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Lesson 23 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Lesson 23: Base Angles of Isosceles Triangles Student Outcomes ๏ง Students examine two different proof techniques via a familiar theorem. ๏ง Students complete proofs involving properties of an isosceles triangle. Lesson Notes In Lesson 23, students study two proofs to demonstrate why the base angles of isosceles triangles are congruent. The first proof uses transformations, while the second uses the recently acquired understanding of the SAS triangle congruency. The demonstration of both proofs highlight the utility of the SAS criteria. Encourage students to articulate why the SAS criteria is so useful. The goal of this lesson is to compare two different proof techniques by investigating a familiar theorem. Be careful not to suggest that proving an established fact about isosceles triangles is somehow the significant aspect of this lesson. However, if necessary, you can help motivate the lesson by appealing to history. Point out that Euclid used SAS and that the first application he made of it was in proving that base angles of an isosceles triangle are congruent. This is a famous part of the Elements. Today, proofs using SAS and proofs using rigid motions are valued equally. Classwork Opening Exercise (6 minutes) Opening Exercise Describe the additional piece of information needed for each pair of triangles to satisfy the SAS triangle congruence criteria. a. Given: ๐จ๐ฉ = ๐ซ๐ช ฬ ฬ ฬ ฬ , ๐ซ๐ช ฬ ฬ ฬ ฬ โฅ ๐ช๐ฉ ฬ ฬ ฬ ฬ , or ๐โ ๐ฉ = ๐โ ๐ช ฬ ฬ ฬ ฬ โฅ ๐ฉ๐ช ๐จ๐ฉ b. Prove: โณ ๐จ๐ฉ๐ช โ โณ ๐ซ๐ช๐ฉ Given: ๐จ๐ฉ = ๐น๐บ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ โฅ ๐น๐บ ๐จ๐ฉ ๐ช๐ฉ = ๐ป๐บ, or ๐ช๐ป = ๐ฉ๐บ Prove: Lesson 23: โณ ๐จ๐ฉ๐ช โ โณ ๐น๐บ๐ป Base Angles of Isosceles Triangles This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 200 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 23 M1 GEOMETRY Exploratory Challenge (25 minutes) Exploratory Challenge Today we examine a geometry fact that we already accept to be true. We are going to prove this known fact in two ways: (1) by using transformations and (2) by using SAS triangle congruence criteria. Here is isosceles triangle ๐จ๐ฉ๐ช. We accept that an isosceles triangle, which has (at least) two congruent sides, also has congruent base angles. Label the congruent angles in the figure. Now we prove that the base angles of an isosceles triangle are always congruent. Prove Base Angles of an Isosceles are Congruent: Transformations While simpler proofs do exist, this version is included to reinforce the idea of congruence as defined by rigid motions. Allow 15 minutes for the first proof. Prove Base Angles of an Isosceles are Congruent: Transformations Given: Isosceles โณ ๐จ๐ฉ๐ช, with ๐จ๐ฉ = ๐จ๐ช Prove: ๐โ ๐ฉ = ๐โ ๐ช Construction: Draw the angle bisector โโโโโโ ๐จ๐ซ of โ ๐จ, where ๐ซ is the intersection of the ฬ ฬ ฬ ฬ . We need to show that rigid motions maps point ๐ฉ to point ๐ช and bisector and ๐ฉ๐ช point ๐ช to point ๐ฉ. โกโโโโ . Through the reflection, we want to demonstrate two pieces of information that map Let ๐ be the reflection through ๐จ๐ซ โโโโโ , and (2) ๐จ๐ฉ = ๐จ๐ช. โโโโโโ maps to ๐จ๐ช ๐ฉ to point ๐ช and vice versa: (1) ๐จ๐ฉ Since ๐จ is on the line of reflection, โกโโโโ ๐จ๐ซ, ๐(๐จ) = ๐จ. Reflections preserve angle measures, so the measure of the reflected โโโโโโ ) = โโโโโ angle ๐(โ ๐ฉ๐จ๐ซ) equals the measure of โ ๐ช๐จ๐ซ; therefore, ๐(๐จ๐ฉ ๐จ๐ช. Reflections also preserve lengths of segments; ฬ ฬ ฬ ฬ still has the same length as ๐จ๐ฉ ฬ ฬ ฬ ฬ . By hypothesis, ๐จ๐ฉ = ๐จ๐ช, so the length of the reflection is therefore, the reflection of ๐จ๐ฉ also equal to ๐จ๐ช. Then ๐(๐ฉ) = ๐ช. Using similar reasoning, we can show that ๐(๐ช) = ๐ฉ. โโโโโ and ๐(๐ฉ๐ช โโโโโโ ) = ๐ช๐ฉ โโโโโโ . Again, since reflections preserve angle measures, the โโโโโโ ) = ๐ช๐จ Reflections map rays to rays, so ๐(๐ฉ๐จ measure of ๐(โ ๐จ๐ฉ๐ช) is equal to the measure of โ ๐จ๐ช๐ฉ. We conclude that ๐โ ๐ฉ = ๐โ ๐ช. Equivalently, we can state that โ ๐ฉ โ โ ๐ช. In proofs, we can state that โbase angles of an isosceles triangle are equal in measureโ or that โbase angles of an isosceles triangle are congruent.โ Lesson 23: Base Angles of Isosceles Triangles This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 201 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 23 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Prove Base Angles of an Isosceles are Congruent: SAS Allow 10 minutes for the second proof. Prove Base Angles of an Isosceles are Congruent: SAS Given: Isosceles โณ ๐จ๐ฉ๐ช, with ๐จ๐ฉ = ๐จ๐ช Prove: โ ๐ฉ โ โ ๐ช โโโโโโ of โ ๐จ, where ๐ซ is the intersection of the Construction: Draw the angle bisector ๐จ๐ซ bisector and ฬ ฬ ฬ ฬ ๐ฉ๐ช. We are going to use this auxiliary line towards our SAS criteria. Allow students five minutes to attempt the proof on their own. ๐จ๐ฉ = ๐จ๐ช Given ๐จ๐ซ = ๐จ๐ซ Reflexive property ๐โ ๐ฉ๐จ๐ซ = ๐โ ๐ช๐จ๐ซ Definition of an angle bisector โณ ๐จ๐ฉ๐ซ โ โณ ๐จ๐ช๐ซ SAS โ ๐ฉ โ โ ๐ช Corresponding angles of congruent triangles are congruent. Exercises (10 minutes) Note that Exercise 4 asks students to use transformations in the proof. This is intended to reinforce the notion that congruence is defined in terms of rigid motions. Exercises 1. ฬ ฬ ฬ ฬ bisects ๐ฒ๐ณ ฬ ฬ ฬ ฬ ๐ฑ๐ฒ = ๐ฑ๐ณ; ๐ฑ๐น ฬ ฬ ฬ ฬ โฅ ๐ฒ๐ณ ฬ ฬ ฬ ฬ ๐ฑ๐น Given: Prove: ๐ฑ๐ฒ = ๐ฑ๐ณ Given โ ๐ฒ โ โ ๐ณ Base angles of an isosceles triangle are congruent. ๐ฒ๐น = ๐น๐ณ Definition of a segment bisector โด โณ ๐ฑ๐น๐ฒ โ โณ ๐ฑ๐น๐ณ SAS โ ๐ฑ๐น๐ฒ โ โ ๐ฑ๐น๐ณ Corresponding angles of congruent triangles are congruent. ๐โ ๐ฑ๐น๐ฒ + ๐โ ๐ฑ๐น๐ณ = ๐๐๐° Linear