Lesson 25: Congruence Criteria for Triangles—AAS and HL

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1 GEOMETRY

Student Outcomes . Students learn why any two triangles that satisfy the AAS 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)

Opening Exercise

Write a proof for the following question. Once done, compare your proof with a neighbor’s.

Given: 푫푬 = 푫푮, 푬푭 = 푮푭

Prove: 푫푭̅̅̅̅ is the angle bisector of ∠푬푫푮

푫푬 = 푫푮 Given

푬푭 = 푮푭 Given

푫푭 = 푫푭 Reflexive property

△ 푫푬푭 ≅ △ 푫푮푭 SSS

∠푬푫푭 ≅ ∠푮푫푭 Corresponding angles of congruent triangles are congruent.

푫푭̅̅̅̅ is the angle bisector of ∠푬푫푮. Definition of an angle bisector

Exploratory Challenge (23 minutes) The included proofs of AAS and HL are not transformational; rather, they follow from ASA and SSS, already proved.

Exploratory Challenge

Today we are going to examine three possible triangle congruence criteria, Angle-Angle-Side (AAS), Side-Side-Angle (SSA), and Angle-Angle-Angle (AAA). Ultimately, only one of the three possible criteria ensures congruence.

ANGLE-ANGLE-SIDE TRIANGLE CONGRUENCE CRITERIA (AAS): Given two triangles △ 푨푩푪 and △ 푨′푩′푪′. If 푨푩 = 푨′푩′ (Side), 풎∠푩 = 풎∠푩′ (Angle), and 풎∠푪 = 풎∠푪′ (Angle), then the triangles are congruent.

PROOF:

Consider a pair of triangles that meet the AAS 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 angle of each triangle?

Lesson 25: Congruence Criteria for Triangles—AAS and HL 215

This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1 GEOMETRY

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 criteria can we use to determine if the pair of triangles are congruent?

ASA

Therefore, the AAS criterion is actually an extension of the ASA triangle congruence criterion.

Note that when using the Angle-Angle-Side triangle congruence criteria as a reason in a proof, you need only to state the congruence and AAS.

HYPOTENUSE-LEG TRIANGLE CONGRUENCE CRITERIA (HL): Given two right triangles △ 푨푩푪 and △ 푨′푩′푪′ with right angles 푩 and 푩′. If 푨푩 = 푨′푩′ (Leg) and 푨푪 = 푨′푪′ (Hypotenuse), then the triangles are congruent.

PROOF:

As with some of our other proofs, we do not start at the very beginning, but imagine that a congruence exists so that triangles have been brought together such that 푨 = 푨′ and 푪 = 푪′; the hypotenuse acts as a common side to the transformed triangles.

Similar to the proof for SSS, we add a construction and draw ̅푩푩̅̅̅̅′.

△ 푨푩푩′ is isosceles by definition, and we can conclude that base angles 풎∠푨푩푩′ = 풎∠푨푩′푩. Since ∠푪푩푩′ and ∠푪푩′푩 are both the complements of equal angle measures (∠푨푩푩′ and ∠푨푩′푩), they too are equal in measure. Furthermore, since 풎∠푪푩푩′ = 풎∠푪푩′푩, the sides of △ 푪푩푩′ opposite them are equal in measure: 푩푪 = 푩′푪′.

Then, by SSS, we can conclude △ 푨푩푪 ≅ △ 푨′푩′푪′. Note that when using the Hypotenuse-Leg triangle congruence criteria as a reason in a proof, you need only to state the congruence and HL.

Lesson 25: Congruence Criteria for Triangles—AAS and HL 216

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1 GEOMETRY

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 ퟏퟏ and ퟗ, as well as the non-included angle of ퟐퟑ°. Yet, the triangles are not congruent.

Examine the composite made of both triangles. The sides of length ퟗ each have been dashed to show their possible locations.

The triangles that satisfy the conditions of SSA cannot guarantee congruence criteria. In other words, two triangles under SSA criteria may or may not be congruent; therefore, we cannot categorize SSA as congruence criterion.

ANGLE-ANGLE-ANGLE (AAA): A correspondence exists between △ 푨푩푪 and △ 푫푬푭. Trace △ 푨푩푪 onto patty paper, and line up corresponding vertices.

