Download Congruence Criteria for Triangles—SAA and HL

Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
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Lesson 25: Congruence Criteria for Triangles—SAA and HL
Student Outcomes
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Students learn why any two triangles that satisfy the SAA or HL congruence criteria must be congruent.
Students learn why any two triangles that meet the AAA or SSA criteria are not necessarily congruent.
Classwork
Opening Exercise (7 minutes)
Write a proof for the following question. Once done, compare your proof with that of a neighbor’s.
Given: ࡰࡱ = ࡰࡳ, ࡱࡲ = ࡳࡲ
Prove: ࡰࡲ is the angle bisector of ‫ࡳࡰࡱס‬
Proof:
ࡰࡱ = ࡰࡳ
Given
Common side
ࡱࡲ = ࡳࡲ
Given
ࡰࡲ = ࡰࡲ
ᇞ ࡰࡱࡲ =ᇞ ࡰࡳࡲ
SSS
‫ࡲࡰࡳס = ࡲࡰࡱס‬
ࡰࡲ is the angle bisector of ‫ࡳࡰࡱס‬
Corr. ‫ס‬s of ؆ ᇞs
Defn. of angle bisector
Discussion (25 minutes)
Today we are going to examine three possible triangle congruence criteria, Side-Angle-Angle (SAA) and Side-Side-Angle
(SSA), and Angle-Angle-Angle (AAA). Ultimately, only one of the three possible criteria will actually ensure congruence.
Side-Angle-Angle triangle congruence criteria (SAA): Given two triangles ࡭࡮࡯ and ࡭Ԣ࡮Ԣ࡯Ԣ. If ࡭࡮ = ࡭Ԣ࡮Ԣ (Side),
‫࡮ס = ࡮ס‬Ԣ (Angle), and ‫࡯ס = ࡯ס‬Ԣ (Angle), then the triangles are congruent.
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Congruence Criteria for Triangles—SAA and HL
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Proof
Consider a pair of triangles that meet the SAA criteria. If you knew that two angles of one triangle corresponded to and
were equal in measure to two angles of the other triangle, what conclusions can you draw about the third angles of each
triangle?
Since the first two angles are equal in measure, the third angles must also be equal in measure.
Given this conclusion, which formerly learned triangle congruence criterion can we use to determine the pair of triangles
as congruent?
ASA
Therefore, the SAA criterion is actually an extension of the
triangle congruence criteria.
ASA
Hypotenuse-Leg triangle congruence criteria (HL): Given two right triangles ࡭࡮࡯ and ࡭Ԣ࡮Ԣ࡯Ԣ with right angles ‫ ࡮ס‬and
‫࡮ס‬Ԣ. If ࡭࡮ = ࡭Ԣ࡮Ԣ (Leg) and ࡭࡯ = ࡭Ԣ࡯Ԣ (Hypotenuse), then the triangles are congruent.
Proof
As with some of our other proofs, we will not start at the very beginning, but imagine that a congruence exists so that
triangles have been brought together such that ࡭ = ࡭ᇱ and ࡯ = ࡯Ԣ; the hypotenuse acts as a common side to the
transformed triangles.
Similar to the proof for SSS, we add a construction and draw ࡮࡮Ԣ.
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Congruence Criteria for Triangles—SAA and HL
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GEOMETRY
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ᇞ ࡭࡮࡮Ԣ is isosceles by definition, and we can conclude that base angles ‫࡮࡮࡭ס‬ᇱ = ‫࡮࡭ס‬Ԣ࡮. Since ‫࡮࡮࡯ס‬ᇱ and ‫࡮࡯ס‬ᇱ ࡮ are
both the complements of equal angle measures (‫࡮࡮࡭ס‬ᇱ and ‫࡮࡭ס‬Ԣ࡮), they too are equal in measure. Furthermore, since
‫࡮࡮࡯ס‬ᇱ = ‫࡮࡯ס‬ᇱ ࡮, the sides of ᇞ ࡯࡮࡮ᇱ opposite them are equal in measure; ࡮࡯ = ࡮Ԣ࡯Ԣ.
