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Proving Congruence: ASA, AAS
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SECTION 4-5Proving Congruence: ASA, AAS
Wednesday, February 8, 2012
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:
Wednesday, February 8, 2012
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:
Wednesday, February 8, 2012
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
Wednesday, February 8, 2012
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.
Wednesday, February 8, 2012
EXAMPLE 1Prove the following.
Prove: WRL ≅EDLGiven: L is the midpoint of WE, WR ED
Wednesday, February 8, 2012
EXAMPLE 1Prove the following.
Prove: WRL ≅EDLGiven: L is the midpoint of WE, WR ED
1. L is the midpoint of WE, WR ED
Wednesday, February 8, 2012
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
Wednesday, February 8, 2012
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
Wednesday, February 8, 2012
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
Wednesday, February 8, 2012
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
Wednesday, February 8, 2012
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
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
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
Wednesday, February 8, 2012
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
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
Wednesday, February 8, 2012
EXAMPLE 2Prove the following.
Prove: LN ≅ MNGiven: ∠NKL ≅ ∠NJM, KL ≅ JM
Wednesday, February 8, 2012
EXAMPLE 2Prove the following.
Prove: LN ≅ MNGiven: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM
Wednesday, February 8, 2012
EXAMPLE 2Prove the following.
Prove: LN ≅ MNGiven: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
Wednesday, February 8, 2012
EXAMPLE 2Prove the following.
Prove: LN ≅ MNGiven: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
2. ∠JNK ≅ ∠KNJ
Wednesday, February 8, 2012
EXAMPLE 2Prove the following.
Prove: LN ≅ MNGiven: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
2. ∠JNK ≅ ∠KNJ 2. Reflexive
Wednesday, February 8, 2012
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
Wednesday, February 8, 2012
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
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
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
Wednesday, February 8, 2012
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.
Wednesday, February 8, 2012
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.
Wednesday, February 8, 2012
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.
Wednesday, February 8, 2012
EXAMPLE 3
Wednesday, February 8, 2012
EXAMPLE 3
a2 + b2 = c2
Wednesday, February 8, 2012
EXAMPLE 3
a2 + b2 = c2
42 + 32 = (PO)2
Wednesday, February 8, 2012
EXAMPLE 3
a2 + b2 = c2
42 + 32 = (PO)2
16 + 9 = (PO)2
Wednesday, February 8, 2012
EXAMPLE 3
a2 + b2 = c2
42 + 32 = (PO)2
16 + 9 = (PO)2
25 = (PO)2
Wednesday, February 8, 2012
EXAMPLE 3
a2 + b2 = c2
42 + 32 = (PO)2
16 + 9 = (PO)2
25 = (PO)2
25 = (PO)2
Wednesday, February 8, 2012
EXAMPLE 3
a2 + b2 = c2
42 + 32 = (PO)2
16 + 9 = (PO)2
25 = (PO)2
25 = (PO)2
PO = 5 cm
Wednesday, February 8, 2012
CHECK YOUR UNDERSTANDING
Review p. 276 #1-5
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PROBLEM SET
Wednesday, February 8, 2012
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