Based on your observations, why isn’t AAA categorized as congruence criteria? 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, AAS, HL

List the criteria that do not determine congruence here:

SSA, AAA

Lesson 25: Congruence Criteria for Triangles—AAS and HL 217

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1 GEOMETRY

Examples (8 minutes)

Examples

1. Given: 푩푪̅̅̅̅ ⊥ 푪푫̅̅̅̅, 푨푩̅̅̅̅ ⊥ 푨푫̅̅̅̅ , 풎∠ퟏ = 풎∠ퟐ Prove: △ 푩푪푫 ≅ △ 푩푨푫

풎∠ퟏ = 풎∠ퟐ Given

푨푩̅̅̅̅ ⊥ 푨푫̅̅̅̅ Given

푩푪̅̅̅̅ ⊥ 푪푫̅̅̅̅ Given

푩푫 = 푩푫 Reflexive property

풎∠ퟏ + 풎∠푪푫푩 = ퟏퟖퟎ° Linear pairs form supplementary angles.

풎∠ퟐ + 풎∠푨푫푩 = ퟏퟖퟎ° Linear pairs form supplementary angles.

풎∠푪푫푩 = 풎∠푨푫푩 If two angles are equal in measure, then their supplements are equal in measure.

풎∠푩푪푫 = 풎∠푩푨푫 = ퟗퟎ° Definition of perpendicular line segments

△ 푩푪푫 ≅ △ 푩푨푫 AAS

2. Given: 푨푫̅̅̅̅ ⊥ 푩푫̅̅̅̅̅, 푩푫̅̅̅̅̅ ⊥ 푩푪̅̅̅̅, 푨푩 = 푪푫 Prove: △ 푨푩푫 ≅ △ 푪푫푩

푨푫̅̅̅̅ ⊥ 푩푫̅̅̅̅̅ Given

푩푫̅̅̅̅̅ ⊥ 푩푪̅̅̅̅ Given

△ 푨푩푫 is a right triangle. Definition of perpendicular line segments

△ 푪푫푩 is a right triangle. Definition of perpendicular line segments

푨푩 = 푪푫 Given

푩푫 = 푫푩 Reflexive property

△ 푨푩푫 ≅ △ 푪푫푩 HL

Closing (2 minutes)

. ANGLE-ANGLE-SIDE TRIANGLE CONGRUENCE CRITERIA (AAS): Given two triangles △ 퐴퐵퐶 and △ 퐴′퐵′퐶′. If 퐴퐵 = 퐴′퐵′ (Side), 푚∠퐵 = 푚∠퐵′ (Angle), and 푚∠퐶 = 푚∠퐶′ (Angle), then the triangles are congruent. . HYPOTENUSE-LEG TRIANGLE CONGRUENCE CRITERIA (HL): Given two right triangles △ 퐴퐵퐶 and △ 퐴′퐵′퐶′ with right angles 퐵 and 퐵′. If 퐴퐵 = 퐴′퐵′ (Leg) and 퐴퐶 = 퐴′퐶′ (Hypotenuse), then the triangles are congruent. . The AAS and HL criteria implies the existence of a congruence that maps one triangle onto the other. . Triangles that meet either the Angle-Angle-Angle or the Side-Side-Angle criteria do not guarantee congruence.

Exit Ticket (5 minutes)

Lesson 25: Congruence Criteria for Triangles—AAS and HL 218

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1 GEOMETRY

Name Date

Lesson 25: Congruence Criteria for Triangles—AAS and HL

Exit Ticket

1. Sketch an example of two triangles that meet the AAA criteria but are not congruent.

2. Sketch an example of two triangles that meet the SSA criteria that are not congruent.

Lesson 25: Congruence Criteria for Triangles—AAS and HL 219

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1 GEOMETRY

Exit Ticket Sample Solutions

1. Sketch an example of two triangles that meet the AAA criteria but are not congruent.

Responses should look something like the example below.

