16 + 9 = (PO)2
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EXAMPLE 3
a2 + b2 = c2
42 + 32 = (PO)2
16 + 9 = (PO)2
25 = (PO)2
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EXAMPLE 3
a2 + b2 = c2
42 + 32 = (PO)2
16 + 9 = (PO)2
25 = (PO)2
25 = (PO)2
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EXAMPLE 3
a2 + b2 = c2
42 + 32 = (PO)2
16 + 9 = (PO)2
25 = (PO)2
25 = (PO)2
PO = 5 cm
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CHECK YOUR UNDERSTANDING
Review p. 276 #1-5
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PROBLEM SET
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PROBLEM SET
p. 277 #7-23 odd, 35
“There is only one you...Don’t you dare change just because you’re outnumbered.” - Charles Swindoll
Wednesday, February 8, 2012