Then by SSS, we can conclude ᇞ ࡭࡮࡯ ؆ᇞ ࡭Ԣ࡮Ԣ࡯Ԣ.
Criteria that do not determine two triangles as congruent: SSA and AAA
Side-Side-Angle (SSA): Observe the diagrams below. Each triangle has a set of adjacent sides of measures 11 and 9, as
well as the non-ŝŶĐůƵĚĞĚĂŶŐůĞŽĨϮϯȗ͘zĞƚ͕ƚŚĞƚƌŝĂŶŐůĞƐĂƌĞŶŽƚĐŽŶŐƌƵĞŶƚ͘
Examine the composite made of both triangles. The sides of lengths 9 each have been dashed to show their possible
locations.
The pattern of SSA cannot guarantee congruence criteria. In other words, two triangles under SSA criteria might be
congruent, but they might not be; therefore we cannot categorize SSA as congruence criteria.
Angle-Angle-Angle (AAA): A correspondence exists between triangles ᇞ ࡭࡮࡯ and ᇞ ࡰࡱࡲ. Trace ᇞ ࡭࡮࡯ onto patty paper
and line up corresponding vertices.
Based on your observations, why can’t we categorize AAA as congruence criterion? Is there any situation in which AAA
does guarantee congruence?
Even though the angle measures may be the same, the sides can be proportionally larger; you can have similar triangles in
addition to a congruent triangle.
List all the triangle congruence criteria here:
SSS, SAS, ASA, SAA, HL
List the criteria that do not determine congruence here:
SSA, AAA
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Congruence Criteria for Triangles—SAA and HL
7/10/13
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Examples (8 minutes)
1.
Given:
Prove:
‫ס‬૚ = ‫ס‬૛
࡭࡮ ٣ ࡭ࡰ
࡮࡯ ٣ ࡯ࡰ, ࡭࡮ ٣ ࡭ࡰ , ‫ס‬૚ = ‫ס‬૛
ᇞ ࡮࡯ࡰ ؆ ο ࡮࡭ࡰ
࡮࡯ ٣ ࡯ࡰ
Given
BD = BD
Common Side
‫ס‬૚ + ‫ = ࡮ࡰ࡯ס‬૚ૡ૙°
Straight Angle
‫ס‬૛ + ‫ = ࡮ࡰ࡭ס‬૚ૡ૙°
Straight Angle
‫ ࡮ࡰ࡯ס‬؆ ‫࡮ࡰ࡭ס‬
2.
Given
Given
Given:
Prove:
࡭ࡰ ٣ ࡮ࡰ
࡮ࡰ ٣ ࡮࡯
Substitution
࡭ࡰ ٣ ࡮ࡰ, ࡮ࡰ ٣ ࡮࡯, ࡭࡮ = ࡯ࡰ
ᇞ ࡭࡮ࡰ ؆ ο ࡯ࡰ࡮
Given
Given
࡭࡮ = ࡯ࡰ
Given
BD = BD
Common Side
ᇞ ࡭࡮ࡰ ؆ ο ࡯ࡰ࡮
HL
Exit Ticket (5 minutes)
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Congruence Criteria for Triangles—SAA and HL
7/10/13
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Name ___________________________________________________
Date____________________
Lesson 25: Congruence Criteria for Triangles—SAA and HL
Exit Ticket
1.
Sketch an example of two triangles that meet the AAA criteria that are not congruent.
2.
Sketch an example of two triangles that meet the SSA criteria that are not congruent.
Lesson 25:
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Congruence Criteria for Triangles—SAA and HL
7/10/13
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Lesson 25
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GEOMETRY
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Exit Ticket Sample Solutions
1.
Sketch an example of two triangles that meet the AAA criteria that are not congruent.
Responses should look something like the example below.
2.
Sketch an example of two triangles that meet the SSA criteria that are not congruent.