pairs form supplementary angles. ๐(๐โ ๐ฑ๐น๐ฒ) = ๐๐๐° Substitution property of equality ๐โ ๐ฑ๐น๐ฒ = ๐๐° Division property of equality ฬ ฬ ฬ ฬ โด ฬ ฬ ฬ ฬ ๐ฑ๐น โฅ ๐ฒ๐ณ Definition of perpendicular lines Lesson 23: Base Angles of Isosceles Triangles This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 202 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 23 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 2. ๐จ๐ฉ = ๐จ๐ช, ๐ฟ๐ฉ = ๐ฟ๐ช ฬ ฬ ฬ ฬ ๐จ๐ฟ bisects โ ๐ฉ๐จ๐ช Given: Prove: ๐จ๐ฉ = ๐จ๐ช Given ๐โ ๐จ๐ฉ๐ช = ๐โ ๐จ๐ช๐ฉ Base angles of an isosceles triangle are equal in measure. ๐ฟ๐ฉ = ๐ฟ๐ช Given ๐โ ๐ฟ๐ฉ๐ช = ๐โ ๐ฟ๐ช๐ฉ Base angles of an isosceles triangle are equal in measure. ๐โ ๐จ๐ฉ๐ช = ๐โ ๐จ๐ฉ๐ฟ + ๐โ ๐ฟ๐ฉ๐ช, ๐โ ๐จ๐ช๐ฉ = ๐โ ๐จ๐ช๐ฟ + ๐โ ๐ฟ๐ช๐ฉ Angle addition postulate ๐โ ๐จ๐ฉ๐ฟ = ๐โ ๐จ๐ฉ๐ช โ ๐โ ๐ฟ๐ฉ๐ช, 3. ๐โ ๐จ๐ช๐ฟ = ๐โ ๐จ๐ช๐ฉ โ ๐โ ๐ฟ๐ช๐ฉ Subtraction property of equality ๐โ ๐จ๐ฉ๐ฟ = ๐โ ๐จ๐ช๐ฟ Substitution property of equality โด โณ ๐จ๐ฉ๐ฟ โ โณ ๐จ๐ช๐ฟ SAS โ ๐ฉ๐จ๐ฟ โ โ ๐ช๐จ๐ฟ Corresponding parts of congruent triangles are congruent. โด ฬ ฬ ฬ ฬ ๐จ๐ฟ bisects โ ๐ฉ๐จ๐ช. Definition of an angle bisector Given: ๐ฑ๐ฟ = ๐ฑ๐, ๐ฒ๐ฟ = ๐ณ๐ Prove: โณ ๐ฑ๐ฒ๐ณ is isosceles ๐ฑ๐ฟ = ๐ฑ๐ Given ๐ฒ๐ฟ = ๐ณ๐ Given ๐โ ๐ = ๐โ ๐ Base angles of an isosceles triangle are equal in measure. ๐โ ๐ + ๐โ ๐ = ๐๐๐° Linear pairs form supplementary angles. ๐โ ๐ + ๐โ ๐ = ๐๐๐° Linear pairs form supplementary angles. ๐โ ๐ = ๐๐๐° โ ๐โ ๐ Subtraction property of equality ๐โ ๐ = ๐๐๐° โ ๐โ ๐ Subtraction property of equality ๐โ ๐ = ๐โ ๐ Substitution property of equality โด โณ ๐ฑ๐ฒ๐ฟ โ โณ ๐ฑ๐๐ณ SAS ฬ ฬ ฬ ฬ ๐ฑ๐ฒ โ ฬ ฬ ฬ ๐ฑ๐ณ Corresponding parts of congruent triangles are congruent. โด โณ ๐ฑ๐ฒ๐ณ is isosceles. Definition of an isosceles triangle Lesson 23: Base Angles of Isosceles Triangles This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 203 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 23 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY 4. Given: โณ ๐จ๐ฉ๐ช, with ๐โ ๐ช๐ฉ๐จ = ๐โ ๐ฉ๐ช๐จ Prove: ๐ฉ๐จ = ๐ช๐จ (Converse of base angles of isosceles triangle) Hint: Use a transformation. PROOF: We can prove that ๐จ๐ฉ = ๐จ๐ช by using rigid motions. Construct the perpendicular bisector ฬ ฬ ฬ ฬ . Note that we do not know whether the point ๐จ is on ๐. If we did, we would ๐ to ๐ฉ๐ช know immediately that ๐จ๐ฉ = ๐จ๐ช, since all points on the perpendicular bisector are equidistant from ๐ฉ and ๐ช. We need to show that ๐จ is on ๐, or equivalently, that the reflection across ๐ takes ๐จ to ๐จ. Let ๐๐ be the transformation that reflects the โณ ๐จ๐ฉ๐ช across ๐. Since ๐ is the perpendicular bisector, ๐๐ (๐ฉ) = ๐ช and ๐๐ (๐ช) = ๐ฉ. We do not know where the transformation takes ๐จ; for now, let us say that ๐๐ (๐จ) = ๐จโฒ . Since โ ๐ช๐ฉ๐จ โ โ ๐ฉ๐ช๐จ and rigid motions preserve angle measures, after applying ๐๐ to โ ๐ฉ๐ช๐จ, we