45o

o 35o 110o 35o 110

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: 푨푩̅̅̅̅ ⊥ 푩푪̅̅̅̅, 푫푬̅̅̅̅ ⊥ 푬푭̅̅̅̅, 푩푪̅̅̅̅ ∥ 푬푭̅̅̅̅, 푨푭 = 푫푪 Prove: △ 푨푩푪 ≅ △ 푫푬푭

푨푩̅̅̅̅ ⊥ 푩푪̅̅̅̅ Given

푫푬̅̅̅̅ ⊥ 푬푭̅̅̅̅ Given

푩푪̅̅̅̅ ∥ 푬푭̅̅̅̅ Given

푨푭 = 푫푪 Given

풎∠푩 = 풎∠푬 = ퟗퟎ° Definition of perpendicular lines

풎∠푪 = 풎∠푭 When two parallel lines are cut by a transversal, the alternate interior angles are equal in measure.

푭푪 = 푭푪 Reflexive property

푨푭 + 푭푪 = 푭푪 + 푪푫 Addition property of equality

푨푪 = 푫푭 Segment addition

△ 푨푩푪 ≅ △ 푫푬푭 AAS

Lesson 25: Congruence Criteria for Triangles—AAS and HL 220

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 25 M1 GEOMETRY

2. In the figure, 푷푨̅̅̅̅ ⊥ 푨푹̅̅̅̅ and 푷푩̅̅̅̅ ⊥ 푹푩̅̅̅̅ and 푹 is equidistant from 푷푨⃡ and 푷푩⃡ . Prove that 푷푹̅̅̅̅ bisects ∠푨푷푩.

푷푨̅̅̅̅ ⊥ 푨푹̅̅̅̅ Given

푷푩̅̅̅̅ ⊥ 푩푹̅̅̅̅ Given

푹푨 = 푹푩 Given

풎∠푨 = 풎∠푹 = ퟗퟎ° Definition of perpendicular lines

△ 푷푨푹, △ 푷푩푹 are right triangles. Definition of right triangle

푷푹 = 푷푹 Reflexive property

△ 푷푨푹 ≅ △ 푷푩푹 HL

∠푨푷푹 ≅ ∠푹푷푩 Corresponding angles of congruent triangles are congruent.

푷푹̅̅̅̅ bisects ∠푨푷푩. Definition of an angle bisector

3. Given: ∠푨 ≅ ∠푷, ∠푩 ≅ ∠푹, 푾 is the midpoint of 푨푷̅̅̅̅ Prove: 푹푾̅̅̅̅̅ ≅ 푩푾̅̅̅̅̅

∠푨 ≅ ∠푷 Given

∠푩 ≅ ∠푹 Given

푾 is the midpoint of 푨푷̅̅̅̅. Given

푨푾 = 푷푾 Definition of midpoint

△ 푨푾푩 ≅ △ 푷푾푹 AAS

푹푾̅̅̅̅̅ ≅ 푩푾̅̅̅̅̅ Corresponding sides of congruent triangles are congruent.

4. Given: 푩푹 = 푪푼, rectangle 푹푺푻푼 Prove: △ 푨푹푼 is isosceles

푩푹 = 푪푼 Given Rectangle 푹푺푻푼 Given 푩푪̅̅̅̅ ∥ 푹푼̅̅̅̅ Definition of a rectangle 풎∠푹푩푺 = 풎∠푨푹푼 When two para. lines are cut by a trans., the corr. angles are equal in measure. 풎∠푼푪푻 = 풎∠푨푼푹 When two para. lines are cut by a trans., the corr. angles are equal in measure. 풎∠푹푺푻 = ퟗퟎ°, 풎∠푼푻푺 = ퟗퟎ° Definition of a rectangle 풎∠푹푺푩 + 풎∠푹푺푻 = ퟏퟖퟎ Linear pairs form supplementary angles. 풎∠푼푻푪 + 풎∠푼푻푺 = ퟏퟖퟎ Linear pairs form supplementary angles. 풎∠푹푺푩 = ퟗퟎ°, 풎∠푼푻푪 = ퟗퟎ° Subtraction property of equality △ 푩푹푺 and △ 푻푼푪 are right triangles. Definition of right triangle 푹푺 = 푼푻 Definition of a rectangle △ 푩푹푺 ≅ △ 푻푼푪 HL 풎∠푹푩푺 = 풎∠푼푪푻 Corresponding angles of congruent triangles are equal in measure. 풎∠푨푹푼 = 풎∠푨푼푹 Substitution property of equality △ 푨푹푼 is isosceles. If two angles in a triangle are equal in measure, then it is isosceles.

Lesson 25: Congruence Criteria for Triangles—AAS and HL 221