Responses should look something like the example below.
Problem Set Sample Solutions
Use your knowledge of triangle congruence criteria to write proofs for each of the following problems.
1.
Given:
Prove:
࡭࡮ ٣ ࡮࡯
ࡰࡱ ٣ ࡱࡲ
࡭࡮ ٣ ࡮࡯, ࡰࡱ ٣ ࡱࡲ, ࡮࡯ ‫ࡲࡱ צ‬, ࡭ࡲ = ࡰ࡯
ᇞ ࡭࡮࡯ ؆ ࡰࡱࡲ
࡮࡯ ‫ࡲࡱ צ‬
Given
࡭ࡲ = ࡰ࡯
Given
FC=FC
Common Side
AF + FC = FC +CD
Substitution
ο࡭࡮࡯ ؆ οࡰࡱࡲ
SAA
‫ࡲס = ࡯ס‬
2.
Given
Given
Alt. Int. ‫࢙ס‬
In the figure, ࡼ࡭ ٣ ࡭ࡾ and ࡼ࡮ ٣ ࡮ࡾ and ࡾ is equidistant from the lines ࡼ࡭ and ࡼ࡮. Prove that ࡼࡾ bisects ‫࡮ࡼ࡭ס‬.
ࡼ࡭ ٣ ࡭ࡾ
Given
RA = RB
Given
PR= ࡼࡾ
Common Side
ࡼ࡮ ٣ ࡮ࡾ
Given
οࡼ࡭ࡾ ؆ οࡼ࡮ࡾ
‫ ࡾࡼ࡭ס‬؆ ‫࡮ࡼࡾס‬
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HL
Corr. ‫ ࢌ࢕ ࢙ס‬؆ ο
Congruence Criteria for Triangles—SAA and HL
7/10/13
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Lesson 25
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GEOMETRY
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3.
Given:
Prove:
‫ࡼס = ࡭ס‬
‫ࡾס = ࡮ס‬
‫ࡼס = ࡭ס‬, ‫ࡾס = ࡮ס‬, ࢃ is the midpoint of ࡭ࡼ
ࡾࢃ = ࡮ࢃ
Given
ࢃ is the midpoint of ࡭ࡼ
Given
AW = PW
Def. of midpoint
ο࡭ࢃ࡮ ؆ οࡼࢃࡾ
SAA
ࡾࢃ = ࡮ࢃ
4.
Given
Given:
Prove:
࡮ࡾ = ࡯ࢁ
Corr. Sides of ؆ ο࢙
࡮ࡾ = ࡯ࢁ, rectangle RSTU
ᇞ ࡭ࡾࢁ is isosceles
Given
rectangle RSTU
Given
BC‫ࢁࡾ צ‬
Def. of Rect.
‫ૢ = ࢀࡿࡾס‬૙°, ‫ૢ = ࡿࢀࢁס‬૙°
Def. of a Rect.
‫ࢁࡾ࡭ס = ࡿ࡮ࡾס‬
‫ࡾࢁ࡭ס = ࢀ࡯ࢁס‬
‫ ࡮ࡿࡾס‬+ ‫ = ࢀࡿࡾס‬૚ૡ૙
‫ ࡯ࢀࢁס‬+ ‫ = ࡿࢀࢁס‬૚ૡ૙
Corr. ‫࢙ס‬
Corr. ‫࢙ס‬
Straight Angle
Straight Angle
RS = UT
Def. of Rect.
ο࡮ࡾࡿ ؆ οࢀࢁ࡯
HL
‫ࢀ࡯ࢁס = ࡿ࡮ࡾס‬
‫ࡾࢁ࡭ס = ࢁࡾ࡭ס‬
ᇞ ࡭ࡾࢁ is isosceles
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Corr. ‫ ࢌ࢕ ࢙ס‬؆ ο࢙
Substitution
Base ‫ ࢙ס‬converse
Congruence Criteria for Triangles—SAA and HL
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