get that โ ๐ช๐ฉ๐จ โ โ ๐ช๐ฉ๐จโฒ . โ ๐ช๐ฉ๐จ and โ ๐ช๐ฉ๐จโฒ share a ray ๐ฉ๐ช, are of equal measure, and ๐จ and ๐จโฒ are both in the same halfโโโโโโ and โโโโโโโ โโโโโโ . plane with respect to line ๐ฉ๐ช. Hence, ๐ฉ๐จ ๐ฉ๐จโฒ are the same ray. In particular, ๐จโฒ is a point somewhere on ๐ฉ๐จ Likewise, applying ๐๐ to โ ๐ช๐ฉ๐จ gives โ ๐ฉ๐ช๐จ โ โ ๐ฉ๐ช๐จโฒ , and for the same reasons in the previous paragraph, ๐จโฒ must โโโโโ . Therefore, ๐จโฒ is common to both ๐ฉ๐จ โโโโโ . โโโโโโ and ๐ช๐จ also be a point somewhere on ๐ช๐จ The only point common to both โโโโโโ ๐ฉ๐จ and โโโโโ ๐ช๐จ is point ๐จ; thus, ๐จ and ๐จโฒ must be the same point, i.e., ๐จ = ๐จโฒ . Hence, the reflection takes ๐จ to ๐จ, which means ๐จ is on the line ๐ and ๐๐ (๐ฉ๐จ) = ๐ช๐จโฒ = ๐ช๐จ, or ๐ฉ๐จ = ๐ช๐จ. 5. Given: ฬ ฬ ฬ ฬ โฅ ๐ฟ๐ ฬ ฬ ฬ ฬ ฬ is the angle bisector of โ ๐ฉ๐๐จ, and ๐ฉ๐ช ฬ ฬ ฬ ฬ โณ ๐จ๐ฉ๐ช, with ๐ฟ๐ Prove: ๐๐ฉ = ๐๐ช ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ๐ฉ๐ช โฅ ๐ฟ๐ Given ๐โ ๐ฟ๐๐ฉ = ๐โ ๐ช๐ฉ๐ When two parallel lines are cut by a transversal, the alternate interior angles are equal. ๐โ ๐ฟ๐๐จ = ๐โ ๐ฉ๐ช๐ When two parallel lines are cut by a transversal, the corresponding angles are equal. ๐โ ๐ฟ๐๐จ = ๐โ ๐ฟ๐๐ฉ Definition of an angle bisector ๐โ ๐ช๐ฉ๐ = ๐โ ๐ฉ๐ช๐ Substitution property of equality ๐๐ฉ = ๐๐ช When the base angles of a triangle are equal in measure, the triangle is isosceles. Closing (1 minute) ๏ง Would you prefer to prove that the base angles of an isosceles triangle are equal in measure using transformations or by using SAS? Why? Exit Ticket (3 minutes) Lesson 23: Base Angles of Isosceles Triangles This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 204 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 23 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Name Date Lesson 23: Base Angles of Isosceles Triangles Exit Ticket For each of the following, if the given congruence exists, name the isosceles triangle and the pair of congruent angles for the triangle based on the image above. 1. ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ๐ด๐ธ โ ๐ฟ๐ธ 2. ฬ ฬ ฬ ฬ โ ฬ ฬ ฬ ฬ ๐ฟ๐ธ ๐ฟ๐บ 3. ฬ ฬ ฬ ฬ โ ๐ฟ๐ ฬ ฬ ฬ ฬ ๐ด๐ 4. ฬ ฬ ฬ ฬ โ ฬ ฬ ฬ ฬ ๐ธ๐ ๐บ๐ 5. ฬ ฬ ฬ ฬ ๐๐บ โ ฬ ฬ ฬ ฬ ๐ฟ๐บ 6. ฬ ฬ ฬ ฬ ๐ด๐ธ โ ฬ ฬ ฬ ฬ ๐๐ธ Lesson 23: Base Angles of Isosceles Triangles This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 205 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 23 NYS COMMON CORE MATHEMATICS CURRICULUM M1 GEOMETRY Exit Ticket Sample Solutions For each of the following, if the given congruence exists, name the isosceles triangle and the pair of congruent angles for the triangle based on the image above. 1. ฬ ฬ ฬ ฬ โ ๐ณ๐ฌ ฬ ฬ ฬ ฬ ๐จ๐ฌ 2. โณ ๐จ๐ฌ๐ณ, โ ๐ฌ๐จ๐ณ โ โ ๐ฌ๐ณ๐จ 3. โณ ๐ณ๐ฌ๐ฎ, โ ๐ณ๐ฌ๐ฎ โ โ ๐ณ๐ฎ๐ฌ ฬ ฬ ฬ ฬ ๐จ๐ต โ ฬ ฬ ฬ ฬ ๐ณ๐ต 4. โณ ๐ต๐ณ๐จ, โ ๐ต๐ณ๐จ โ โ ๐ต๐จ๐ณ 5. ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ โ ๐ณ๐ฎ ๐ณ๐ฌ ฬ ฬ ฬ ฬ ๐ฌ๐ต โ ฬ ฬ ฬ ฬ ฬ ๐ฎ๐ต โณ ๐ต๐ฎ๐ฌ, โ ๐ต๐ฎ๐ฌ โ โ ๐ต๐ฌ๐ฎ ฬ ฬ ฬ ฬ ฬ โ ๐ณ๐ฎ ฬ ฬ ฬ ฬ ๐ต๐ฎ 6. โณ ๐ฎ๐ณ๐ต, โ ๐ฎ๐ต๐ณ โ โ ๐ฎ๐ณ๐ต ฬ ฬ ฬ ฬ โ ๐ต๐ฌ ฬ ฬ ฬ ฬ ๐จ๐ฌ โณ ๐จ๐ฌ๐ต, โ ๐ฌ๐ต๐จ โ โ ๐ฌ๐จ๐ต Problem Set Sample Solutions 1. Given: ๐จ๐ฉ = ๐ฉ๐ช, ๐จ๐ซ = ๐ซ๐ช Prove: โณ ๐จ๐ซ๐ฉ and โณ ๐ช๐ซ๐ฉ are right triangles ๐จ๐ฉ = ๐ฉ๐ช Given โณ ๐จ๐ฉ๐ช is isosceles. Definition of isosceles triangle ๐โ ๐จ = ๐โ ๐ช Base angles of an isosceles triangle are equal in measure. ๐จ๐ซ = ๐ซ๐ช Given โณ ๐จ๐ซ๐ฉ โ โณ ๐ช๐ซ๐ฉ SAS ๐โ ๐จ๐ซ๐ฉ = ๐โ ๐ช๐ซ๐ฉ Corresponding angles of congruent triangles are equal in measure. ๐โ ๐จ๐ซ๐ฉ + ๐โ ๐ช๐ซ๐ฉ = ๐๐๐° Linear pairs form supplementary angles. ๐(๐โ ๐จ๐ซ๐ฉ) = ๐๐๐° Substitution property of equality ๐โ ๐จ๐ซ๐ฉ = ๐๐° Division property of equality ๐โ ๐ช๐ซ๐ฉ = ๐๐° Transitive property โณ ๐จ๐ซ๐ฉ and โณ ๐ช๐ซ๐ฉ are right triangles. Definition of a right triangle Lesson 23: Base Angles of Isosceles Triangles This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 206 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 23 M1 GEOMETRY 2. 3. Given: ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ โ๐ช๐ฌ ๐จ๐ช = ๐จ๐ฌ and ๐ฉ๐ญ Prove: ๐จ๐ฉ = ๐จ๐ญ ๐จ๐ช = ๐จ๐ฌ Given โณ ๐จ๐ช๐ฌ is isosceles. Definition of isosceles triangle ๐โ ๐จ๐ช๐ฌ = ๐โ ๐จ๐ฌ๐ช Base angles of an isosceles triangle are equal in measure. ฬ ฬ ฬ ฬ ๐ฉ๐ญ โ ฬ ฬ ฬ ฬ ๐ช๐ฌ Given ๐โ ๐จ๐ช๐ฌ = ๐โ ๐จ๐ฉ๐ญ If parallel lines are cut by a transversal, then corresponding angles are equal in measure. ๐โ ๐จ๐ญ๐ฉ = ๐โ ๐จ๐ฌ๐ช If parallel lines are cut by a transversal, then corresponding angles are equal in measure. ๐โ ๐จ๐ญ๐ฉ = ๐โ ๐จ๐ฉ๐ญ Transitive property โณ ๐จ๐ฉ๐ญ is isosceles. If the base angles of a triangle are equal in measure, the triangle is isosceles. ๐จ๐ฉ = ๐จ๐ญ Definition of an isosceles triangle ฬ ฬ ฬ ฬ โ ๐จ๐ฉ ฬ ฬ ฬ ฬ . In your own words, describe how transformations and the In the diagram, โณ ๐จ๐ฉ๐ช is isosceles with ๐จ๐ช properties of rigid motions can be used to show that โ ๐ช โ โ ๐ฉ. Answers will vary. A possible response would summarize the key elements of the proof from the lesson, such as the response below. It can be shown that โ ๐ช โ โ ๐ฉ using a reflection across the bisector of โ ๐จ. The two angles formed by the bisector of โ ๐จ would be mapped onto one another since they share a ray, and rigid motions preserve angle measure. Since segment length is also preserved, ๐ฉ is mapped onto ๐ช and vice versa. Finally, since rays are taken to rays and rigid motions preserve angle โโโโโ is mapped onto ๐ฉ๐จ โโโโโโ is mapped onto ๐ฉ๐ช โโโโโโ , and โ ๐ช is โโโโโโ , ๐ช๐ฉ measure, ๐ช๐จ mapped onto โ ๐ฉ. From this, we see that โ ๐ช โ โ ๐ฉ. Lesson 23: Base Angles of Isosceles Triangles This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015 207 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
